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On Parabolic Stochastic
Integro-Differential Equations
Existence, Regularity, and Numerics
James-Michael Leahy
Doctor of Philosophy
University of Edinburgh
June 2015

Declaration
I hereby declare that this thesis was composed by myself and that the work contained
herein is my own, except where explicitly stated otherwise. Further, I declare that this
work has not been submitted for any other degree or professional qualification.
James-Michael Leahy
June 2015
iii

Abstract
In this thesis, we study the existence, uniqueness, and regularity of systems of degener-
ate linear stochastic integro-differential equations (SIDEs) of parabolic type with adapted
coefficients in the whole space. We also investigate explicit and implicit finite difference
schemes for SIDEs with non-degenerate diffusion. The class of equations we consider
arise in non-linear filtering of semimartingales with jumps.
In Chapter 2, we derive moment estimates and a strong limit theorem for space inverses
of stochastic flows generated by Lévy driven stochastic differential equations (SDEs) with
adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and
the change of variable formula. As an application of some basic properties of flows of
Weiner driven SDEs, we prove the existence and uniqueness of classical solutions of lin-
ear parabolic second order stochastic partial differential equations (SPDEs) by partition-
ing the time interval and passing to the limit. The methods we use allow us to improve
on previously known results in the continuous case and to derive new ones in the jump
case. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutions
of degenerate SIDEs using the method of stochastic characteristics. More precisely, we
use Feynman-Kac transformations, conditioning, and the interlacing of space inverses of
stochastic flows generated by SDEs with jumps to construct solutions.
In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear
stochastic evolution equations driven by jump processes in a Hilbert scale using the varia-
tional framework of stochastic evolution equations and the method of vanishing viscosity.
As an application, we establish the existence and uniqueness of solutions of degenerate
linear stochastic integro-differential equations in the L2-Sobolev scale.
Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5.
Specifically, we study the rate of convergence of an explicit and an implicit-explicit finite
difference scheme for linear SIDEs and show that the rate is of order one in space and
order one-half in time.
Key words: Stochastic flows, stochastic differential equations (SDEs), Lévy processes, strong-limit the-
orem, stochastic partial differential equations (SPDEs), degenerate parabolic type, parabolic stochastic
integro-differential equations (SIDEs), partial integro-differential equations (PIDEs), non-local operators,
method of stochastic characteristics, Ito-Wentzell formula, stochastic evolution equations, vanishing viscos-
ity, finite difference schemes
AMS subject classifications (MSC2010). 60H10, 60J75, 60F15, 60H15, 34F05, 35F40, 35R60, 35K15,
35K65, 60H20, 45K05, 65M06, 65M12
v

Lay Summary
In this thesis, we investigate a class of equations governing random processes that depend
on time and space. These equations are a generalization of deterministic parabolic partial
differential equations (PDEs). The prototypical example of a parabolic PDE is the heat
equation, which is an equation that governs the evolution of heat in some medium in time
and space. Randomness can be introduced to the heat equation through a random source
and sink of heat at certain points of time and space, a random initial heat profile, and
a random conductivity coefficient of the medium. The heat equation is an example of
a diffusion equation, which is an equation that describes the density of particles as they
diffuse according to some law. The physical law that governs the heat equation is called
Fick’s law and it describes only a special type of diffusion. The equations that we study,
even if we take them to be deterministic, allow for a much wider range of diffusions than
the prototypical heat equation.
When investigating an equation, a natural first question to ask is whether there exists
a solution to the equation in some well-defined sense. Moreover, if there exists a solution,
then one ought to ask whether it is unique, and if so, what properties it has. The equations
we consider have range of possible inputs, and thus we want to know how the properties of
the solution depend on the inputs. There are many properties of the equations to explore,
but we consider only a property that is referred to as regularity. Regularity is characterized
by a class of spaces to which the solution and inputs belong that are ordered by inclusion;
smaller spaces correspond to higher regularity. It is natural to expect that if you take more
regular inputs, then the solution should be more regular. This is the case for the equations
we consider.
It is rare for the solution of an equation we study to have a closed form expression of
time, space, and randomness. Nevertheless, it is still possible to prove that there exists
a unique solution and to study the regularity of that solution. In fact, there are different
approaches to accomplish this objective and each way has its own advantage. We will dis-
cuss two such approaches in this thesis. Since we can not expect a closed form expression
for the solutions, it also practical and interesting to develop some method to approximate
the solutions and to prove that the approximate solutions are close in some well-defined
sense to the true solution that we know exists. In the final chapter, we describe a simple
approximation scheme for a special subclass of the equations considered. The approxi-
mate solutions are defined on some countable set of points in time and space and satisfy
an equation that can be solved by simple algebraic manipulations for some realization of
the random variables in the equation.
vii

I dedicate this thesis to my family.
ix

Acknowledgements
First, I would like to thank my supervisor, István Gyöngy, for the support and guidance
he has provided me during my time at the University of Edinburgh. I am indebted to
Remigijus Mikulevicˇius for the opportunity to work with him at the University of Southern
California in the Fall of 2013 and to collaborate with him on numerous projects. I have
also benefited from having Kontantinos Dareiotis as a friend, flatmate, and collaborator.
The research contained in this thesis would not have been possible without funding
from the Principal’s Career Development Scholarship and the Edinburgh Global Research
Scholarship and support from Gill Law, Graduate School Administrator.
I have been fortunate enough to participate in numerous discussions with other students
that have directly contributed to my academic development. In particular, I would like to
mention: Eric Hall, Máté Gerencsèr, Brian Hamilton, Marc Ryser, David Cottrell, Andrea
Meireles Rodrigues, Sebastian Vollmer, Chaman Kumar, and José Rodríguez Villarreal.
In fact, it was Marc Ryser who introduced me to Stochastic Partial Differential Equations
and suggested I apply to the University of Edinburgh to work with István Gyöngy.
I would like to thank my friends and family, especially my mother, grandparents, and
Aunt Mary Ann and Uncle Greg, for supporting all my endeavours. Last, I am grateful for
the love, patience, and encouragement Bihotz Barrenetxea Dominguez has shown me.
xi

Contents
Declaration iii
Abstract v
Lay Summary vii
Acknowledgements xi
1 Introduction 1
2 Properties of space inverses of stochastic flows 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Properties of stochastic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Homeomorphism property of flows . . . . . . . . . . . . . . . . . . . 14
2.3.2 Moment estimates of inverse flows: Proof of Theorem 2.2.1 . . . . . 18
2.3.3 Strong limit of a sequence of flows: Proof of Theorem 2.2.3 . . . . . 23
2.4 Classical solutions of degenerate SPDEs: Proof of Theorem 2.2.4 . . . . . . 27
2.5 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 The method of stochastic characteristics for parabolic SIDEs 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Proof of main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 Proof of uniqueness for Theorem 3.2.2 . . . . . . . . . . . . . . . . . 48
3.3.2 Small jump case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Adding free and zero-order terms. . . . . . . . . . . . . . . . . . . . . 57
3.3.4 Adding uncorrelated part (Proof of Theorem 3.2.2) . . . . . . . . . . 63
3.3.5 Proof of Theorem 3.2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.1 Martingale and point measure moment estimates . . . . . . . . . . . 69
3.4.2 Optional projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.3 Estimates of Hölder continuous functions . . . . . . . . . . . . . . . . 71
3.4.4 Stochastic Fubini theorem . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.5 Itô-Wentzell formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xiii
4 The L2-Sobolev theory for parabolic SIDEs 89
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Degenerate linear stochastic evolution equations . . . . . . . . . . . . . . . . 91
4.2.1 Basic notation and definitions . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.3 Proof of Theorem 4.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 The L2-Sobolev theory for degenerate SIDEs . . . . . . . . . . . . . . . . . . 106
4.3.1 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.2 Proof of Theorem 4.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5 A finite difference scheme for non-degenerate parabolic SIDEs 131
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5 Proof of the main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6 Conclusions and future work 165
References 167
xiv
Chapter 1
Introduction
The subject of this thesis is parabolic linear stochastic integro-differential equations (SIDEs).
These equations arise in the study of non-linear filtering of semimartingales, scaled lim-
its of particle systems, flows of stochastic diffeomorphisms, and mathematical biology,
physics, and finance. For the author, the strongest impetus to the advancement of the
theory of SIDEs is the problem of non-linear filtering of jump diffusions.
The first derivation of the filtering equations for semimartingales with jumps is due to
B. Grigelionis in [Gri72]. The reduced form filtering equations, which are linear SIDEs,
were derived in [Gri76]. A proof of existence and uniqueness of solutions to the reduced
form filtering equations was given in [Tin77b] by E. Tinfavicˇius under the assumption of
non-degenerate stochastic parabolicity. While in [Tin77b] essentially the variational ap-
proach of SPDEs was used, one important aspect of this approach was missing, namely
Itô’s formula for the square of the norm for jump processes. Such a result is used to
conclude that the variational solution of a stochastic evolution equation is càdlàg with
values in the pivot space and to get an energy estimate of the supremum in time of the
norm of the solution in the pivot space. In [GK81], I. Gyöngy and N.V. Krylov derived
Itô’s formula for the square of the norm in the jump case, and the corresponding exis-
tence and uniqueness result for monotone stochastic evolution equations with jumps was
studied in [Gyö82]. In [Gri82], B. Grigelionis applied the result in [GK81] to complete
the variational existence and uniqueness result for the reduced form equations in the non-
degenerate setting that began in [Tin77b]. The reader that is interested in the derivation of
the reduced form filtering equations for semimartingales with jumps is advised to consult
the article [Gri82] and the review article [GM11].
It is also worth mentioning that the stochastic evolution equations driven by addi-
tive càdlàg martingale noise were studied by G. Pistone and M. Métivier in Section 5 of
[MP76] using the semigroup approach. One important aspect of the reduced form filter-
ing equation is that, in general, the principal part of the operators in the drift of these
equations depend on space and time and are random. The standard semigroup approach to
stochastic equations in infinite dimensions cannot treat these type of equations, since if the
semigroup generated by the principal part is random, the stochastic convolution does not
make sense as an ordinary Itô integral. In a recent work, M. Pronk and M. Veraar [PV14]
showed that it is possible to extend the semigroup approach to treat equations where the
1
2 Chapter 1. Introduction
principal part of stochastic evolution equation is random. However, at the time of writing,
there still seems to be some limitations in this approach; for example, some regularity in
time is needed for the diffusion coefficient in the stochastic heat equation when applying
their theory. We emphasize that this is not needed in the variational approach.
This thesis is dedicated to the the proof of existence, uniqueness, and regularity of
fully degenerate linear SIDEs. Specifically, we derive a theory for these equations in
Hölder spaces using the method of stochastic characteristics and a theory in L2-Sobolev
spaces using the variational approach of stochastic evolution equations and the method of
vanishing viscosity. We also investigate finite difference approximations of SIDEs with
non-degenerate diffusion.
Let us state the general form of the equation that we will investigate in this thesis. Let
(Ω,F ,F = (Ft)t≥0,P) be a complete filtered probability space satisfying the usual condi-
tions of right-continuity and completeness. For each real number T > 0, we let RT and PT
be the F-progressive and F-predictable sigma-algebra on Ω × [0,T ], respectively. For our
driving processes, we take a sequence wϱt , t ≥ 0, ϱ ∈ N, of independent one-dimensional F-
adapted Wiener processes and a F-adapted Poisson random measure p(dt, dz) on (R+×Z1,
B(R+)⊗Z1) with intensity measure π1(dz)dt, where (Z1,Z1, π1) is a sigma-finite measure
space. Denote by q(dt, dz) = p(dt, dz) − π1(dz)dt the compensated Poisson random mea-
sure. Let D1, E1,V1 ∈ Z be disjoint Z1-measurable subsets such that D1 ∪ E1 ∪ V1 = Z1
and π(V1) < ∞. Let (Z2,Z2, π2) be a sigma-finite measure space and D2, E2 ∈ Z2 be dis-
jointZ2-measurable subsets such that D2 ∪ E2 = Z2.
Fix an arbitrary positive real number T > 0 and integers d1, d2 ≥ 1. Let α ∈ (0, 2] and
let φ : Ω × Rd1 → Rd2 be F0 ⊗ B(Rd1)-measurable. We consider the system of SIDEs on
[0,T ] × Rd1 given by
dult =
(
(L1;lt +L
2;l
t )ut + 1[1,2](α)b
i;ll¯
t ∂iu
l¯
t + c
ll¯
t u
l¯
t + f
l
t
)
dt +
(
N lϱt ut + g
lϱ
t
)
dwϱt
+
∫
Z1
(
I1;lt,z ut− + h
l
t(z)
)
[1D1(z)q(dt, dz) + 1E1∪V1(z)p(dt, dz)], t ≤ T,
ul0 = φ
l, l ∈ {1, . . . , d2}, SIDE
where for ϕ ∈ C∞c (R
d1; Rd2), k ∈ {1, 2}, and l ∈ {1, . . . , d2},
Lk;lt ϕ(x) : = 1{2}(α)a
k;i j
t (x)∂i jϕ
l(x) +
∫
Dk
ρ
k;ll¯
t (x, z)
(
ϕl¯(x + Hkt (x, z)) − ϕ
l¯(x)
)
πk(dz)
+
∫
Dk
(
ϕl(x + Hkt (x, z)) − ϕ
l(x) − 1(1,2](α)H
k;i
t (x, z)∂iϕ
l(x)
)
πk(dz)
+ 1{2}(k)
∫
E2
(
(Ill¯d2 + ρ
2;ll¯
t (x, z))ϕ
l¯(x + H2t (x, z)) − ϕ
l(x)
)
π2(dz),
N lϱt ϕ(x) : = 1{2}(α)σ
iϱ
t (x)∂iϕ
l(x) + υll¯ϱt (x)ϕ
l¯(x), ϱ ∈ N,
3Ilt,zϕ(x) : = (I
ll¯
d2 + ρ
1;ll¯
t (x, z))ϕ
l¯(x + H1t (x, z)) − ϕ
l(x),
and
∫
Dk
(
|Hkt (x, z)|
α + |ρkt (x, z)|
2
)
πk(dz) +
∫
Ek
(
|Hkt (x, z)|
1∧α + |ρkt (x, z)|
)
πk(dz) < ∞.
The summation convention with respect to repeated indices i, j ∈ {1, . . . , d1}, l¯ ∈ {1, . . . ,
d2}, and ϱ ∈ N is used here and below. The d2 × d2 dimensional identity matrix is denoted
by Id2 . For a subset A of a larger set X, 1A denotes the {0, 1}-valued function taking the
value 1 on the set A and 0 on the complement of A. We assume that for each k ∈ {1, 2},
σkt (x) = (σ
k;iϱ
t (ω, x))1≤i≤d1, ϱ∈N, bt(x) = (b
i
t(ω, x))1≤i≤d1 , ct(x) = (c
ll¯
t (ω, x))1≤l,l¯≤d2 ,
υkt (x) = (υ
k;ll¯ϱ
t (ω, x))1≤l,l¯≤d2, ϱ∈N, ft(x) = ( f
i
t (ω, x))1≤i≤d2 , gt(x) = (g
iϱ
t (ω, x))1≤i≤d2, ϱ∈N,
are random fields on Ω× [0,T ]×Rd1 that are RT ⊗B(Rd1)-measurable. For each k ∈ {1, 2},
we assume that
Hkt (x, z) = (H
k;i
t (ω, x, z))1≤i≤d1 , ρ
k
t (x, z) = (ρ
k;ll¯
t (ω, x, z))1≤l,l¯≤d2 ,
are random fields on Ω×[0,T ]×Rd1×Zk that are PT ⊗B(Rd1)⊗Zk-measurable. Moreover,
we assume that ht(x, z) = (hit(ω, x, z))1≤i≤d2 is a random field on Ω × [0,T ] × R
d1 × Z1 that
is PT ⊗ B(Rd1) ⊗Z1-measurable.
This thesis is organized as follows. Chapter 2 is dedicated to establishing some prop-
erties of space inverses of stochastic flows generated by an SDEs with jumps. These
properties play an important role in Chapter 3 when we construct classical solutions of
(SIDE). Specifically, in Chapter 2, we derive moment estimates and a strong limit theorem
for space inverses of stochastic flows generated by jump SDEs with adapted coefficients in
weighted Hölder norms. We also give a relatively simple, but novel, proof of existence and
uniqueness of classical solutions of a degenerate SPDE (i.e. (SIDE) with a1;i j = σiϱσ jϱ,
a2 ≡ H1 ≡ H2 ≡ ρ1 ≡ ρ2 ≡ b ≡ c ≡ f ≡ υ ≡ g ≡ h ≡ 0, and φ(x) = x). Our method of
proof allows us to prove existence and uniqueness when σ can degenerate on set of pos-
itive probability under less regularity than done previously. If σ is non-degenerate, then
much more can be done, and we refer the reader to the seminal work [FGP10].
The focus of Chapter 3 is to give a complete proof of existence and uniqueness of clas-
sical solutions of (SIDE) with ai j = σiϱσ jϱ + σ2;iϱσ2; jϱ and with Hölder assumptions on
the coefficients and data using Feynman-Kac transformations, conditioning, and the inter-
lacing of space-inverses of stochastic flows associated with the equations. We emphasize
that σ,σ2,H, and H2 can vanish on a set of positive probability.
In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear
4 Chapter 1. Introduction
stochastic evolution equations driven by jump processes in a Hilbert scale using the varia-
tional framework of stochastic evolution equations and the method of vanishing viscosity.
This result extends the work of B. Rozovskii in [Roz90] from the continuous case to the
jump case. As an application of our result, we derive unique solutions of (SIDE) with
a1;i j ≡ a2;i j ≡ σ ≡ g ≡ 0, E2 = E1 = V1 = ∅, and α ∈ (1, 2) in the integer L2-Sobolev scale.
We confine ourself to an equation without continuous noise and diffusion because the the-
ory of Sobolev solutions for equations in this case are well-studied (see, e.g. [KR82],
[Roz90], and [GGK14]). The results derived in Chapters 1, 2, and 3 are the fruit of a
collaborative effort with Remigijus Mikulevicˇius at the University of Southern California;
both R. Mikulevicˇius and the author have contributed substantially to the development of
the ideas contained in these chapters.
Finally, in Chapter 5, we consider finite difference schemes for (SIDE) with d2 = 1,
H1t (x, z) = z, ρ
1 = 1, E1 = V1 = ∅, α = 2, h ≡ H2 ≡ ρ2 ≡ 0, and ai j − σiϱσ jϱ > 0.
We show that the L2(Ω)-pointwise rate of convergence is of order one in space and order
one-half in time. The results given in this chapter are the fruit of a collaborative effort with
Konstantinos Dareiotis at the University of Edinburgh; both K. Dareiotis and the author
have contributed substantially to the development of the ideas contained in this chapter.
Basic Notation
Let N be the set of natural numbers and N0 = {0, 1, . . .} be the set of natural numbers
including zero. Let Z be the set of integers. For a topological space (X,X), we denote the
Borel sigma-field on X by B(X).
For each integer n ≥ 1, let Rn be the n-dimensional Euclidean space and for each
x ∈ Rn, denote by |x| the Euclidean norm of x = (x1, . . . , xn). We set R = R1 and for
two elements of x, y ∈ R, we denote by x ∨ y the maximum of x and y and by x ∧ y, the
minimum of x and y. Let R+ denote the set of non-negative elements of R1.
For an integer n ≥ 1 and for each i ∈ {1, . . . , n}, we let ∂i = ∂∂xi be the spatial derivative
operator with respect to the direction xi and write ∂i j = ∂i∂ j for each i, j ∈ {1, . . . , n}. We
also denote the identity operator by ∂0. For a once differentiable function f = ( f 1 . . . , f n) :
Rn → Rn, we denote the gradient of f by ∇ f = (∂ j f i)1≤i, j≤d1 . For a multi-index γ =
(γ1, . . . , γd) ∈ {0, 1, 2, . . . , }d1 of length |γ| := γ1 + · · · + γd, denote by ∂γ the operator
∂γ = ∂
γ1
1 · · · ∂
γd
d , where ∂
0
i is the identity operator for all i ∈ {1, . . . , d1}. For i ∈ {1, . . . , d},
let ∂−i = −∂i.
For integers d1, d2 ≥ 1, we denote by C∞c (R
d1; Rd2) the space of Rd2-valued infinitely
differentiable functions with compact support in Rd1 .
For each integer n ≥ 1, the norm of an element x of ℓ2(Rn), the space of square-
summable Rn-valued sequences, is denoted by |x|. We set ℓ2 = ℓ2(R). For integers n ≥ 1
5and a once differentiable function f = ( f 1ϱ, . . . , f nϱ)ϱ∈N : Rn → ℓ2(Rn), we denote the
gradient of f by ∇ f = (∂ j f iϱ)1≤i, j≤n,ϱ∈N and understand it as a function from Rd1 to ℓ2(Rn
2
).
For a Fréchet space χ and fixed time T > 0, we denote by D([0,T ]; χ) the space of
χ-valued càdlàg (continuous from the right and limits from the left) functions on [0,T ]
and by C([0,T ]2; χ) the space of χ-valued continuous functions on [0,T ] × [0,T ]. The
spaces D([0,T ]; χ) and C([0,T ]2; χ) are Fréchet spaces when endowed with the set of the
supremum (in time) seminorms, which we assume, unless otherwise specified.
The notation N = N(·, · · · , ·) is used to denote a positive constant depending only on
the quantities appearing in the parentheses. In a given context, the same letter is often
used to denote different constants depending on the same parameter. Moreover, for any
function f defined on a set X and taking valued in a linear space Y with zero element 0Y ,
the notation f ≡ 0 indicates that f (x) = 0Y for all x ∈ X.

Chapter 2
Properties of space inverses of stochastic
flows
2.1 Introduction
Let (Ω,F ,F = (Ft)t≥0,P) be a complete filtered probability space satisfying the usual con-
ditions of right-continuity and completeness. Let wϱt , t ≥ 0, ϱ ∈ N, be a sequence of inde-
pendent one-dimensional F-adapted Wiener processes. For a sigma-finite measure space
(Z,Z, π), we let p(dt, dz) be an F-adapted Poisson random measure on (R+×Z,B(R+)⊗Z)
with intensity measure π(dz)dt and denote by q(dt, dz) = p(dt, dz) − π(dz)dt the compen-
sated Poisson random measure. For each real number T > 0, we let RT and PT be the
F-progressive and F-predictable sigma-algebra on Ω × [0,T ], respectively.
Fix a real number T > 0 and an integer d ≥ 1. For each stopping time τ ≤ T , consider
the flow Xt = Xt(τ, x), (t, x) ∈ [0,T ] × Rd, generated by the SDE
dXt = bt(Xt)dt + σ
ϱ
t (Xt)dw
ϱ
t +
∫
Z
Ht(Xt−, z)q(dt, dz), τ < t ≤ T,
Xt = x, t ≤ τ, (2.1.1)
where bt(x) = (bit(ω, x))1≤i≤d and σt(x) = (σ
iϱ
t (ω, x)1≤i≤d,ϱ∈N are R
d-valued RT ⊗ B(Rd)-
measurable functions defined on Ω × [0,T ] × Rd and Ht(x, z) = (Hit(ω, x, z))1≤i≤d is an
Rd-valued PT ⊗ B(Rd) ⊗ Z-measurable function defined on Ω × [0,T ] × Rd × Z. The
summation convention with respect to the repeated index ϱ is used here and below.
In this chapter, under Hölder regularity conditions on the coefficients b, σ, and H, we
provide a simple and direct derivation of moment estimates of the space inverse of the
flow, denoted X−1t (τ, x), in weighted Hölder norms. This is done by applying the Sobolev
embedding theorem and the change of variable formula. Using a similar method, we es-
tablish a strong limit theorem in weighted Hölder norms for a sequence of flows X(n)t (τ, x)
and their inverses X(n);−1t (τ, x) corresponding to a sequence of coefficients (b
(n), σ(n),H(n))
converging in weighted Hölder norms. Furthermore, as an application of the diffeomor-
phism property of flow, we give a direct derivation of the linear second order degenerate
SPDE governing the inverse flow X−1t (τ, x) when H ≡ 0. Specifically, for each τ ≤ T ,
7
8 Chapter 2. Properties of space inverses of stochastic flows
consider the stochastic flow Yt = Yt(τ, x), (t, x) ∈ [0,T ] × Rd, generated by the SDE
dYt = bt(Yt)dt + σ
ϱ
t (Yt)dw
ϱ
t , τ < t ≤ T,
Yt = x, t ≤ τ.
Assume that b and σ have linear growth, bounded first and second derivatives, and that
the second derivatives of b and σ are α-Hölder for some α > 0. By partitioning the time
interval and using Taylor’s theorem, the Sobolev embedding theorem, and some basic
properties of the flow and its inverse, we show that ut(x) = ut(τ, x) := Y−1t (τ, x), (t, x) ∈
[0,T ] × Rd is the unique classical solution of the SPDE given by
dut(x) =
(
1
2
σ
iϱ
t (x)σ
jϱ
t (x)∂i jut(x) − bˆ
i
t(x)∂iut(x)
)
dt − σiϱt (x)∂iut(x)dw
ϱ
t , τ < t ≤ T,
ut(x) = x, t ≤ τ, (2.1.2)
where
bˆit(x) = b
i
t(x) − σ
jϱ
t (x)∂ jσ
iϱ
t (x).
In Chapter 3, we will use all of the properties of the flow Xt(τ, x) that are established in
this first chapter in order to derive the existence and uniqueness of classical solutions of
linear parabolic SIDEs.
One of the earliest works to investigate the homeomorphism property of flows
of SDEs with jumps is by P. Meyer in [Mey81]. In [Mik83], R. Mikulevicˇius extended
the properties found in [Mey81] to SDEs driven by arbitrary continuous martingales and
random measures. Many other authors have since expanded upon the work in [Mey81],
see for example [FK85, Kun04, MB07, QZ08, Zha13, Pri14] and references therein. In
[Kun04, Kun86b], H. Kunita studied the diffeomorphism property of the flow Xt(s, x),
(s, t, x) ∈ [0,T ]2 × Rd, and in the setting of deterministic coefficients, he showed that for
each fixed t, the inverse flow X−1t (s, x), (s, x) ∈ [t,T ] × R
d, solves a backward SDE. By
estimating the associated backward SDE, one can obtain moment estimates and a strong
limit theorem for the inverse flow in essentially the same way that moment estimates are
obtained for the direct flow (see, e.g. [Kun86b]). However, this method of deriving mo-
ment estimates and a strong limit theorem for the inverse flow uses a time reversal, and
thus ,requires that the coefficients are deterministic. In the case H ≡ 0, numerous authors
have investigated properties of the inverse flow with random coefficients. In Chapter 2 of
[Bis81], Lemma 2.1 and 2.2 of [OP89], and Section 6.1 and 6.2 of [Kun96], properties
of Y−1t (τ, x) (i.e. moment estimates, strong limit theorem, and that it solves (2.1.2)) are
established by first showing that the inverse flow solves the Stratonovich form SDE for
2.1. Introduction 9
Zt = Zt(τ, x), (t, x) ∈ [0,T ] × Rd, given by
dZt(x) = −Ut(Zt(x))bt(x)dt − Ut(Zt(x))σ
ϱ
t (x) ◦ dw
ϱ
t , τ < t ≤ T, (2.1.3)
Z0(x) = x, τ < t,
where Ut(x) = Ut(τ, x) = ∇Yt(τ, x)−1. In order to obtain a strong solution of (2.1.3), condi-
tions are imposed to ensure that ∇Ut(x) is locally-Lipschitz in x. In the degenerate setting,
the third derivative of bt and σt need to be α-Hölder for some α > 0 to ensure that ∇Ut(x)
is locally-Lipschitz in x. In direct contrast to this approach, we first derive properties of
the inverse flow under the very assumptions that guarantee Yt(τ, x) is a diffeomorphism
(i.e. when the first derivative of the coefficients are α-Hölder for some α > 0), and then
we derive the existence of the equation without resorting to the SDE interpretation of the
SPDE.
Classical solutions of (2.1.2) have been constructed in [Bis81, Kun96] by directly
showing that Y−1t (τ, x) solves (2.1.3). As we have mentioned above, this approach requires
the third derivatives of bt and σt to be α-Hölder for some α > 0. Yet another approach
to deriving existence of classical solutions of (2.1.2) is using the method of time reversal
(see, e.g. [Kun96, DPT98]). While this method only requires that the second derivatives
of bt and σt are α-Hölder for some α > 0, it does impose that the coefficients are deter-
ministic. In [KR82], N.V. Krylov and B.L. Rozvskii derived the existence and uniqueness
of generalized solutions of degenerate second order linear parabolic SPDEs in Sobolev
spaces using variational approach of SPDEs and the method of vanishing viscosity (see,
also, [GGK14] and Chapter 4, Section 2, Theorem 1 in [Roz90]). Thus, by appealing to
the Sobolev embedding theorem, this theory can be used to obtain classical solutions of
degenerate linear SPDEs. Proposition 1 of Chapter 5, Section 2, in [Roz90] shows that
if σ is uniformly bounded and four-times continuously differentiable in x with uniformly
bounded derivatives and b is uniformly bounded and three-times continuously differen-
tiable with uniformly bounded derivatives, then there exists a classical solution of (2.1.2)
and ut(x) = Y−1t (x). This is more regularity than we require and we are also able to obtain
solutions in the entire Hölder scale.
This chapter is organized as follows. In Section 2.2, we state our notation and main
results. Section 2.3 is devoted to deriving moment estimates and a strong limit theorem for
the space inverse of a stochastic flow generated by a Lévy driven SDE. In Section 2.4, we
show that Y−1t (τ, x) is the unique classical solution of (2.1.2). In Section 2.5, the appendix,
auxiliary facts that are used throughout the chapter are discussed.
10 Chapter 2. Properties of space inverses of stochastic flows
2.2 Statement of main results
Let us describe some notation that will be used in this chapter. Elements of Rd are un-
derstood as column vectors and elements of Rd
2
are understood as matrices of dimension
d × d. We denote the transpose of an element x ∈ Rd by x∗. For a Banach space V with
norm | · |V , domain (i.e. open connected set) Q of Rd, and continuous function f : Q→ V ,
we define
| f |0;Q;V = sup
x∈Q
| f (x)|V
and
[ f ]β;Q;V = sup
x,y∈Q,x,y
| f (x) − f (y)|V
|x − y|βV
, β ∈ (0, 1].
For any real number β ∈ R, we write β = [β]− + {β}+, where [β]− is an integer and
{β}+ ∈ (0, 1]. For a Banach space V with norm | · |V , real number β > 0, and domain
Q of Rd, we denote by Cβ(Q; V) the Banach space of all bounded continuous functions
f : Q→ V having finite norm
| f |β;Q;V :=
∑
|γ|≤[β]−
|∂γ f |0;Q;V +
∑
|γ|=[β]−
[∂γ f ]{β}+;Q;V .
When Q = Rd and V = Rn or V = ℓ2(Rn) for any integer n ≥ 1, we drop the subscripts
Q and V from the norm | · |β;Q;V and write | · |β. For a Banach space V and for each β > 0,
denote by Cβloc(R
d; V) the Fréchet space of continuous functions f : Rd → V satisfying
f ∈ Cβ(Q; V) for all bounded domains Q ⊂ Rd. We call a function f : Rd → Rd a
Cβloc(R
d; Rd)-diffeomorphism if f is a homeomorphism and both f and its inverse f −1 are
in Cβloc(R
d; Rd).
If we do not specify to which space the parameters ω, t, x, y, z and n belong, then we
mean ω ∈ Ω, t ∈ [0,T ], x, y ∈ Rd, z ∈ Z, and n ∈ N.
Let r1(x) =
√
1 + |x|2, x ∈ Rd. For each real number β > 1, we introduce the following
regularity condition on the coefficients b, σ, and H.
Assumption 2.2.1 (β). (1) There is a constant N0 > 0 such that for all (ω, t, z) ∈ Ω ×
[0,T ] × Z,
|r−11 bt|0 + |∇bt|β−1 + |r
−1
1 σt|0 + |∇σt|β−1 ≤ N0 and |r
−1
1 Ht(z)|0 + |∇Ht(z)|β−1 ≤ Kt(z),
2.2. Statement of main results 11
where K : Ω × [0,T ] × Z → R+ is a PT ⊗Z-measurable function satisfying
Kt(z) +
∫
Z
Kt(z)
2π(dz) ≤ N0,
for all (ω, t, z) ∈ Ω × [0,T ] × Z.
(2) There are constants η ∈ (0, 1) and Nκ > 0 such that for all (ω, t, x, z) ∈ {(ω, t, x, z) ∈
Ω × [0,T ] × Rd × Z : |∇Ht(ω, x, z)| > η},
| (Id + ∇Ht(x, z))−1 | ≤ Nκ.
The following theorem shows that if Assumption 2.2.1 (β) holds for some β > 1, then
for any β′ ∈ [1, β], the solution Xt(τ, x) of (2.1.1) has a modification that is a C
β′
loc(R
dRd)
-diffeomorphism and the p-th moments of the weighted β′-Hölder norms of the inverse
flow are bounded. This theorem will be proved in the next section.
Theorem 2.2.1. Let Assumption 2.2.1(β) hold for some β > 1.
(1) For any stopping time τ ≤ T and β′ ∈ [1, β), there exists a modification of the strong
solution Xt(τ, x) of (2.1.1), also denoted by Xt(τ, x), such that P-a.s. the mapping
Xt(τ, ·) : Rd → Rd is a C
β′
loc(R
d; Rd)-diffeomorphism, X·(τ, ·), X−1· (τ, ·) ∈ D([0,T ];
Cβ
′
loc(R
d; Rd)), and X−1t− (τ, ·) coincides with the inverse of Xt−(τ, ·). Moreover, for all
ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
t≤T
|r−(1+ϵ)1 Xt(τ)|
p
0
]
+ E
[
sup
t≤T
|r−ϵ1 ∇Xt(τ)|
p
β′−1
]
≤ N
and a constant N = N(d, p,N0,T, β′, η,Nκ, ϵ) such that
E
[
sup
t≤T
|r−(1+ϵ)1 X
−1
t (τ)|
p
0
]
+ E
[
sup
t≤T
|r−ϵ1 ∇X
−1
t (τ)|
p
β′−1
]
≤ N. (2.2.1)
(2) If H ≡ 0, then for all β′ ∈ (1, β), P-a.s. X·(·, ·), X−1· (·, ·) ∈ C([0,T ]
2;Cβ
′
loc(R
d; Rd)) and
for all ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
s,t≤T
|r−(1+ϵ)1 Xt(s)|
p
0
]
+ E
[
sup
s,t≤T
|r−ϵ1 ∇Xt(s)|
p
β′−1
]
≤ N
and
E
[
sup
s,t≤T
|r−(1+ϵ)1 X
−1
t (s)|
p
0
]
+ E
[
sup
s,t≤T
|r−ϵ1 ∇X
−1
t (s)|
p
β′−1
]
≤ N.
Remark 2.2.2. The estimate (2.2.1) is used in Chapter 3 to take the optional projection
of a linear transformation of the inverse flow of a jump SDE driven by two independent
12 Chapter 2. Properties of space inverses of stochastic flows
Weiner processes and two independent Poisson random measures relative to the filtration
generated by one of the Weiner processes and Poisson random measures.
Now, let us state our strong limit theorem for a sequence of flows, which will also
be proved in the next section. We will use this strong limit theorem in [LM14a] to show
that the inverse flow of a jump SDE solves a parabolic SIDE. For each n, consider the
stochastic flow X(n)t = X
(n)
t (τ, x), (t, x) ∈ [0,T ] × R
d, generated by the SDE
dX(n)t = b
(n)
t (X
(n)
t )dt + σ
(n)lϱ
t (X
(n)
t )dw
ϱ
t +
∫
Z
H(n)t (X
(n)
t− , z)q(dt, dz), τ ≤ t ≤ T,
X(n)t = x, t ≤ τ.
Here we assume that for all n, b(n), σ(n), and H(n) satisfy the same measurability conditions
as b, σ, and H, respectively.
Theorem 2.2.3. Let Assumption 2.2.1(β) hold for some β > 1 and assume that b(n), σ(n),
and H(n) satisfy Assumption 2.2.1 (β) uniformly in n ∈ N. Moreover, assume that
dPdt − lim
n→∞
(
|r−11 b
(n)
t − r
−1
1 bt|0 + |∇b
(n)
t − ∇bt|β−1
)
= 0,
dPdt − lim
n→∞
(
|r−11 σ
(n)
t − r
−1
1 σt|β−1 + |∇σ
(n)
t − ∇σt|0
)
= 0,
and for all (ω, t, z) ∈ Ω × [0,T ] × Z and n ∈ N,
|r−11 H
(n)
t (z) − r
−1
1 Ht(z)|0 + |∇H
(n)
t (z) − ∇Ht(z)|β−1 ≤ K
(n)(t, z),
where (K(n)t (z))n∈N is a sequence of R+-valued PT ⊗ Z measurable functions defined on
Ω × [0,T ] × Z satisfying for all (ω, t, z) ∈ Ω × [0,T ] × Z and n ∈ N,
K(n)t (z) +
∫
Z
K(n)t (z)
2π(dz) ≤ N0
and
dPdt − lim
n→∞
∫
Z
K(n)t (z)
2π(dz) = 0.
Then for any stopping time τ ≤ T, β′ ∈ [1, β), and all ϵ > 0, and p ≥ 2, we have
lim
n→∞
(
E
[
sup
t≤T
|r−(1+ϵ)1 X
(n)
t (τ) − r
−(1+ϵ)
1 Xt(τ)|
p
0
]
+ E
[
sup
t≤T
|r−ϵ1 ∇X
(n)
t (τ) − r
−ϵ
1 ∇Xt(τ)|
p
β′−1
])
= 0,
lim
n→∞
E
[
sup
t≤T
|r−(1+ϵ)1 X
(n);−1
t (τ) − r
−(1+ϵ)
1 X
−1
t (τ)|
p
0
]
= 0,
and
lim
n→∞
E
[
sup
t≤T
|r−ϵ1 ∇X
(n);−1
t (τ) − r
−ϵ
1 ∇X
−1
t (τ)|
p
β′−1
]
= 0.
2.2. Statement of main results 13
Let us introduce our class of solutions for the equation (2.1.1). LetOT be the F-optional
sigma-algebra on Ω× [0,T ]. For a each number β′ > 2, let Cβ
′
cts(R
d; Rd) be the linear space
of all random fields v : Ω × [0,T ] × Rd → Rd such that v is OT ⊗ B(Rd)-measurable and
P-a.s. r−λ1 (·)v·(·) is a C([0,T ];C
β′(Rd; Rd)) for a real number λ > 0.
We introduce the following assumption for a real number β > 2.
Assumption 2.2.2 (β). There is a constant N0 such that for all (ω, t) ∈ Ω × [0,T ],
|r−11 bt|0 + |r
−1
1 σt|0 + |∇bt|β−1 + |∇σt|β−1 ≤ N0.
Theorem 2.2.4. Let Assumption 2.2.2(β) hold for some β > 2. Then for any stopping time
τ ≤ T and β′ ∈ [1, β), there exists a unique process u(τ) in Cβ
′
cts(R
d; Rd) that solves (2.1.2).
Moreover, P-a.s. ut(τ, x) = Y−1t (τ, x) for all (t, x) ∈ [0,T ]×R
d and for all ϵ > 0 and p ≥ 2,
there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
s,t≤T
|r−(1+ϵ)1 ut(s)|
p
0
]
+ E
[
sup
s,t≤T
|r−ϵ1 ∇ut(s)|
p
β′−1
]
≤ N.
Remark 2.2.5. It is clear by the proof of this theorem that if σ ≡ 0, then we only need to
assume that Assumption 2.2.2 (β) holds for some β > 1.
Now let us consider the SPDE given by
du¯t(x) =
(
1
2
σ
iϱ
t (x)σ
jϱ
t (x)∂i ju¯t(x) + b
i
t(x)∂iu¯t(x)
)
dt + σiϱt (x)∂iu¯t(x)dw
ϱ
t , τ < t ≤ T,
u¯t(x) = x, t ≤ τ. (2.2.2)
This SPDE differs from the one given in (2.1.2) by the first-order coefficient in the
drift. In order to obtain an existence and uniqueness theorem for this equation, we have to
impose additional assumptions on σ.
We introduce the following assumption for a real number β > 2.
Assumption 2.2.3 (β). There is a constant N0 > 0 such that for all (ω, t) ∈ Ω × [0,T ],
|r−11 bt|0 + |∇bt|β−1 + |σt|β+1 ≤ N0.
For each τ ≤ T , consider the stochastic flow Yˆt = Yˆt(τ, x), (t, x) ∈ [0,T ]×Rd, generated
by the SDE
dY¯t = −bˆt(Y¯t)dt − σ
ϱ
t (Y¯t)dw
ϱ
t , τ < t ≤ T,
Yt = x, t ≤ τ.
14 Chapter 2. Properties of space inverses of stochastic flows
If Assumption 2.2.3(β) holds for some β > 2, then for all (ω, t, x) ∈ Ω × [0,T ] × Rd,
|bˆt(x)| ≤ |bt(x)| + |σt(x)|∇σt(x)| ≤ N0(N0 + 1) + N0|x|
and
|∇bˆt|β−1 ≤ |∇bt|β−1 + |σt|β−1|∇2σt|β−1 + |∇σt|2β−1 ≤ N0 + 2N
2
0 ,
which immediately implies the following corollary of Theorem 2.2.4.
Corollary 2.2.6. If Assumption 2.2.3(β) holds for some β > 2, then for any stopping time
τ ≤ T and β′ ∈ [1, β), there exists a unique process u¯(τ) in Cβ
′
cts(R
d; Rd) that solves (2.2.2).
Moreover, P-a.s. u¯t(τ, x) = Y¯−1t (τ, x) for all (t, x) ∈ [0,T ]×R
d and for all ϵ > 0 and p ≥ 2,
there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
s,t≤T
|r−(1+ϵ)1 u¯t(s)|
p
0
]
+ E
[
sup
s,t≤T
|r−ϵ1 ∇u¯t(s)|
p
β′−1
]
≤ N.
2.3 Properties of stochastic flows
2.3.1 Homeomorphism property of flows
In this subsection, we collect some results about flows of jump SDEs that we will need. In
particular, we present sufficient conditions that guarantee the homeomorphism property of
flows of jump SDEs. First, let us introduce the following assumption, which is the usual
linear growth and Lipschitz condition on the coefficients b, σ, and H of the SDE (2.1.1).
Assumption 2.3.1. There is a constant N0 > 0 such that for all (ω, t, x, y) ∈ Ω×[0,T ]×R2d,
|bt(x)| + |σt(x)| ≤ N0(1 + |x|),
|bt(x) − bt(y)| + |σt(x) − σt(y)| ≤ N0|x − y|.
Moreover, for all (ω, t, x, y, z) ∈ Ω × [0,T ] × R2d × Z,
|Ht(x, z)| ≤ K1(t, z)(1 + |x|),
|Ht(x, z) − Ht(y, z)| ≤ K2(t, z)|x − y|,
where K1,K2 : Ω × [0,T ] × Z → R+ are PT ⊗Z-measurable functions satisfying
K1(t, z) + K2(t, z) +
∫
Z
(
K1(t, z)
2 + K2(t, z)
2
)
π(dz) ≤ N0,
for all (ω, t, z) ∈ Ω × [0,T ] × Z.
2.3. Properties of stochastic flows 15
It is well-known that under this assumption, there exists a unique strong solution
Xt(s, x) of (2.1.1) (see e.g. Theorem 3.1 in [Kun04]). We will also make use of the follow-
ing assumption.
Assumption 2.3.2. For all (ω, t, x, z) ∈ Ω × [0,T ] × Rd × Z, Ht(x, z) is differentiable in x,
and there are constants η ∈ (0, 1) and Nκ > 0 such that for all (ω, t, x, z) ∈ {(ω, t, x, z) ∈
Ω × [0,T ] × Rd × Z : |∇Ht (ω, x, z)| > η},
∣∣∣(Id + ∇Ht(x, z))−1
∣∣∣ ≤ Nκ.
The coming lemma shows that under Assumptions 2.3.1 and 2.3.2, the mapping x +
Ht(x, z) from Rd to Rd is a diffeomorphism and the gradient of inverse map is bounded.
Lemma 2.3.1. Let Assumptions 2.3.1 and 2.3.2 hold. For all (ω, t, z) ∈ Ω× [0,T ]× Z, the
mapping H˜t(·, z) : Rd → Rd defined by H˜t(x, z) := x + Ht(x, z) is a diffeomorphism and
|H˜−1t (x, z)| ≤ N¯N0 + N¯ |x| and |∇H˜
−1
t (x, z)| ≤ N¯,
where N¯ := (1 − η)−1 ∨ N0.
Proof. (1) On the set (ω, t, x, z) ∈ {(ω, t, x, z) ∈ Ω × [0,T ] × Rd × Z : |∇Ht(ω, x, z)| ≤ η},
we have
|κt(ω, x, z)| ≤
∣∣∣∣∣∣∣
Id +
∞∑
n=1
(−1)n[∇Ht(ω, x, z)]n
∣∣∣∣∣∣∣
≤
1
1 − η
.
It follows from Assumption 2.3.2 that for all ω, t, x, and z, the mapping ∇H˜t(x, z) has a
bounded inverse. Therefore, by Theorem 0.2 in [DMGZ94], the mapping H˜t(·, z) : Rd →
Rd is a global diffeomorphism. Moreover, for all ω, t, x and z,
|H˜−1t (x, z) − H˜
−1
t (y, z)| ≤ N¯ |x − y|,
which yields
|H˜t(x, z) − H˜t(y, z)| ≥ N¯−1|x − y| =⇒ |H˜t(x, z)| + K1(t, z) ≥ N¯−1|x|,
and hence
|H˜−1t (x, z)| ≤ N¯K1(t, z) + N¯|x| ≤ N¯N0 + N¯ |x|.
□
The following estimates are essential in the proof of the homeomorphic property of
the flow and the derivation of moment estimates of the inverse flow. We refer the reader
to Theorem 3.2 and Lemmas 3.7 and 3.9 in [Kun04] and Lemma 4.5.6 in [Kun97] (H ≡ 0
case) for the proof of the following lemma.
16 Chapter 2. Properties of space inverses of stochastic flows
Lemma 2.3.2. Let Assumption 2.3.1 hold.
(1) For all p ≥ 2, there is a constant N = N(p,N0,T ) such that for all s, s¯ ∈ [0,T ] and
x, y ∈ Rd,
E
[
sup
t≤T
r1(Xt(s, x))
p
]
≤ Nr1(x)p, (2.3.1)
E
[
sup
t≤T
|Xt(s, x) − Xt(s, y)|p
]
≤ N|x − y|p. (2.3.2)
(2) If Assumption 2.3.2 holds, then for any p ∈ R, there is a constant N = N(p,N0,T,
η,Nκ) such that for all s ∈ [0,T ] and x, y ∈ Rd,
E
[
sup
t≤T
r1(Xt(s, x))
p
]
≤ Nr1(x)p, (2.3.3)
and
E
[
sup
t≤T
|Xt(s, x) − Xt(s,Y)|p
]
≤ N|x − y|p. (2.3.4)
In the next proposition, we collect some facts about the homeomorphic property of
the flow. Let us mention that the homeomorphism property has been shown in [QZ08] to
hold under a log-Lipschitz condition on the coefficients. The key idea is to use Bihari’s
inequality instead of Gronwall’s inequality, but we do not pursue this here.
Proposition 2.3.3. Let Assumptions 2.3.1 and 2.3.2 hold.
(1) There exists a modification of the strong solution Xt(s, x), (s, t, x) ∈ [0,T ]2 × Rd,
of (2.1.1), also denoted by Xt(s, x), that is càdlàg in s and t and continuous in x.
Moreover, for any stopping time τ ≤ T, P-a.s. for all t ∈ [0,T ], the mappings
Xt(τ, ·), Xt−(τ, ·) : Rd → Rd are homeomorphisms and the inverse of Xt(τ, ·), denoted
by X−1t (τ, ·), is càdlàg in t and continuous in x, and X
−1
t− (τ, ·) coincides with the inverse
of Xt−(τ, ·). In particular, if (xn)n≥1 is a sequence in Rd such that limn→∞ xn = x for
some x ∈ Rd, then P-a.s.
lim
n→∞
sup
t≤T
|X−1t (τ, xn) − X
−1
t (τ, x)| = 0.
Furthermore, for all β′ ∈ [0, 1), P-a.s. X(τ, ·) ∈ D([0,T ];Cβ
′
loc(R
d; Rd)) and for all
ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
t≤T
|r−(1+ϵ)1 Xt(τ)|
p
β′
]
≤ N. (2.3.5)
(2) If H ≡ 0, then P-a.s. for all s, t ∈ [0,T ], the Xt(s, x) and X−1t (s, x) are continuous in
s, t, and x. Moreover, for all β′ ∈ [0, 1], P-a.s. X(·, ·) ∈ C([0,T ]2;Cβ
′
loc(R
d; Rd)) and for
2.3. Properties of stochastic flows 17
all ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
s,t≤T
|r−(1+ϵ)1 Xt(s)|
p
β′
]
≤ N. (2.3.6)
Proof. (1) Owing to Assumptions 2.3.1 and 2.3.2, by Lemma 2.3.1, for all ω, t and z, the
process H˜t(x, z) := x + Ht(x, z) is a homeomorphism (in fact, it is a diffeomorphism) in x
and H˜−1t (x, z) has linear growth and is Lipschitz. This implies that assumptions of Theorem
3.5 in [Kun04] hold and hence there is modification of Xt(s, x), denoted Xt(s, x), such that
for all s ∈ [0,T ], P-a.s. for all t ∈ [0,T ], Xt(s, ·) is a homeomorphism. Following [Kun04],
for each (s, t, x) ∈ [0,T ]2 × Rd, we set
X¯t(s, x) =



x t ≤ s
Xt(0, X−1s (0, x)) t ≥ s,
(2.3.7)
and remark that P-a.s. X¯t(s, x) is càdlàg in s and t and continuous in x, and P-a.s. for
all (s, t) ∈ [0,T ]2, X¯t(s, ·) is a homeomorphism, and X¯t(s, x) is a version of Xt(s, x) (the
equation started at s). Fix a stopping time τ ≤ T . We will now show that X¯t(τ, x) =
X¯t(s, x)|s=τ (i.e. X¯t(s, x) evaluated at s = τ) is a version of Xt(τ, x). Define the sequence of
stopping times (τn)n≥1 by
τn =
n−1∑
k=1
kT
n
1{ (k−1)T
n ≤τ<
kT
n
} + T1{
τ≥ (n−1)Tn
}.
For each n and x, let X(n)t = X
(n)
t (x) = X¯t(τn, x), t ∈ [0,T ]. It follows that for all n, t, and x,
P-a.s. for all k ∈ {1, . . . , n},
X(n)t (x)1{τn= kTn } = Xt
(
kT
n
, x
)
1{τn= kTn },
and hence
X(n)t (x)1{τn= kTn } = 1{τn= kTn }x + 1{τn= kTn }
∫
] kTn ,
kT
n ∨t]
br(X
(n)
r (x))dr
+ 1{τn= kTn }
∫
] kTn ,
kT
n ∨t]
σϱr (X
(n)
r (x))dw
ϱ
r
+ 1{τn= kTn }
∫
] kTn ,
kT
n ∨t]
∫
Z
Hr(X
(n)
r (x), z)q(dr, dz).
Since Ω is the disjoint union of the sets
{
τn =
kT
n
}
, k ∈ {1, . . . , n}, it follows that X(n)t (x)
18 Chapter 2. Properties of space inverses of stochastic flows
solves
X(n)t (x) = x +
∫
]τn,τn∨t]
br(X
(n)
r (x))dr +
∫
]τn,τn∨t]
σϱr (X
(n)
r (x))dw
ϱ
r
+
∫
]τn,τn∨t]
∫
Z
Hr(X
(n)
r (x), z)q(dr, dz).
Thus, by uniqueness, we find that for all t and x, P-a.s. X¯t(τn, x) = X
(n)
t (x) = Xt(τn, x).
It is easy to check that for all t and x, P-a.s. Xt(τn, x) converges to Xt(τ, x) as n tends to
infinity. Since X¯t(s, x) is càdlàg in s, X¯t(τn, x) converges to X¯t(τ, x) as n tends to infinity.
Therefore, X¯t(τ, x) is a version of Xt(τ, x) for all t and x. We identify Xt(s, x) and X¯t(s, x)
for all (s, t, x) ∈ [0,T ]2 × Rd. Using Lemma 2.3.2(1) and Corollary 2.5.3, we obtain that
P-a.s X·(τ, ·) ∈ D([0,T ];C
β′
loc(R
d; Rd)) and that the estimate (2.3.5) holds. Note here that
for all β ≥ 0, the Fréchet spaces D([0,T ];Cβloc(R
d; Rd)) and Cβloc(R
d; D([0,T ]; Rd)) are
equivalent. It follows from the proof of Theorem 3.5 in [Kun04] that for every stopping
time τ¯ ≤ T , P-a.s.
lim
|x|→∞
inf
t≤T
|Xt(τ¯, x)| = ∞. (2.3.8)
Let (tn) ⊆ [0,T ] and (xn) ⊆ Rd be convergent sequences with limits t and x, respectively.
First, assume tn < t for all n. By (2.3.8), for every stopping time τ¯ ≤ T , P-a.s. the sequence(
X−1tn (τ¯, xn)
)
is uniformly bounded. Since P-a.s. X·(τ, ·) ∈ D([0,T ];Cβ(Rd; Rd)), β′ ∈ (0, 1),
we have
lim
n→∞
(
Xt−(τ¯, X
−1
tn (τ¯, xn)) − Xt−(τ¯, X
−1
t− (τ¯, x)
)
= lim
n→∞
(
Xt−(τ¯, X
−1
tn (τ¯, xn)) − x
)
= lim
n→∞
(
Xtn(τ¯, X
−1
tn (τ¯, xn)) − x
)
= lim
n→∞
(xn − x) = 0,
which implies
lim
n→∞
X−1tn (τ¯, xn) = X
−1
t− (τ¯, x).
A similar argument is used for tn > t. (2) It follows from the definition (2.3.7) that X¯t(s, x)
and X¯−1t (s, x) are continuous in s, t, and x. Moreover, applying Lemma 2.3.2(1) and Corol-
lary 2.5.3, we conclude that P-a.s. X·(·, ·) ∈ C([0,T ]2;C
β′
loc(R
d; Rd)) and that the estimate
(2.3.6) holds. The continuity of Xs(τ, x) with respect to s plays an important role in the
proof of Theorem 2.2.4. □
2.3.2 Moment estimates of inverse flows: Proof of Theorem 2.2.1
In this subsection, under Assumption 2.2.1(β), β ≥ 1, we derive moment estimates for the
flow Xt(τ, x) and its inverse X−1t (τ, x) in weighted Hölder norms and complete the proof of
Theorem 2.2.1. In particular, we will apply Corollaries 2.5.2 and 2.5.3 with the Banach
2.3. Properties of stochastic flows 19
spaces V = D([0,T ]; Rd) and V = C([0,T ]2; Rd).
Proposition 2.3.4. Let Assumption 2.2.1(β) hold for some β > 1
(1) For all stopping time sτ ≤ T and β′ ∈ [1, β), P-a.s. ∇X·(τ, ·) ∈ D([0,T ];C
β′−1
loc (R
d; Rd))
and for all ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
t≤T
|r−ϵ1 ∇Xt(τ)|
p
β′−1
]
≤ N. (2.3.9)
Moreover, for all p ≥ 2, there is a constant N = N(d, p,N0, β,T ) such that for all
multi-indices γ with 1 ≤ |γ| ≤
[
β
]
and all x ∈ Rd,
E
[
sup
t≤T
|∂γXt(τ, x)|p
]
≤ N (2.3.10)
and for all multi-indices γ with |γ| = [β]− and all x, y ∈ Rd,
E
[
sup
t≤T
|∂γXt(τ, x) − ∂γXt(τ, y)|p
]
≤ N |x − y|{β}
+ p. (2.3.11)
(2) If H ≡ 0, then for all β′ ∈ [1, β), P-a.s. ∇X·(·, ·) ∈ C([0,T ]2;C
β′−1
loc (R
d; Rd)) and for all
ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that
E
[
sup
s,t≤T
|r−(1+ϵ)1 ∇Xt(s)|
p
β′−1
]
≤ N.
Moreover, for all p ≥ 2, there is a constant N = N(d, p,N0,T, β) such that for all
multi-indices γ with |γ| = [β]− and all s, s¯ ∈ [0,T ] and x ∈ Rd,
E
[
sup
t≤T
|∂γXt(s, x) − ∂γXt(s, x)|p
]
≤ N |s − s|p/2. (2.3.12)
Proof. (1) Fix a stopping time τ ≤ T and write Xt(τ, x) = Xt(x). First, let us assume
that [β]− = 1. It follows from Theorem 3.4 in [Kun04] that P-a.s. for all t, Xt(τ, ·) is
continuously differentiable and Ut = ∇Xt(τ, x) satisfies
dUt = ∇bt(Xt)Utdt + ∇σ
ϱ
t (Xt−)Utdw
ϱ
t +
∫
Z
∇Ht(Xt−, z)Ut−q(dt, dz), τ < t ≤ T,
∇Xt = Id, t ≤ τ, (2.3.13)
where Id is the d × d-dimensional identity matrix. Taking λ = 0 in the estimates (3.10)
and (3.11) in Theorem 3.3 in [Kun04], we obtain (2.3.10) and (2.3.11). Then applying
Corollary 2.5.3 with V = D([0,T ]; Rd), we have that X·(·) ∈ D([0,T ];C
β′
loc(R
d; Rd)) and
20 Chapter 2. Properties of space inverses of stochastic flows
that (2.3.9) holds. The proof for [β]− > 1 follows by induction (see, e.g. the proof of
Theorem 6.4 in [Kun97]).
(2) The estimate (2.3.12) is given in Theorem 4.6.4 in [Kun97] in equation (19). The
remaining items of part (2) then follow in exactly the same way as part (1) with the only
exception being that we apply Corollary 2.5.3 with V = C([0,T ]2; Rd). □
Lemma 2.3.5. Let Assumption 2.2.1(β) hold for some β > 1.
(1) For any stopping time τ ≤ T and β′ ∈ [1, β), P-a.s. ∇X·(τ, ·)−1 ∈ D([0,T ];C
β′−1
loc
(Rd; Rd)) and for all p ≥ 2, there is a constant N = N(d, p,N0,T, η,Nκ) such that for
all x, y ∈ Rd
E
[
sup
t≤T
|∇Xt(τ, x)−1|p
]
≤ N (2.3.14)
and
E
[
sup
t≤T
|∇Xt(τ, x)−1 − ∇Xt(τ, y)−1|p
]
≤ N|x − y|((β−1)∧1)p. (2.3.15)
(2) If H ≡ 0, then for all β′ ∈ [1, β), P-a.s. ∇X·(·, ·)−1 ∈ C([0,T ]2;C
β′−1
loc (R
d; Rd)) and for
all p ≥ 2, there is a constant N = N(d, p,N0,T ) such that for all s, s¯ ∈ [0,T ] and
x ∈ Rd,
E
[
sup
t≤T
|∇Xt(s, x)−1 − ∇Xt(s, x)−1|p
]
≤ N|s − s¯|p/2.
Proof. (1) Let τ ≤ T be a fixed stopping time and write Xt(τ, x) = Xt(x). Using Itô’s
formula (see also Lemma 3.12 in [Kun04]), we deduce that U¯t = ∇Xt(x)−1 satisfies
dU¯t = U¯t
(
∇σϱt (Xt−)∇σ
ϱ
t (Xt−(τ)) − ∇bt(Xt)
)
dt − U¯t∇σ
ϱ
t (Xt)dw
ϱ
t
−
∫
Z
U¯t−∇Ht(Xt−, z)(Id + ∇Ht(Xt−, z))−1q(dt, dz)
+
∫
Z
U¯t∇Ht(Xt−, z)2(Id + ∇Ht(Xt−, z))−1π(dz)dt, τ < t ≤ T,
U¯t = Id, t ≤ τ. (2.3.16)
Since matrix inversion is a smooth mapping, the coefficients of the linear equation (2.3.16)
satisfy the same assumptions as the coefficients of the linear equation (2.3.13), and hence
the derivation of the estimates (2.3.14) and (2.3.15) proceed in the same way as the anal-
ogous estimates for (2.3.13). To see that P-a.s. X·(·)−1 ∈ D([0,T ];C
β′−1
loc (R
d; Rd)), we
only need to note that P-a.s. X·(·) ∈ D([0,T ];C
β′−1
loc (R
d; Rd)) and that matrix inversion is a
smooth mapping. Part (2) follows with the obvious changes. □
As an immediate corollary, we obtain the diffeomorphism property of the flow Xt(τ, x)
under Assumption 2.2.1(β), β > 1.
2.3. Properties of stochastic flows 21
Corollary 2.3.6. Let Assumption 2.2.1(β) hold.
(1) For all stopping times τ ≤ T and β′ ∈ [1, β) the mapping Xt(τ, ·) : Rd → Rd is a
Cβ
′
loc(R
d; Rd)-diffeomorphism, P-a.s. X·(τ, ·), X−1· (τ, ·) ∈ D([0,T ];C
β′
loc(R
d; Rd)) and for
all t ∈ [0,T ], X−1t− (τ) coincides with the inverse of Xt−(τ).
(2) If H ≡ 0, then for all β′ ∈ [1, β), P-a.s. X·(·, ·), X−1· (·, ·) ∈ C([0,T ]
2,Cβ
′
loc(R
d; Rd)).
Proof. (1) Fix a stopping time τ ≤ T and write Xt(τ, x) = Xt(x). It follows from Propo-
sitions 2.3.3 and 2.3.4 that P-a.s. for all t, the mappings Xt(·), Xt−(·) : Rd → Rd are
homeomorphisms and X·(·) ∈ D([0,T ];C
β′
loc(R
d; Rd)). Moreover, by Lemma 2.3.5, P-a.s.
for all t and x, the matrix ∇Xt (τ, x) has an inverse. Therefore, by Hadamard’s Theorem
(see, e.g., Theorem 0.2 in [DMGZ94]), P-a.s. for all t, Xt(·) is a diffeomorphism. Using
the chain rule, P-a.s. for all t and x,
∇X−1t (x) = ∇Xt(X
−1
t (x))
−1. (2.3.17)
Since, by Lemma 2.3.5, P-a.s. [∇X·(·)]−1 ∈ D([0,T ];C
β′−1
loc (R
d; Rd)) and we know that
P-a.s. for all t, X−1t (·) is differentiable, it follows from (2.3.17) that P-a.s.
∇X·(X−1· (·))
−1 ∈ D([0,T ];C(β
′−1)∧1
loc (R
d; Rd)).
We proceed inductively to complete the proof. Making the obvious changes in the proof
of part (1), we obtain part (2). □
We conclude with a derivation of Hölder moment estimates of the inverse flow X−1t (τ, x),
which will complete the proof of Theorem 2.2.1.
Proof of Theorem 2.2.1. (1) Fix a stopping time τ ≤ T and write Xt(τ, x) = Xt(x). Fix
ϵ > 0. First, let us assume that
[
β
]−
= 1. Set Jt(x) = | det∇Xt(x)|. It is clear from (2.3.10)
that for all p ≥ 2 and x, there is a constant N = N(d, p,N0,T ) such that
E[sup
t≤T
|Jt(x)|p] ≤ N. (2.3.18)
By the mean value theorem, for all x and y and p¯ ∈ R, we have
|r1(x) p¯ − r1(y) p¯| ≤ |p¯|(r1(x) p¯−1 + r1(y) p¯−1)|x − y|. (2.3.19)
Using the change of variable (x¯, y¯) = (X−1t (x), X
−1
t (y)), Fatou’s lemma, Fubini’s theorem,
Hölder’s inequality, and the inequalities (2.3.3), (2.3.18), (2.3.19), (2.3.2), and (2.3.4), for
all δ ∈ (0, 1] and p > d
ϵ
, we conclude that there is a constant N = N(d, p,N0,T, δ, η,Nκ, ϵ)
22 Chapter 2. Properties of space inverses of stochastic flows
such that
E sup
t≤T
∫
Rd
|r1(x)−(1+ϵ)X−1t (x)|
pdx ≤
∫
Rd
|x¯|pE sup
t≤T
[r1(Xt(x¯))
−p(1+ϵ)Jt(x¯)]dx¯
≤ NE
∫
Rd
r1(x¯)
−pϵdx¯ ≤ N
and
E sup
t≤T
∫
|x−y|<1
|r−(1+ϵ)1 (x)X
−1
t (x) − r
−(1+ϵ)
1 (y)X
−1
t (y)|
p
|x − y|2d+δp
dxdy
≤
∫
|x¯−y¯|<1
E sup
t≤T


r−p(1+ϵ)1 (Xt(x¯))|x¯ − y¯|
pJt(x¯)Jt(y¯)
|Xt(x¯) − Xt(y¯)|2d+δp

 dx¯dy¯
+
∫
|x¯−y¯|<1
E sup
t≤T


|y¯|p|r−(1+ϵ)1 (Xt(x¯)) − r
−(1+ϵ)
1 (Xt(y¯))|
pJt(x¯)Jt(y¯)
|Xt(x¯) − Xt(y¯)|2d+δp

 dx¯dy¯
≤ N
∫
|x¯−y¯|<1
r1(x¯)−p(1+ϵ)
|x¯ − y¯|2d−(1−δ)p
dx¯dy¯ + N
∫
|x¯−y¯|<1
r1(x¯)−p(1+ϵ) + r1(y¯)−p(1+ϵ)
|x¯ − y¯|2d−(1−δ)p
dx¯dy¯ ≤ N.
Similarly, making use of the inequalities (2.3.3), (2.3.18), (2.3.19), (2.3.2), (2.3.4), (2.3.14),
and (2.3.15), for all p > d
ϵ
∨ d
β−β′ ∨
d
2−β′ , we get
E sup
t≤T
∫
Rd
|r−ϵ(x)∇X−1t (x)|
pdx ≤
∫
Rd
E sup
t≤T
[r1(Xt(x¯))
−pϵ |[∇Xt(x¯)]−1|pJt(x¯)]dx¯
≤ NE
∫
Rd
r1(x¯)
−pϵdx¯ ≤ N
and
E sup
t≤T
∫
|x−y|<1
|r−ϵ1 (x)∇X
−1
t (x) − r
−ϵ
1 (y)∇X
−1
t (y)|
p
|x − y|2d+(β′−1)p
dxdy
≤
∫
|x¯−y¯|<1
E sup
t≤T
[
|r−ϵ1 (Xt(x¯))[∇Xt(x¯)]
−1 − r−ϵ1 (Xt(y¯))[∇Xt(y¯)]
−1|pJt(x¯)Jt(y¯)
|Xt(x¯) − Xt(y¯)|2d+(β
′−1)p
]
dx¯dy¯
≤ N
∫
|x¯−y¯|<1
r1(x¯)−pϵ
|x¯ − y¯|2d−(β−β′)p
dx¯dy¯ + N
∫
|x¯−y¯|<1
r1(x¯)−pϵ + r1(y¯)−pϵ
|x¯ − y¯|2d−(2−β′)p
dx¯dy¯ ≤ N,
where N = N(d, p,N0,T, β′, η,Nκ, ϵ) is a positive constant. Therefore, combining the
above estimates and applying Corollary 2.5.2, we find that for all p ≥ 2, there is f a
constant N = N(d, p,N0,T, β′, η,Nκ, ϵ), such that
E
[
sup
t≤T
|r−(1+ϵ)1 X
−1
t (τ)|
p
0
]
+ E
[
sup
t≤T
|r−ϵ1 ∇X
−1
t (τ)|
p
β′−1
]
≤ N.
It is well-known that the the inverse map I on the set of invertible d × d-dimensional
2.3. Properties of stochastic flows 23
matrices is infinitely differentiable and for all n, there is a constant N = N(n, d) such
that for all invertible matrices M, the nth derivative of I evaluated at M, denoted I(n)(M),
satisfies
∣∣∣I(n)(M)
∣∣∣ ≤ N |M−n−1| ≤ N
∣∣∣M−1
∣∣∣
n+1
.
We claim that for all n and every multi-index γ with |γ| = n, the components of ∂γX−1t (x)
are a polynomial in terms of the entries of [∇Xt(X−1t (x))]
−1 and ∂γ
′
∇Xt(X−1t (x)) for all
multi-indices γ′ with 1 ≤ |γ′| ≤ n − 1. Assume that statement holds for some n. By the
chain rule, for all ω, t, and x, we have
∇(∇Xt(X−1t (x))
−1) = I(1)(∇Xt(X−1t (x)))∇
2Xt(X
−1
t (x))∇Xt(X
−1
t (x))
−1
and for all multi-indices γ with 1 ≤ |γ′| ≤ n − 1, we have
∇(∂γ
′
∇Xt(X−1t (x))) = ∂
γ′∇2Xt(X−1t (x))∇Xt(X
−1
t (x))
−1,
where ∇2Xt(X−1t (x)) is the tensor of second-order derivatives of Xt(·) evaluated at X
−1
t (x).
This implies that for every multi-index γ with |γ| = n + 1, the components of ∂γX−1t (x)
are a polynomial in terms of the entries of ∇Xt(X−1t (x))
−1 and ∂γ
′
∇Xt(X−1t (x)) for all multi-
indices γ′ with 1 ≤ |γ′| ≤ n. By induction, the claim is true. Therefore, for [β]− ≥ 2, using
(2.3.10) and (2.3.11), we obtain the moment estimates for the inverse flow in the almost
exact same way we did for [β]− = 1. Making the obvious changes in the proof of part (1),
we obtain part (2). This completes the proof of Theorem 2.2.1. □
2.3.3 Strong limit of a sequence of flows: Proof of Theorem 2.2.3
Proof of Theorem 2.2.3. Let τ ≤ T be a fixed stopping time and write Xt(x) = Xt(τ, x). For
each n, let
Z(n)t (x) = X
(n)
t (x) − Xt(x), (t, x) ∈ [0,T ] × R
d.
Throughout the proof we denote by (δn)n≥1 a deterministic sequence with δn → 0 as n→ ∞
that may change from line to line. Let N = N(p,N0,T ) be a positive constant, which may
change from line to line. By virtue of Theorem 2.1 in [Kun04] and (2.3.1), for all p ≥ 2
and t, x and n, we have
E
[
sup
s≤t
|Z(n)s (x)|
p
]
≤ NE
∫
]0,t]
|Z(n)s (x)|
pds + Nδnr1(x)
p.
24 Chapter 2. Properties of space inverses of stochastic flows
Since the right-hand-side is finite by (2.3.1), applying Gronwall’s lemma we find that for
all x and n,
E
[
sup
t≤T
|Z(n)t (x)|
p
]
≤ Nδnr1(x)p. (2.3.20)
Similarly, by (2.3.10), for all x and n, we have
E
[
sup
t≤T
|∇Z(n)t (x)|
p
]
≤ Nδn.
Using (2.3.10), for all x, y, and n, we obtain
E
[
sup
t≤T
|Z(n)t (x) − Z
(n)(y)|p
]
≤ |x − y|pE sup
t≤T
∫ 1
0
|∇Z(n)t (y + θ(x − y))|
pdθ ≤ N|x − y|p.
It follows immediately from (2.3.11) that for all x, y, and n,
E
[
sup
t≤T
]|∇Z(n)t (x) − ∇Z
(n)
t (y)|
p
]
≤ N|x − y|(β−1)∨1.
Thus, by Corollary 2.5.4, we have
lim
n→∞
(
E
[
sup
t≤T
|r−(1+ϵ)1 X
(n)
t − r
−(1+ϵ)
1 Xt|
p
0
]
+ E
[
sup
t≤T
|r−ϵ1 ∇X
(n)
t − r
−ϵ
1 ∇Xt|
p
0
])
= 0. (2.3.21)
Owing to a standard interpolation inequality for Hölder spaces (see, e.g. Lemma 6.32 in
[GT01]), for all δ ∈ (0, 1) and β¯ ∈ (β′, β), there is a constant N(δ) such that
E
[
sup
t≤T
|r−ϵ1 ∇X
(n)
t − r
−ϵ
1 ∇Xt|
p
β′−1
]
≤ δE
[
sup
t≤T
|r−ϵ1 ∇X
(n)
t − ∇Xt|
p
β¯−1
]
+ CδE
[
sup
t≤T
|r−ϵ1 ∇X
(n)
t − ∇Xt|
p
0
]
,
and hence since
sup
n∈N
E
[
sup
t≤T
|rε1∇X
(n)|p
β¯−1
]
+ E
[
sup
t≤T
|rε1∇Xt|
p
β¯−1
]
< ∞,
we have
lim
n→∞
E[sup
t≤T
|r−ϵ1 ∇X
(n)
t − r
−ϵ
1 ∇Xt|
p
β′−1] = 0.
By Theorem 2.2.1, Corollary 2.5.4, and the interpolation inequality for Hölder spaces used
above, in order to show
lim
n→∞
E
[
sup
t≤T
|r−(1+ϵ)1 X
(n);−1
t (τ) − r
−(1+ϵ)
1 X
−1
t (τ)|
p
0
]
= 0
2.3. Properties of stochastic flows 25
lim
n→∞
E
[
sup
t≤T
|r−ϵ1 ∇X
(n);−1
t (τ) − r
−ϵ
1 ∇X
−1
t (τ)|
p
β′−1
]
= 0,
it suffices to show that for all x,
dP − lim
n→∞
sup
t≤T
|X(n);−1t (x) − X
−1
t (x)| = 0 (2.3.22)
and
dP − lim
n→∞
sup
t≤T
|∇X(n);−1t (x) − ∇X
−1
t (x)| = 0. (2.3.23)
For each n, define
Θ
(n)
t (x) = r1(X
(n)
t (x))
−1 − r1(Xt(x))−1, (t, x) ∈ [0,T ] × Rd.
For all ω, t, x, and n, we have
|Θ(n)t (x)| ≤ r1(X
(n)
t (x))
−1r1(Xt(x))
−1|Z(n)t (x)|,
and hence using Hölder’s inequality, (2.3.4), and (2.3.20), we obtain that for all p ≥ 2, x,
there is a constant N = N(p,N0,T, η,Nκ) such that for all n,
E
[
sup
t≤T
|Θ(n)t (x)|
p
]
≤ Nr1(x)−pδn,
where N = N(p,N0,T, η,Nκ) is a constant. Furthermore, since
|∇Θ(n)t (x)| ≤ r1(X
(n)
t (x))
−2|∇X(n)t (x)| + r1(Xt(x))
−2|∇X(n)t (x)|,
for all ω, t, x, and n, applying (2.3.4) and (2.3.10), for all p ≥ 2, x, and n, we get
E
[
sup
t≤T
|r1(x)Θ
(n)
t (x) − r1(y)Θ
(n)
t (y)|
p
]
≤ N|x − y|p.
Then owing to Corollary 2.5.4, for all p ≥ 2,
lim
n→∞
E
[
sup
t≤T
|Θ(n)t |
p
0
]
= 0. (2.3.24)
We claim that for all R > 0,
dP − lim
n→∞
E(n,R) := dP − lim
n→∞
sup
t≤T
|X(n);−1t − X
−1
t |0;{|x|≤R} = 0. (2.3.25)
Fix R > 0. It is enough to show that every subsequence of E(n) = E(n,R) has a sub-
subsequence converging to 0, P-a.s.. Owing to (2.3.21) and (2.3.24), for a given subse-
26 Chapter 2. Properties of space inverses of stochastic flows
quence (E(nk)), we can always find sub-subsequence (still denoted (E(nk)) to avoid double
indices) such that P-a.s.,
lim
k→∞
sup
t≤T
|X(nk)t − Xt|β′;{|x|≤R¯} = 0, ∀ R¯ > 0, (2.3.26)
and
lim
k→∞
sup
t≤T
|r1(X
(nk)
t (x))
−1 − r1(Xt(x))−1|0 = 0.
Fix an ω for which both limits are zero. We will prove that
lim
k→∞
sup
t≤T
|X(nk);−1t (ω) − X
−1
t (ω)|0;{|x|≤R} = 0. (2.3.27)
Suppose, by contradiction, that (2.3.27) is not true. Then there exists an ε > 0 and a
subsequence of (nk) (still denoted (nk)) such that tnk → t− (or tnk → t+) and xnk → x as
k → ∞ with
∣∣∣xnk
∣∣∣ ≤ R such that (dropping ω),
|X(nk);−1tnk (xnk) − X
−1
tnk
(xnk)| ≥ ε. (2.3.28)
Arguing by contradiction and using (2.3.3), we have
sup
k∈N
|X(nk);−1tnk (xnk)| < ∞. (2.3.29)
Applying (2.3.29), (2.3.26), and the fact that X·(·), X−1· (·) ∈ D([0,T ];C
β′
loc(R
d; Rd)) , we
obtain
lim
k→∞
(
Xt−(X
(nk);−1
tnk
(xnk)) − Xt−(X
−1
tnk
(xnk))
)
= lim
k→∞
(
Xt−(X
(nk);−1
tnk
(xnk)) − xnk
)
= lim
k→∞
(
Xt−(X
(nk);−1
tnk
(xnk)) − X
(nk)
tnk
(X(nk);−1tnk (xnk))
)
= lim
k→∞
(
Xt−(X
(nk);−1
tnk
(xnk)) − Xtnk (X
(nk);−1
tnk
(xnk))
)
+ lim
k→∞
(
Xtnk (X
(nk);−1
tnk
(xnk)) − X
(nk)
tnk
(X(nk);−1tnk (xnk))
)
= 0,
which contradicts (2.3.28), and hence proves (2.3.27), (2.3.25), and (2.3.22). For each n,
define
U¯ (n)t = U¯
(n)(t, x) = ∇X(n)t (x)
−1 and U¯(t) = U¯(t, x) = ∇Xt(x)−1, (t, x) ∈ [0,T ] × Rd.
2.4. Classical solutions of degenerate SPDEs: Proof of Theorem 2.2.4 27
Using (2.3.14) and (2.3.15) and repeating the arguments given above, for all p ≥ 2, we get
lim
n→∞
E
[
sup
t≤T
|r−ϵ1 U¯
(n)
t − r
−ϵ
1 U¯t|
p
β′−1
]
= 0.
Then (2.3.3) and (2.3.25) imply that for all R > 0,
dP − lim
n→∞
sup
t≤T
|∇X(n);−1t (x) − ∇X
−1
t (x)|0;{|x|≤R}
= dP − lim
n→∞
sup
t≤T
|∇X(n)t (X
(n);−1
t (x))
−1 − ∇Xt(X−1t (x))
−1|0;{|x|≤R} = 0,
which yields (2.3.23) and completes the proof. □
2.4 Classical solutions of degenerate SPDEs: Proof of The-
orem 2.2.4
Proof of Theorem 2.2.4. Fix a stopping time τ ≤ T . By virtue of Theorem 2.2.1, we only
need to show that Y−1(τ) = Y−1t (τ, x) solves (2.1.2) and is the unique solution. Suppose we
have shown that Y−1(s, x), s ∈ [0,T ], solves (2.1.2) (i.e. where τ = s is deterministic). It is
then a routine argument to conclude that Y−1(τ′) solves (2.1.2) for finite-valued stopping
times τ′ ≤ T . We can then use an approximation argument as in the proof of Proposition
2.3.3 to show that Y−1(τ) = Y−1t (τ, x) solves (2.1.2). Thus, it suffices to take τ deterministic.
Let ut(x) = ut(s, x) = Y−1t (s, x), (s, t, x) ∈ [0,T ]
2 × Rd. Fix (s, t, x) ∈ [0,T ]2 × Rd with
s < t and write Yt(x) = Yt(s, x). Let ((tMn )0≤n≤M)1≤M≤∞ be a sequence of partitions of the
interval [s, t] such that for all M > 0, (tMn )0≤n≤M has mesh size (t − s)/M. Fix M and set
(tn)0≤n≤M = (tMn )0≤n≤M. Immediately, we obtain
ut(x) − x =
M−1∑
n=0
(utn+1(x) − utn(x)). (2.4.1)
We will use Taylor’s theorem to expand each term in the sum on the right-hand-side of
(2.4.1). By Taylor’s theorem, for all n and y, we have
utn+1(Ytn+1(y)) − utn(Ytn+1(y)) = y − utn(Ytn+1(y)) = utn(Ytn(y)) − utn(Ytn+1(y))
= ∇utn(Ytn(y))(Ytn(y) − Ytn+1(y)) − (Ytn(y) − Ytn+1(y))
∗Θn(Ytn(y))(Ytn(y) − Ytn+1(y)),
(2.4.2)
where
Θi jn (z) =
∫ 1
0
(1 − θ)∂i jutn
(
z + θ(Ytn+1(Y
−1
tn (z)) − z)
)
dθ.
28 Chapter 2. Properties of space inverses of stochastic flows
Since for all n, Ytn+1(s, x) = Ytn+1(tn,Ytn(s, x)), we have
Ytn+1(Y
−1
tn (x)) = Ytn+1(tn, x)
and hence substituting y = Y−1tn (x) into (2.4.2), for all n, we get
utn+1(x) − utn(x) = An + Bn, (2.4.3)
where
An := ∇utn(x)(x − Ytn+1(tn, x)) − (x − Ytn+1(tn, x))
∗Θi jn (x)(x − Ytn+1(tn, x))
and
Bn := (utn+1(x) − utn(x)) − (utn+1(Ytn+1(tn, x)) − utn(Ytn+1(tn, x))).
Applying Taylor’s theorem once more, for all n, we obtain
Bn = Cn + Dn, (2.4.4)
where
Cn := (∇utn+1(x) − ∇utn(x))(x − Ytn+1(tn, x)),
Dn := −(x − Ytn+1(tn, x))
∗Θ˜n(x)(x − Yti+1(ti, x))),
and
Θ˜n(x)
i j :=
∫ 1
0
(1 − θ)∂i j(utn+1 − utn)(x + θ(Ytn+1(tn, x) − x))dθ.
Thus, combining (2.4.1), (2.4.3), and (2.4.4), P-a.s. we have
ut(x) − x =
M−1∑
n=0
(An + Cn + Dn). (2.4.5)
Now, we will derive the limit of the right-hand-side of (2.4.5).
Claim 2.4.1. (1)
dP − lim
M→∞
M−1∑
n=0
An = −
∫
]s,t]
[
1
2
σiϱr (x)σ
jϱ
r (x)∂i jur(x) + b
i
r(x)∂iur(x)]dr
−
∫
]s,t]
σiϱr (x)∂iur(x)dw
ϱ
r ;
(2) dP − limM→∞
∑M−1
n=0 Dn = 0;
(3) dP − limM→∞
∑M−1
n=0 Cn =
∫
]s,t]
σ
jϱ
r (x)∂ jσ
iϱ
r (x)∂iur(x)dr +
∫
]s,t]
σ
iϱ
r (x)σ
jϱ
r (x)∂i jur(x)dr.
2.4. Classical solutions of degenerate SPDEs: Proof of Theorem 2.2.4 29
Proof of Claim 2.4.1. (1) For all n, we have
∇utn(x)
(
x − Ytn+1(tn, x)
)
= −
∫
]tn,tn+1]
bir(x)∂iutn(x)dr −
∫
]tn,tn+1]
σiϱr (x)∂iutn(x)dw
ϱ
r
+ R(1)n + R
(2)
n ,
where
R(1)n :=
∫
]tn,tn+1]
(
bir(x) − b
i
r(Yr(tn, x))
)
∂iutn(x)dr
and
R(2)n :=
∫
]tn,tn+1]
[σiϱr (x) − σ
iϱ
r (Yr(tn, x))]∂iutn(x)dw
ϱ
r .
Since b and σ are Lipschitz, there is a constant N = N(N0,T ) such that
M−1∑
n=0
∣∣∣R1n
∣∣∣ ≤ N sup
s≤r≤t
|∇ur(x)| sup
|r1−r2 |≤ tM
|x − Yr1(r2, x)|
and
∫
]s,t]
∣∣∣∣∣∣∣
M−1∑
n=0
1]tn,tn+1](r)
(
σi·r (x) − σ
i·
r (Yr(tn, x))
)
∂iutn(x)
∣∣∣∣∣∣∣
2
ds
≤ N sup
s≤r≤t
|∇ur(x)|2 sup
|r1−r2 |≤ tM
|x − Yr1(r2, x)|
2.
Owing to the joint continuity of Yt(s, x) in s and t and the dominated convergence theorem
for stochastic integrals, we obtain
dP − lim
M→∞
M−1∑
n=0
(R(1)n + R
(2)
n ) = 0. (2.4.6)
In a similar way, this time using the continuity of ∇ut(x) in t and the linear growth of b
and σ, we find
dP − lim
M→∞
M−1∑
n=0
(
−
∫
]tn,tn+1]
bir(x)∂iutn(x)dr −
∫
]tn,tn+1]
σiϱr (x)∂iutn(x)dw
ϱ
r
)
= −
∫
]s,t]
br(x)∂iur(x)dr −
∫
]s,t]
σϱr (x)∂iur(x)dw
ϱ
r .
For all n, we have
−(x − Ytn+1(tn, x))
∗Θn(x)(x − Ytn+1(tn, x)) = S
(1)
n + S
(2)
n ,
30 Chapter 2. Properties of space inverses of stochastic flows
where S (1)n (t, x) has only drdr and drdw
ϱ
r terms and where
S (2)n : = −
1
2
(∫
]tn,tn+1]
σiϱr (Yr(tn, x))dw
ϱ
r
)
∂i jutn(x)
(∫
]tn,tn+1]
σ jϱr (Yr(tn, x))dw
ϱ
r
)
−
(∫
]tn,tn+1]
σiϱr (Yr(tn, x))dw
ϱ
r
) (
Θi jn (x) −
1
2
∂i jutn(x)
) (∫
]tn,tn+1]
σ jϱr (Yr(tn, x))dw
ϱ
r
)
.
Since
∣∣∣∣∣Θ
i j
n (x) −
1
2
∂i jutn(x)
∣∣∣∣∣ =
∣∣∣∣∣∣
∫ 1
0
(1 − θ)(∂i jutn(x + θ(Ytn+1(tn, x) − x)) − ∂i jutn(x))dθ
∣∣∣∣∣∣
≤ N sup
|r1−r2 |≤ tM ,θ∈(0,1)
|∂i jur1(x + θ(Yr2(r1, x) − x)) − ∂i jur1(x))|,
proceeding as in the derivation of (2.4.6) and using the joint continuity of ∂i jut(x) in t and
x, the continuity of Yt(s, x) in s and t, the explicit form of the quadratic variation of the
stochastic integral (i.e. part (5) of Chapter 2, Section 2, Theorem 2 in [LS89]), and the
stochastic dominated convergence theorem, we obtain
dP − lim
M→∞
M−1∑
n=0
S (2)n = −
1
2
∫
]0,t]
σiϱr (x)σ
jϱ
r (x)∂i jur(x)dr.
Similarly, making use of the properties stated in Theorem 2.2.1(2), we have
dP − lim
M→∞
M−1∑
n=0
S (1)n = 0,
which completes the proof of part (1). The proof of part (2) is similar to the proof of part
(1), so we proceed to the proof of part (3). We know that for all n, Ytn+1(x) = Ytn+1(tn,Ytn(x)).
Thus, for all n, we have utn+1(x) = utn(Y
−1
tn+1(tn, x)), and hence by the chain rule,
∇utn+1(x) = ∇utn(Y
−1
tn+1(tn, x))∇Y
−1
tn+1(tn, x). (2.4.7)
By (2.4.7) and Taylor’s theorem, for all n, we have
Cn = (∇utn+1(x) − ∇utn(x))(x − Ytn+1(tn, x))
= ∇utn(Y
−1
tn+1(tn, x))(∇Y
−1
tn+1(tn, x) − Id)(x − Ytn+1(tn, x))
+(Y−1tn+1(tn, x) − x)
∗Θ˜n(x)(x − Ytn+1(tn, x)) =: En + Fn,
where
Θ˜i jn (x) :=
∫ 1
0
∂i jutn(x + θ(Y
−1
tn+1(tn, x) − x))dθ.
2.4. Classical solutions of degenerate SPDEs: Proof of Theorem 2.2.4 31
Using Itô’s formula, for all n, we get (see, also, Lemma 3.12 in [Kun04]),
∇Ytn+1(tn, x)
−1 = Id −
∫
]tn,tn+1]
∇Yr(tn, x)−1∇σϱr (Yr(tn, x))dw
ϱ
r
+
∫
]tn,tn+1]
∇Yr(tn, x)−1
(
∇σϱr (Yr(tn, y))∇σ
ϱ
r (Yr(tn, x)) − ∇br(Yr(tn, x))
)
dr,
and hence
∇Y−1tn+1(tn) − Id = ∇Y
−1
tn+1(tn,Y
−1
tn+1(tn, x)) − Id =: G
(1)
tn,tn+1(Y
−1
tn+1(tn, x)) + G
(2)
tn,tn+1(Y
−1
tn+1(tn, x)),
where for y ∈ Rd,
G(1)tn,tn+1(y) :=
∫
]tn,tn+1]
∇Yr(tn, z)−1
(
∇σϱr (Yr(tn, y))∇σ
ϱ
r (Yr(tn, y)) − ∇br(Yz(tn, y))
)
dr
and
G(2)tn,tn+1(z) := −
∫
]tn,tn+1]
∇Yr(tn, y)−1∇σϱr (Yr(tn, y))dw
ϱ
r .
By the Burkholder-Davis-Gundy inequality, Hölder’s inequality, and the inequalities (2.3.2),
(2.3.14), and (2.3.15), for all p ≥ 2, there is a constant N = N(p, d,N0,T ) such that for all
x1 and x2,
E
[
|G(2)tn,tn+1(x1)|
p
]
≤ NM−p/2+1
∫
]tn,tn+1]
E
[
|∇Yr(tn, x1)−1|p|∇σr(Yr(tn, x1)|p
]
dr ≤ NM−p/2
and
E
[
|G(2)tn,tn+1(x1) −G
(2)
tn,tn+1(x2)|
p
]
≤ NM−p/2+1
∫
]tn,tn+1]
E
[
|∇Yr(tn, x1)−1 − ∇Yr(tn, x2)−1|p
]
dr
+NM−p/2+1
∫
]tn,tn+1]
(
E
[
|∇Yr(tn, x1)−1|2p
])1/2 (
E
[
|Yr(tn, x1) − Yr(tn, x2)|2p
])1/2
dr
≤ NM−p/2|x − y|p.
Thus, by Corollary 2.5.3, we obtain that for all p ≥ 2, ϵ > 0, and δ < 1, there is a constant
N = N(p, d, δ,N0,T ) such that
E
[
|r−ϵG(2)tn,tn+1 |
p
δ
]
≤ NM−p/2. (2.4.8)
For all n, we have
En = ∇utn(Y
−1
tn+1(tn, x))G
(1)
tn,tn+1(Y
−1
tn+1(tn, x))(x − Ytn+1(tn, x))
+ ∇utn(x)G
(2)
tn,tn+1(x)(x − Ytn+1(tn, x))
32 Chapter 2. Properties of space inverses of stochastic flows
+ ∇utn(Y
−1
tn+1(tn, x))(G
(2)
tn,tn+1(Y
−1
tn+1(tn, x)) −G
(2)
tn,tn+1(x))(x − Ytn+1(tn, x))
+ (∇utn(Y
−1
tn+1(tn, x) − ∇utn(x))G
(2)
tn,tn+1(x)(x − Ytn+1(tn, x))
One can easily check that
dP − lim
M→∞
M−1∑
n=0
∇utn(Y
−1
tn+1(tn, x))G
(1)
tn,tn+1(Y
−1
tn+1(tn, x))(x − Ytn+1(tn, x)) = 0. (2.4.9)
Since ∇ut(x) is jointly continuous in t and x and Y−1t (s, x) is jointly in s and t, we have
dP − lim
M→∞
sup
n
|∇utMn (Y
−1
tMn+1
(tMn , x)) − ∇utMn (x)| = 0.
Moreover, using Hölder’s inequality, (2.4.8), and (2.3.1), we get
sup
M
E
M−1∑
n=0
|G(2)tn,tn+1(x)|x − Ytn+1(tn, x)| < ∞,
and hence
dP − lim
M→∞
M−1∑
n=0
(∇utn(Y
−1
tn+1(tn, x)) − ∇utn(x))G
(2)
tn,tn+1(x)(x − Ytn+1(tn, x)) = 0. (2.4.10)
We claim that
dP − lim
M→∞
M−1∑
n=0
∇utn(Y
−1
tn+1(tn, x))
(
G(2)tn,tn+1(Y
−1
tn+1(tn, x)) −G
(2)
tn,tn+1(x)
)
(x − Ytn+1(tn, x)) = 0.
(2.4.11)
Set
JM =
M−1∑
n=0
|G(2)tn,tn+1(Y
−1
tn+1(tn, x)) −G
(2)
tn,tn+1(x)|x − Ytn+1(tn, x)|.
For all δ¯, ϵ ∈ (0, 1), we have
P(JM > δ¯) ≤ P
(
JM > δ¯, max
n
|Y−1tn+1(tn, x) − x| ≤ ϵ
)
+ P
(
max
n
|Y−1tn+1(tn, x) − x| > ϵ
)
.
By virtue of (2.4.8), there is a deterministic constant N = N(x) independent of M such
that for all ω ∈ V M := {maxn |Y−1tn+1(tn, x) − x| ≤ ϵ},
JM ≤ Nϵδ
M−1∑
n=0
[r−ϵ1 G
(2)
tn,tn+1]δ|x − Ytn+1(tn, x)|,
2.4. Classical solutions of degenerate SPDEs: Proof of Theorem 2.2.4 33
which implies that
E1V M J
M ≤ NϵδE
M−1∑
n=0
(
[r−ϵ1 G
(2)
tn,tn+1]
2
δ + |x − Ytn+1(tn, x)|
2
)
≤ Nϵδ
M−1∑
n=0
M−1 ≤ Nϵδ.
Applying Markov’s inequality, we derive
P(JM | > δ¯, max
n
|Y−1tn+1(tn, x) − x| ≤ ϵ) ≤ N
ϵδ
δ¯
,
and hence for all δ¯ > 0,
lim
M→∞
P(JM > δ¯) = 0,
which yields (2.4.11). Owing to (2.4.9), (2.4.10), and (2.4.11) we have
dP − lim
M→∞
M−1∑
n=0
En = lim
M→∞
M−1∑
n=0
∇utn(x)G
(2)
tn,tn+1(x)(x − Ytn+1(tn, x)).
Proceeding as in the proof of part (1) of the claim, we obtain
lim
M→∞
M−1∑
n=0
Kn
= lim
M→∞
M−1∑
n=0
∇utn(x)
∫
]tn,tn+1]
(∇Yr(tn, x)−1 − Id)∇σϱr (x)dW
ϱ
r
∫
]tn,tn+1]
σϱr (x)dW
ϱ
r
+ lim
M→∞
M−1∑
n=0
∇utn(x)
∫
]tn,tn+1]
∇σϱr (x)dw
ϱ
r
∫
]tn,tn+1]
σϱr (x)dw
ϱ
r
=
∫
]s,t]
σ jϱr (x)∂ jσ
iϱ
r (x)∂iur(x)dr. (2.4.12)
It is easy to check that for all n,
Fn = (Y
−1
tn+1(tn, x) − x)
∗Θ˜n(x)(x − Ytn+1(tn, x))
=: (G(3)tn,tn+1(Y
−1
tn+1(tn, x)) + G
(4)
tn,tn+1(Y
−1
tn+1(tn, x)))
∗Θ˜n(x)(x − Ytn+1(tn, x)),
where for y ∈ Rd,
G(3)tn,tn+1(y) := −
∫
]tn,tn+1]
br(Yr(tn, y))dr, G
(4)
tn,tn+1(y) := −
∫
]tn,tn+1]
σϱr (Yr(tn, y))dw
ϱ
r .
34 Chapter 2. Properties of space inverses of stochastic flows
Arguing as in the proof of (2.4.12), we get
dP − lim
M→∞
M−1∑
n=0
Fn =
∫
]s,t]
σiϱr (x)σ
jϱ
r (x)∂i jur(x)dt,
which completes the proof of the claim. □
By virtue of (2.4.5) and Claim 2.4.1, for all s and t with s ≤ t and x, P-a.s.
ut(x) = x +
∫
]s,t]
(
1
2
σiϱr (x)σ
jϱ(x)∂i jur(x) − bˆit(x)∂iur(x)
)
dr −
∫
]s,t]
σiϱr (x)∂iur(x)dw
ϱ
r .
(2.4.13)
Owing to Theorem 2.2.1, u = ut(x) has a modification that is jointly continuous in s and t
and twice continuously differentiable in x. It is easy to check that the Lebesgue integral on
the right-hand-side of (2.4.13) has a modification that is continuous in s, t, and x. Thus, the
stochastic integral on the right-hand-side of (2.4.13) has a modification that is continuous
in s, t, and x, and hence the equality in (2.4.13) holds P-a.s. for all s and t with s ≤ t and x.
This proves that Y−1(τ) = Y−1t (τ, x) solves (2.1.2). However, if u
1(τ), u2(τ) ∈ Cβ
′
cts(R
d; Rd)
are solutions of (2.1.2), then applying the Itô-Wentzell formula (see, e.g. Theorem 9 in
Chapter 1, Section 4.8 in [Roz90]), we get that P-a.s. for all t and x,
u1t (τ,Yt(τ, x)) = x = u
2
t (τ,Yt(τ, x)),
which implies that P-a.s. for all t and x, u1(τ) = Y−1t (τ, x) = u
2(τ). Thus, Y−1(τ) = Y−1t (τ, x)
is the unique solution of (2.1.2) in Cβ
′
cts(R
d; Rd). □
2.5 Appendix
Let V be an arbitrary Banach space. The following lemma and its corollaries are indis-
pensable in this chapter.
Lemma 2.5.1. Let Q ⊆ Rd be an open bounded cube, p ≥ 1, δ ∈ (0, 1], and f be a
V-valued integrable function on Q such that
[
f
]
δ;p;Q;V :=
(∫
Q
∫
Q
| f (x) − f (y)|pV
|x − y|2d+δp
dxdy
)1/p
< ∞.
Then f has a Cδ(Q; V)-modification and there is a constant N = N(d, δ, p) independent of
f and Q such that
[ f ]δ;Q;V ≤ N
[
f
]
δ,p;Q;V
2.5. Appendix 35
and
sup
x∈Q
| f (x)|V ≤ N|Q|δ/d[ f ]δ;p;Q;V + |Q|−1/p
(∫
Q
| f (x)|pVdx
)1/p
,
where |Q| is the volume of the cube.
Proof. If V = R, then the existence of a continuous modification of f and the estimate of
[
f
]
δ;Q follows from Lemma 2 and Exercise 5 in Chapter 10, Section 1, in [Kry08]. The
proof for a general Banach space is the same. For all x ∈ Q, we have
| f (x)|V ≤
1
|Q|
∫
Q
| f (x) − f (y)|Vdy +
1
|Q|
∫
Q
| f (y)|Vdy
≤ N
1
|Q|
[
f
]
δ,p;Q
∫
Q
|x − y|δdy +
1
|Q|
∫
Q
| f (y)|Vdy
≤ N |Q|δ/d[ f ]δ,p;Q + |Q|−1/p
(∫
Q
| f (y)|pVdy
)1/p
,
which proves the second estimate. □
The following is a direct consequence of Lemma 2.5.1.
Corollary 2.5.2. Let p ≥ 1, δ ∈ (0, 1], and f be a V-valued function on Rd such that
| f |δ;p;V :=
(∫
Rd
| f (x)|pVdx +
∫
|x−y|<1
| f (x) − f (y)|pV
|x − y|2d+δp
dxdy
)1/p
< ∞.
Then f has a Cδ(Rd; V)-modification and there is a constant N = N(d, δ, p) independent
of f such that
| f |δ;V ≤ N | f |δ;p;V .
Corollary 2.5.3. Let X be a V-valued random field defined on Rd. Assume that for some
p ≥ 1, l ≥ 0, and β ∈ (0, 1] with βp > d there is a constant N¯ > 0 such that for all
x, y ∈ Rd,
E
[
|X(x)|pV
]
≤ N¯r1(x)lp (2.5.1)
and
E
[
|X(x) − X(y)|pV
]
≤ N¯[r1(x)lp + r1(y)lp]|x − y|βp. (2.5.2)
Then for all δ ∈ (0, β − dp ) and ϵ >
d
p , there exists a C
δ(Rd; V)-modification of r−(l+ϵ)1 X and
a constant N = N(d, p, δ, ϵ) such that
E
[
|r−(l+ϵ)1 X|
p
δ
]
≤ NN¯.
Proof. Fix δ ∈ (0, β − dp ) and ϵ >
d
p . Owing to (2.5.1), there is a constant N = N(d, p, N¯
36 Chapter 2. Properties of space inverses of stochastic flows
, δ, ϵ) such that
∫
Rd
E
[
|r1(x)−(l+ϵ)X(x)|
p
V
]
dx ≤ N¯
∫
Rd
r1(x)
−pϵdx ≤ NN¯.
Appealing to (2.5.2) and (2.3.19), we find that there is a constant N = N(d, p, δ, ϵ) such
that
∫
|x−y|<1
E
[
|r1(x)−(l+ϵ)X(x) − r1(y)−(l+ϵ)X(y)|
p
V
]
|x − y|2d+δp
dxdy
≤ N¯
∫
|x−y|<1
r1(x)−pϵ + r1(y)−pϵ
|x − y|2d−(β−δ)p
dxdy + N¯
∫
|x−y|<1
r1(y)pl|r1(x)−(l+ϵ) − r1(y)−(l+ϵ)|p
|x − y|2d+δp
dxdy
≤ NN¯ + NN¯
∫
|x−y|<1
r1(x)−p(1+ϵ) + r1(y)−p(1+ϵ)
|x − y|2d−(1−δ)p
dxdy ≤ NN¯.
Therefore, E[r−(l+ϵ)1 X]
p
δ,p ≤ NN¯, and hence, by Corollary 2.5.3, r
−(l+ϵ)
1 X has a C
δ(Rd; V)-
modification and the estimate follows immediately. □
Corollary 2.5.4. Let (X(n))n∈N be a sequence of V-valued random fields defined on Rd.
Assume that for some p ≥ 1, l ≥ 0 and β ∈ (0, 1], with βp > d there is a constant N¯ > 0
such that for all x, y ∈ Rd and n ∈ N,
E
[
|X(n)(x)|pV
]
≤ N¯r1(x)lp
and
E
[
|X(n)(x) − X(n)(y)|pV
]
≤ N¯(r1(x)lp + r1(y)lp)|x − y|βp.
Moreover, assume that for all x ∈ Rd, limn→∞ E
[
|X(n)(x)|p
]
= 0. Then for all δ ∈ (0, β − dp )
and ϵ > dp ,
lim
n→∞
E
[
|r−(l+ϵ)1 X
(n)|p
δ
]
= 0.
Proof. Fix δ ∈ (0, β− dp ) and ϵ >
d
p . Using the Lebesgue dominated convergence theorem,
we attain
lim
n→∞
∫
Rd
E
[
|r1(x)−(l+ϵ)X(n)(x)|
p
V
]
dx = 0,
and therefore for all ζ ∈ (0, 1),
lim
n
∫
ζ<|x−y|<1
E
[
|r1(x)−(l+ϵ)Xn(x) − r1(y)−(l+ϵ)Xn(y)|
p
V
]
|x − y|2d+δp
dxdy = 0.
2.5. Appendix 37
By repeating the proof of Corollary 2.5.3, we obtain that there is a constant N such that
∫
|x−y|≤ζ
E
[
|r1(x)−(l+ϵ)X(n)(x) − r1(y)−(l+ϵ)X(n)(y)|
p
V
]
|x − y|2d+δp
dxdy
≤ N¯
∫
|x−y|≤ζ
r1(x)−pϵ + r1(y)−pϵ
|x − y|2d+(δ−β)p
dxdy + N¯
∫
|x−y|≤ζ
r1(x)−p(1+ϵ) + r1(y)−p(1+ϵ)
|x − y|2d+(δ−1)p
dxdy
≤ N¯ζβp−δp−d.
Therefore, limn→∞ E
[
[r−(l+ϵ)1 X]
p
δ,p
]
= 0, and the statement is confirmed. □

Chapter 3
The method of stochastic characteristics
for parabolic SIDEs
3.1 Introduction
Let (Ω,F ,P) be a complete probability space and F˜0 be a sub-sigma-algebra of F . We
assume that this probability space supports a sequence w1;ϱt , t ≥ 0, ϱ ∈ N, of independent
one-dimensional Wiener processes and a Poisson random measure p1(dt, dz) on (R+ × Z1,
B(R+)⊗Z1) with intensity measure π1(dz)dt, where (Z1,Z1, π1) is a sigma-finite measure
space. We also assume that (w1;ϱt )ϱ∈N and p
1(dt, dz) are independent of F0. Let F = (Ft)t≥0
be the standard augmentation of the filtration (F¯t)t≥0, where for each t ≥ 0,
F¯t = σ
(
F˜0, (w1;ϱs )ϱ∈N, p
1([0, s],Γ) : s ≤ t, Γ ∈ Z1
)
.
For each real number T > 0, we let RT , OT , and PT be the F-progressive, F-optional, and
F-predictable sigma-algebra on Ω×[0,T ], respectively. Denote by q1(dt, dz) = p1(dt, dz)−
π1(dz)dt the compensated Poisson random measure. Let D1, E1,V1 ∈ Z be disjoint Z1-
measurable subsets such that D1 ∪ E1 ∪ V1 = Z1 and π(V1) < ∞. Let (Z2,Z2, π2) be a
sigma-finite measure space and D2, E2 ∈ Z2 be disjoint Z2-measurable subsets such that
D2 ∪ E2 = Z2.
Fix an arbitrary positive real number T > 0 and integers d1, d2 ≥ 1. Let α ∈ (0, 2] and
τ ≤ T be a stopping time. Let Fτ be the stopping time sigma-algebra associated with τ
and let φ : Ω × Rd1 → Rd2 be Fτ ⊗ B(Rd1)-measurable. We consider the system of SIDEs
on [0,T ] × Rd1 given by
dult =
(
(L1;lt +L
2;l
t )ut + 1[1,2](α)b
i
t∂iu
l
t + c
ll¯
t u
l¯
t + f
l
t
)
dt +
(
N1;lϱt ut + g
lϱ
t
)
dw1;ϱt
+
∫
Z1
(
I1;lt,z ut− + h
l
t(z)
)
[1D1(z)q
1(dt, dz) + 1E1∪V1(z)p
1(dt, dz)], τ ≤ t ≤ T,
ult = φ
l, t ≤ τ, l ∈ {1, . . . , d2}, (3.1.1)
where for ϕ ∈ C∞c (R
d1; Rd2), k ∈ {1, 2}, and l ∈ {1, . . . , d2},
Lk;lt ϕ(x) : = 1{2}(α)
1
2
σ
k;iϱ
t (x)σ
k; jϱ
t (x)∂i jϕ
l(x) + 1{2}(α)σ
k;iϱ
t (x)υ
k;ll¯ϱ
t (x)∂iϕ
l¯(x)
39
40 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
+
∫
Dk
ρ
k;ll¯
t (x, z)
(
ϕl¯(x + Hkt (x, z)) − ϕ
l¯(x)
)
πk(dz)
+
∫
Dk
(
ϕl(x + Hkt (x, z)) − ϕ
l(x) − 1(1,2](α)H
k;i
t (x, z)∂iϕ
l(x)
)
πk(dz)
+ 1{2}(k)
∫
E2
(
(Ill¯d2 + ρ
2;ll¯
t (x, z))ϕ
l¯(x + H2t (x, z)) − ϕ
l(x)
)
π2(dz),
N1;lϱt ϕ(x) : = 1{2}(α)σ
1;iϱ
t (x)∂iϕ
l(x) + υ1;ll¯ϱt (x)ϕ
l¯(x), ϱ ∈ N,
I1;lt,zϕ(x) : = (I
ll¯
d2 + ρ
1;ll¯
t (x, z))ϕ
l¯(x + H1t (x, z)) − ϕ
l(x),
and
∫
Dk
(
|Hkt (x, z)|
α + |ρkt (x, z)|
2
)
πk(dz) +
∫
Ek
(
|Hkt (x, z)|
1∧α + |ρkt (x, z)|
)
πk(dz) < ∞.
The summation convention with respect to repeated indices i, j ∈ {1, . . . , d1},l¯ ∈ {1, . . . ,
d2}, and ϱ ∈ N is used here and below. The d2 × d2 dimensional identity matrix is denoted
by Id2 . For a subset A of a larger set X, 1A denotes the {0, 1}-valued function taking the
value 1 on the set A and 0 on the complement of A. We assume that for each k ∈ {1, 2},
σkt (x) = (σ
k;iϱ
t (ω, x))1≤i≤d1, ϱ∈N, bt(x) = (b
i
t(ω, x))1≤i≤d1 , ct(x) = (c
ll¯
t (ω, x))1≤l,l¯≤d2 ,
υkt (x) = (υ
k;ll¯ϱ
t (ω, x))1≤l,l¯≤d2, ϱ∈N, ft(x) = ( f
i
t (ω, x))1≤i≤d2 , gt(x) = (g
iϱ
t (ω, x))1≤i≤d2, ϱ∈N,
are random fields on Ω× [0,T ]×Rd1 that are RT ⊗B(Rd1)-measurable. For each k ∈ {1, 2},
we assume that
Hkt (x, z) = (H
k;i
t (ω, x, z))1≤i≤d1 , ρ
k
t (x, z) = (ρ
k;ll¯
t (ω, x, z))1≤l,l¯≤d2 ,
are random fields on Ω × [0,T ] × Rd1 × Zk that are PT ⊗ B(Rd1) ⊗Zk-measurable. More-
over, we assume that ht(x, z) = (hit(ω, x, z))1≤i≤d2 is a random field on Ω × [0,T ] × R
d1 that
is PT ⊗ B(Rd1)-measurable. Systems of linear SIDEs appear in many contexts. They
may be considered as extensions of both first-order symmetric hyperbolic systems and lin-
ear fractional advection-diffusion equations. The equation (3.1.1) also arises in non-linear
filtering of semimartingales as the equation for the unormalized filter of the signal (see,
e.g., [Gri76] and [GM11]). Moreover, (3.1.1) is intimately related to linear transforma-
tions of inverse flows of jump SDEs and it is precisely this connection that we will exploit
to obtain solutions.
Systems of linear SIDEs appear in many contexts. They may be considered as ex-
tensions of both first-order symmetric hyperbolic systems and linear fractional advec-
tion-diffusion equations. The equation (3.1.1) also arises in non-linear filtering of semi-
martingales as the equation for the unormalized filter of the signal (see, e.g., [Gri76] and
3.1. Introduction 41
[GM11]). Moreover, (3.1.1) is intimately related to linear transformations of inverse flows
of jump SDEs and it is precisely this connection that we will exploit to obtain solutions.
There are various techniques available to derive the existence and uniqueness of clas-
sical solutions of linear parabolic SPDEs and SIDEs. One approach is to develop a theory
of weak solutions for the equations (e.g. variational, mild solution, or etc...) and then
study further regularity in classical function spaces via an embedding theorem. We refer
the reader to [Par72, Par75, MP76, KR77, Tin77a, Gyö82, Wal86, DPZ92, Kry99, CK10,
PZ07, Hau05, RZ07, BvNVW08, HØUZ10, LM14b] for more information about weak
solutions of SPDEs driven by continuous and discontinuous martingales and martingale
measures. This approach is especially important in the non-degenerate setting where some
smoothing occurs and has the obvious advantage that it is broader in scope. Another ap-
proach is to regard the solution as a function with values in a probability space and use
the method of deterministic PDEs (i.e. Schauder estimates, see, e.g. [Mik00, MP09]).
A third approach is a direct one that uses solutions of stochastic differential equations.
The direct method allows to obtain classical solutions in the entire Hölder scale while
not restricting to integer derivative assumptions for the coefficients and data. In this
chapter, we derive the existence of a classical solutions of (3.1.1) with regular coefficients
using a Feynman-Kac-type transformation and the interlacing of the space-inverse (first
integrals [KR81]) of a stochastic flow associated with the equation. The construction of
the solution gives an insight into the structure of the solution as well. We prove that the
solution of (3.1.1) is unique in the class of classical solutions with polynomial growth (i.e.
weighted Hölder spaces). As an immediate corollary of our main result, we obtain the ex-
istence and uniqueness of classical solutions of linear partial integro-differential equations
(PIDEs) with random coefficients, since the coefficients σ1, H1, a1, ρ1, and free terms g
and h can be zero. Our work here directly extends the method of characteristics for de-
terministic first-order PDEs and the well-known Feynman-Kac formula for deterministic
second-order PDEs.
In the continuous case (i.e. H1 ≡ 0,H2 ≡ 0, h ≡ 0), the classical solution of (3.1.1)
was constructed in [KR81, Kun81, Kun86a, Roz90] (see references therein as well) us-
ing the first integrals of the associated backward SDE. This method was also used to
obtain classical solutions of (3.1.1) in [DPMT07]. In the references above, the forward
Liouville equation for the first integrals of associated stochastic flow was derived directly.
However, since the backward equation involves a time reversal, the coefficients and in-
put functions are assumed to be non-random. The generalized solutions of (3.1.1) with
d2 = 1, non-random coefficients, non-degenerate diffusion, and finite measures π1 = π2
were discussed in [MB07]. In this chapter, we give a direct derivation of (3.1.1) and all
the equations considered are forward, possibly degenerate, and the coefficients and input
functions are adapted. For other interesting and related developments, we refer the reader
42 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
to [Pri12, Zha13, Pri14], which all concern the fascinating regularizing property of noise
in the case of non-degenerate noise.
This chapter is organized as follows. In Section 3.2, our notation is set forth and the
main results are stated. In Section 3.3, the main theorems are proved. In Section 3.4, the
appendix, auxiliary facts used throughout the chapter are discussed.
3.2 Statement of main results
In this chapter, elements of Rd1 and Rd2 are understood as column vectors and elements of
Rd
2
1 and Rd
2
2 are understood as matrices of dimension d1 × d1 and d2 × d2, respectively. We
also adopt the notation of Chapter 2. If we do not specify to which space the parameters
ω, t, x, y, z and n belong, then we mean ω ∈ Ω, t ∈ [0,T ], x, y ∈ Rd1 , z ∈ Zk, and n ∈ N.
Let us introduce some regularity conditions on the coefficients and free terms. We
consider these assumptions for β¯ > 1 ∨ α and β˜ > α.
Assumption 3.2.1 (β¯). (1) There is a constant N0 > 0 such that for each k ∈ {1, 2} and all
ω, t ∈ Ω × [0,T ],
|r−11 bt|0 + |∇bt|β¯−1 + |r
−1
1 σ
k
t |0 + |∇σ
k
t |β¯−1 ≤ N0.
Moreover, for each k ∈ {1, 2} and all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek),
|r−11 H
k
t (z)|0 ≤ K
k
t (z) and |∇H
k
t (z)|β¯−1 ≤ K¯
k
t (z)
where Kk, K¯k : Ω × [0,T ] × (Dk ∪ Ek) → R+ are PT ⊗ Zk-measurable functions
satisfying
Kkt (z) + K¯
k
t (z) +
∫
Dk
(
Kkt (z)
α + K¯kt (z)
2
)
πk(dz) +
∫
Ek
(
Kkt (z)
1∧α + K¯kt (z)
)
πk(dz) ≤ N0,
for all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek).
(2) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ {(ω, t, x, z) ∈ Ω × [0,T ] ×
Rd1 × (Dk ∪ Ek) : |∇Hkt (ω, x, z)| > η},
∣∣∣∣
(
Id1 + ∇H
k
t (x, z)
)−1∣∣∣∣ ≤ N0.
Assumption 3.2.2 (β˜). There is a constant N0 > 0 such that for each k ∈ {1, 2} and all
(ω, t) ∈ Ω × [0,T ],
|ct|β˜ + |υ
k
t |β˜ + |r
−θ
1 ft|β˜ + |r
−θ
1 gt|β˜ ≤ N0.
3.2. Statement of main results 43
Moreover, for each k ∈ {1, 2} and all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek),
|ρkt (z)|β˜ ≤ l
k
t (z), |r
−θ
1 ht(z)|β˜ ≤ l
k
t (z),
where lk : Ω × [0,T ] × Zk → R+ are PT ⊗Zk-measurable function satisfying
lkt (z) +
∫
Dk
lkt (z)
2πk(dz) +
∫
Ek
lkt (z)π
k(dz) ≤ N0,
for all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek).
Remark 3.2.1. It follows from Lemma 3.4.10 and Remark 3.4.11 that if Assumption
3.2.1(β¯) holds for some β¯ > 1 ∨ α, then for all ω, t, and z ∈ Dk ∪ Ek, x 7→ H˜kt (x, z) :=
x + Hkt (x, z) is a diffeomorphism.
Let Assumptions 3.2.1(β¯) and 3.2.2(β˜) hold for some β¯ > 1 ∨ α and β˜ > α. In our
derivation of a solution of (3.1.1), we first obtain a solution of an equation of a special
form. Specifically, consider the system of SIDEs on [0,T ] × Rd1 given by
duˆlt =
(
(L1;lt +L
2;l
t )uˆt + bˆ
i
t∂iu
l
t + cˆ
ll¯
t u
l¯
t + fˆ
l
t
)
dt +
(
N1;lϱt uˆt + g
lϱ
t
)
dw1;ϱt
+
∫
Z1
(
I1;lt,z uˆt− + h
l
t(z)
)
[1D1(z)q
1(dt, dz) + 1E1(z)p
1(dt, dz)], τ < t ≤ T,
uˆlt = φ
l, t ≤ τ, l ∈ {1, . . . , d2}, (3.2.1)
where
bˆit(x) : = 1[1,2](α)b
i
t(x) +
2∑
k=1
1{2}(α)σ
k; jϱ
t (x)∂ jσ
k;iϱ
t (x)
+
2∑
k=1
1(1,2](α)
∫
Dk
(
Hk;it (x, z) − H
k;i
t (H˜
k;−1
t (x, z), z)
)
πk(dz),
cˆll¯t (x) : = c
ll¯
t (x) +
2∑
k=1
1{2}(α)σ
k; jϱ
t (x)∂ jυ
k;ll¯ϱ
t (x)
+
2∑
k=1
∫
Dk
(ρk;ll¯t (x, z) − ρ
k;ll¯
t (H˜
k;−1
t (x, z), z))π
k(dz),
fˆ lt (x) : = f
l
t (x) + σ
1; jϱ
t (x)∂ jg
lϱ
t (x) +
∫
D1
(
hlt(x, z) − h
l
t(H˜
1;−1
t (x, z), z)
)
π1(dz).
Let (w2;ϱt )ϱ∈N, t ≥ 0, ϱ ∈ N, be a sequence of independent one-dimensional Wiener
processes. Let p2(dt, dz) be a Poisson random measure on ([0,∞) × Z2,B([0,∞) ⊗ Z2)
with intensity measure π2(dz)dt. Extending the probability space if necessary, we take w2
44 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
and p2(dt, dz) to be independent of w1 and p1(dt, dz). Let
Fˆt = σ
(
(w2;ϱs )ϱ∈N, p
2([0, s],Γ) : s ≤ t, Γ ∈ Z2
)
and F˜ =
(
F˜t
)
t≤T
be the standard augmentation of
(
Ft ∨ Fˆt
)
t≤T
. Denote by q2(dt, dz) =
p2(dt, dz) − π2(dz)dt the compensated Poisson random measure. We associate with the
SIDE (3.2.1), the F˜-adapted stochastic flow Xt = Xt(x) = Xt(τ, x), (t, x) ∈ [0,T ] × Rd1 ,
generated by the SDE
dXt = −1[1,2](α)bt(Xt)dt +
2∑
k=1
1{2}(α)σ
k;ϱ
t (Xt)dw
k;ϱ
t
−
2∑
k=1
∫
Dk
Hkt (H˜
k;−1
t (Xt−, z), z)[p
k(dt, dz) − 1(1,2](α)πk(dz)dt]
−
2∑
k=1
∫
Ek
Hkt (H˜
k;−1
t (Xt−, z), z)p
k(dt, dz), τ < t ≤ T,
Xt = x, t ≤ τ, (3.2.2)
and the F˜-adapted random field Φt(x) = Φt(τ, x), (t, x) ∈ [0,T ] × Rd1 , solving the linear
SDE given by
dΦt(x) = (ct(Xt(x))Φt(x) + ft(Xt(x))) dt +
2∑
k=1
υ
k;ϱ
t (Xt(x))Φt(x)dw
k;ϱ
t + g
ϱ
t (Xt(x))dw
1;ϱ
t
+
2∑
k=1
∫
Zk
ρkt (H˜
k;−1
t (Xt−(x), z), z)Φt−(x)[1Dk(z)q
k(dt, dz) + 1Ek(z)p
k(dt, dz)]
+
∫
Z1
ht(H˜
1;−1
t (Xt−(x), z), z)[1D1(z)q
1(dt, dz) + 1E1(z)p
1(dt, dz)], τ < t ≤ T,
Φt(x) = φ(x), t ≤ τ.
The coming theorem is our existence, uniqueness, and representation theorem for
(3.2.1). Let us describe our solution class. For each β′ ∈ (0,∞), denote by Cβ
′
(Rd1; Rd2)
the linear space of all F-adapted random fields v = vt(x) such that P-a.s.
1[τn,τn+1)r
−λn
1 v ∈ D([0,T ];C
β′(Rd1 ,Rd2)),
where (τn)n≥0 is an increasing sequence of F-stopping times with τ0 = 0 and τn = T for
sufficiently large n, and where for each n, λn is a positive Fτn-measurable random variable.
Theorem 3.2.2. Let Assumptions 3.2.1(β¯) and 3.2.2(β˜) hold for some β¯ > 1∨α and β˜ > α.
For any stopping time τ ≤ T and Fτ ⊗ B(Rd1)-measurable random field φ such that for
3.2. Statement of main results 45
some β′ ∈ (α, β¯ ∧ β˜) and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1; Rd2), there exists a unique solution
uˆ = uˆ(τ) of (3.2.1) in Cβ
′
(Rd1; Rd2) and for all (t, x) ∈ [0,T ] × Rd1 , P-a.s.
uˆt(τ, x) = E
[
Φt(τ, X
−1
t (τ, x))|Ft
]
. (3.2.3)
Moreover, for all ϵ > 0 and p ≥ 2, there is a constant N = N(d1, d2, p,N0,T, β′, η1, η2, ϵ,
θ, θ′) such that
E
[
sup
t≤T
|r−θ∨θ
′−ϵ
1 uˆt(τ)|
p
β′
∣∣∣Fτ
]
≤ N(|r−θ
′
1 φ|
p
β′
+ 1). (3.2.4)
Using Itô’s formula, it is easy to verify that if m = 1 and
gt(x) = 0, ht(x) = 0, and ρ
k
t (x, z) ≥ −1,
for all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × (Dk ∪ Ek), k ∈ {1, 2}, then
Φt(x) = Ψt(x)ϕ(x) + Ψt(x)
∫
]τ,τ∨t]
Ψ−1s (x) fs(Xs(x))ds, (3.2.5)
where P-a.s. for all t and x, Ψt(x) := eBt(x), and
Bt(x) :=
∫
[τ,τ∨t]

cs(Xs(x)) −
2∑
k=1
1
2
υk;ϱs (Xs(x))υ
k;ϱ
s (Xs(x))

 ds
+
2∑
k=1
∫
]τ,τ∨t]
υk;ϱs (Xs(x))dw
k;ϱ
s
−
2∑
k=1
∫
]τ,τ∨t]
∫
Dk
(
ln
(
1 + ρks(H˜
k;−1
s (Xs−(x), z), z)
)
− ρks(H˜
k;−1
s (Xs−(x), z), z)
)
πk(dz)ds
2∑
k=1
∫
]τ,τ∨t]
∫
Zk
ln
(
1 + ρks(H˜
k;−1
s (Xs−(x), z), z)
)
[1Dk(z)q
k(ds, dz) + 1Ek(z)p
k(ds, dz)].
The following corollary then follows directly from (3.2.3) and (3.2.5).
Corollary 3.2.3. Let m = 1 and assume that for each k ∈ {1, 2} and all (ω, t, x, z) ∈
Ω × [0,T ] × Rd1 × (Dk ∪ Ek),
gt(x) = 0, ht(x, z) = 0, ρ
k
t (x, z) ≥ −1.
Moreover, let Assumptions 3.2.1(β¯) and 3.2.2(β˜) hold for some β¯ > 1 ∨ α and β˜ > α. Let
τ ≤ T be a stopping time and φ be a Fτ ⊗ B(Rd1)-measurable random field such that for
some β′ ∈ (α, β¯ ∧ β˜) and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1; Rd2).
(1) If for all (ω, t, x) ∈ Ω × [0,T ] × Rd1 , ft(x) ≥ 0 and φ(x) ≥ 0, then the solution uˆ of
(3.1.1) satisfies uˆt(x) ≥ 0, P-a.s. for all (t, x) ∈ [0,T ] × Rd1 .
46 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
(2) If for all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × (Dk ∪ Ek), k ∈ {1, 2}, υkt (x) = 0, ft(x) ≤ 0,
ct(x) ≤ 0, φ(x) ≤ 1, and ρkt (x, z) ≤ 0, then the solution uˆ of (3.1.1) satisfies uˆt(x) ≤ 1,
P-a.s. for all (t, x) ∈ [0,T ] × Rd1 .
Remark 3.2.4. Since L2 can be the zero operator, both Theorem 3.2.2 and Corollary 3.2.3
apply to fully degenerate SIDEs and PIDEs with random coefficients.
Now, let us discuss our existence and uniqueness theorem for (3.1.1). We construct
the solution of u = u(τ) of (3.1.1) by interlacing the solutions of (3.2.1) along a sequence
of large jump moments (see Section 3.3.5). By using an interlacing procedure we are
also able to drop the condition of boundedness of (I + ∇H1t (x, z))
−1 on the set (ω, t, x, z) ∈
{(ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × (D1 ∪ E1) : |∇H1t (ω, x, z)| > η
k}. Also, in order to remove
the terms in bˆ, cˆ, and fˆ that appear in (3.2.1), but not in (3.1.1), we subtract terms from
the relevant coefficients in the flow and the transformation. However, in order to do this,
we need to impose stronger regularity assumptions on some of the coefficients and free
terms. We will introduce parameters µ1, µ2, δ1, δ2 ∈ [0, α2 ] to the regularity assumptions.
These parameters allows for a trade-off of integrability in z and regularity in x of the
coefficients Hkt (x, z), ρ
k
t (x, z), h
k
t (x, z). It is worth mentioning that the removal of terms and
the interlacing procedure are independent of each other and that it is due only to the weak
assumptions on H1 and ρ1 on the set V1 that we do not have moment estimates and a
simple representation property like (3.2.4) for the solution of (3.1.1). However, there is a
representation of the solution and we refer the reader to the proof of the coming theorem
for the description of the representation.
We introduce the following assumptions for β¯ > 1∨α and β˜ > α. For each k ∈ {1, 2}, let
D¯k be the trace sigma-algebra of Dk ∪Ek relative toZk andV1 be the trace sigma-algebra
of V1 relative toZ1.
Assumption 3.2.3 (β¯). (1) There is a constant N0 > 0 such that for each k ∈ {1, 2} and all
(ω, t) ∈ Ω × [0,T ],
|r−11 bt|0 + |∇bt|β¯−1 + |σ
k
t |β¯+1 ≤ N0.
(2) For each k ∈ {1, 2} , there are real-numbers δk, µk ∈ [0, α2 ] such that for all (ω, t) ∈
Ω × [0,T ],
|υkt |β¯+1 ≤ N0, if σ
k
t , 0, |gt|β¯+1 ≤ N0, if σ
1
t , 0,
|Hkt (z)|0 ≤ K
k
t (z), |∇H
k
t (z)|β¯+δk−1 ≤ K¯
k
t (z), ∀z ∈ D
k,
∑
|γ|=[β¯+δk]−
[∂γHkt (z)]{β¯+δk}+ ≤ K˜
k
t (z), ∀z ∈ D
k,
|r−11 H
k
t (z)|0 ≤ K
k
t (z), |∇H
k
t (z)|β¯−1 ≤ K¯
k
t (z), ∀z ∈ E
k,
3.2. Statement of main results 47
|ρkt (z)|β¯+µk ≤ l
k
t (z),
∑
|γ|=[β¯+µk]−
[∂γρkt (z)]{β¯+µk}+ ≤ l˜
k
t (z), ∀z ∈ D
k,
|r−θ1 ht(z)|β¯+µ1 ≤ l
1
t (z),
∑
|γ|=[β¯+µ1]−
[∂γht(z)]{β¯+µ1}+ , ∀z ∈ D
1,
where Kk, K¯k, K˜k, lk, l˜k : Ω×[0,T ]×(Dk∪Ek)→ R+ arePT⊗D¯k-measurable functions
satisfying for all (ω, t, z) ∈ Ω × [0,T ],
Kkt (z) + K¯
k
t (z) + l˜
k
t (z) + l
k
t (z) + l˜
k
t (z) ≤ N0, ∀z ∈ D
k ∪ Ek,
∫
Dk
(
Kkt (z)
α + K¯kt (z)
2 + K˜kt (z)
α
α−δk + K˜kt (z)
2 + lkt (z)
2 + l˜kt (z)
α
α−µk + l˜kt (z)
2
)
πk ≤ N0(dz),
∫
Ek
(
Kkt (z)
1∧α + K¯kt (z)
)
πk(dz) ≤ N0.
(3) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ {(ω, t, x, z) ∈ Ω × [0,T ] ×
Rd1 × Z2 : |∇H2t (ω, x, z)| > η},
∣∣∣∣
(
Id1 + ∇H
2
t (x, z)
)−1∣∣∣∣ ≤ N0.
Assumption 3.2.4 (β˜). (1) There is a constant N0 > 0 such that for each k ∈ {1, 2} and all
(ω, t) ∈ Ω × [0,T ],
|ct|β˜ + |r
−θ
1 ft|β˜ ≤ N0,
|υkt |β˜ ≤ N0, if σ
k
t = 0, |gt|β˜ ≤ N0, if σ
1
t = 0,
|ρk(t, z)|β˜ ≤ l
k
t (z), ∀z ∈ E
k, |r−θ1 ht(z)|β˜ ≤ l
1
t (z), ∀z ∈ E
1,
where for all (ω, t) ∈ Ω × [0,T ],
∫
Ek
lkt (z)π
k(dz) ≤ N0.
(2) There exist processes ξ, ζ : Ω × [0,T ] × V1 → R+ that are PT ⊗ V1measurable and
satisfy
|r−ξt(z)1 H
1
t (z)|β˜∨1 + |r
−ξt(z)
1 ρ
1
t (z)|β˜ + |r
−ξt(z)
1 ht(z)|β˜ ≤ ζt(z),
for all (ω, t, z) ∈ Ω × [0,T ] × V1.
We now state our existence and uniqueness theorem for (3.1.1).
Theorem 3.2.5. Let Assumptions 3.2.3(β¯) and 3.2.4(β˜) hold for some β¯ > 1∨α and β˜ > α.
For any stopping time τ ≤ T and Fτ ⊗ B(Rd1)-measurable random field φ such that for
some β′ ∈ (α, β¯ ∧ β˜) and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1; Rd2), there exists a unique solution
u = u(τ) of (3.1.1) in Cβ
′
(Rd1; Rd2).
48 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
3.3 Proof of main theorems
We will first prove uniqueness of the solution of (3.2.1) in the class Cβ
′
(Rd1; Rd2). The
existence part of the proof of Theorem 3.2.2 is divided into a series of steps. In the first
step, by appealing to the representation theorem we derived for solutions of continuous
SPDEs shown in Theorem 2.4 in [LM14c], we use an interlacing procedure and the strong
limit theorem given in Theorem 2.3 in [LM14c] to show that the space inverse of the flow
generated by a jump SDE (i.e. the SDE (3.2.2) without the uncorrelated noise) solves a
degenerate linear SIDE. Then we linearly transform the inverse flow of a jump SDE to
obtain solutions of degenerate linear SIDEs with free and zero-order terms and an initial
condition. In the last step of the proof of Theorem 3.2.2, we introduce an independent
Wiener process and Poisson random measure as explained above, apply the results we
know for fully degenerate equations, and then take the optional projection. In the last
section, Subsection 3.3.4, we prove Theorem 3.2.5 using an interlacing procedure and
removing the extra terms in bˆ, cˆ and fˆ . The uniqueness of the solution u of (3.1.1) follows
directly from our construction.
3.3.1 Proof of uniqueness for Theorem 3.2.2
Proof of Uniqueness for Theorem 3.2.2. Fix a stopping time τ ≤ T and Fτ ⊗ B(Rd1)-
measurable random field φ such that for some β′ ∈ (α, β¯ ∧ β˜) and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈
Cβ
′
(Rd1; Rd2). In this section, we will drop the dependence of processes on t, x, and z when
we feel it will not obscure the argument. Let uˆ1(τ) and uˆ2(τ) be solutions of (3.2.1) in Cβ
′
.
It follows that v := uˆ1(τ) − uˆ2(τ) solves
dvlt = [(L
1;l
t +L
2;l
t )vt + bˆ
i
t∂iv
l
t + cˆ
ll¯
t v
l¯
t]dt +N
1;ϱ
t v
l
tdw
1;ϱ
t
+
∫
Z1
I1;lt,z vt−[1D1(z)q
1(dt, dz) + 1E1(z)p
1(dt, dz)], τ < t ≤ T,
vlt = 0, t ≤ τ, l ∈ {1, . . . , d2},
and P-a.s.
1[τn,τn+1)(·)r
−λn
1 v ∈ D([0,T ];C
β′(Rd1 ,Rd2)),
where (τn)n≥0 is an increasing sequence of F-stopping times with τ0 = 0 and τn = T for
sufficiently large n, and where for each n, λn is a positive Fτn-measurable random variable.
Clearly, it suffices to take τ1 = τ and λ0 = 0. Thus, vt(x) = 0 for all (ω, t) ∈ [[τ0, τ1)).
Assume that for some n, P-a.s. for all t and x, vt∧τn(x) = 0. We will show that P-a.s. for all
3.3. Proof of main theorems 49
t and x, v˜t(x) := v(τn∨t)∧τn+1(x) = 0. Applying Itô’s formula, for all x, P-a.s. for all t, we find
d|v˜t|2 =
(
2v˜ltL
1;l
t v˜t + |N
1
t v˜t|
2 + 2v˜ltb
i
t∂iv˜
l
t + 2v˜
l
tc
ll¯
t v˜
l¯
t
)
dt
+
(
2v˜ltI
1;l
t,z v˜t +
∫
D1∪E1
|I1;lt,z v˜t|
2π1(dz)
)
dt
+
(
2vltL
2;l
t v˜t + 2v˜
l
tI
2;l
t,z v˜t
)
dt + 2vltN
1;ϱ
t v˜
l
tdw
1;ϱ
t
+
∫
Z1
(
2v˜lt−I
1;l
t,z v˜t− + |I
1;l
t,z v˜t−|
2
)
q1(dt, dz), τn < t ≤ τn+1,
|v˜t|2 = 0, t ≤ τn, l ∈ {1, . . . , d2}, (3.3.1)
where for ϕ ∈ C∞c (R
d1; Rd2), k ∈ {1, 2}, and l ∈ {1, . . . , d2},
Lk;lϕ :=
1
2
σk;iϱσk; jϱ∂i jϕ
l + σk; jϱ∂ jσ
k;iϱ∂iϕ
l + σk;iϱυk;ll¯ϱ∂iϕ
l¯ + σk; jϱ∂ ja
k;ll¯ϱϕl¯
and
Ik;lϕ : =
∫
Dk
(
ρk;ll¯ϕl¯(H˜k) − ρk;ll¯(H˜k;−1)ϕl¯
)
πk(dz)
+
∫
Dk
(
ϕl(H˜k) − ϕl + 1(1,2](α)Fk;i∂iϕl
)
πk(dz)
+
∫
Ek
(
(Ill¯d2 + ρ
k;ll¯)ϕl¯(H˜k) − ϕl
)
πk(dz).
For each ω and t, let
Qt =
∫
Rd1
|v˜t(x)|2r−λ1 (x)dx,
where λ = λn + (d′ + 2)/2 and d′ > d1. Note that
EQt ≤
∫
Rd1
r−d
′
1 (x)dxE|r
−λn
1 v˜t|0 < ∞.
It suffices to show that supt≤T EQt = 0. To this end, we will multiply the equation (3.3.1)
by the weight r−2λ1 = r
−2λn+1
1 r
−d′
1 , integrate in x, and change the order of the integrals in time
and space. Thus, we must verify the assumptions of stochastic Fubini theorem hold (see
Corollary 3.4.13 and Remark 3.4.14 as well) with the finite measure µ(dx) = r−d
′
1 (x)dx on
Rd1 . Since b and σk have linear growth an υk and c are bounded, owing to Lemma 3.4.6,
we easily obtain that there is a constant N = N(d1, d2,N0, λn) such that P-a.s for all t,
∫
Rd1


2∑
k=1
2|r−λn1 v˜||r
−λn−2
1 L
kv˜| + |rλn−11 N
1v˜|2

 r
−d′
1 dx ≤ N sup
t≤T
|r−λn1 v˜|
2
β′ ,
50 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
∫
Rd1
4|r−λn1 v˜|
2|r−λn−11 N
1v˜|2r−d
′
1 dx ≤ N sup
t≤T
|r−λn1 v˜t|
4
β′ ,
and ∫
Rd1
(
2|r−λn1 v˜|r
−λn−1
1 b∂iv˜| + 2|r
−λn
1 v˜||r
−λn
1 cv˜|
)
r−d
′
1 dx ≤ N sup
t≤T
|r−λn1 v˜t|
2
β′ .
For all ϕ ∈ Cαloc(R
d1; Rd2) and all k, ω, t, x, p, and z,
r−p1 (ϕ(H˜
k) − ϕ + 1(1,2](α)Fk;i∂iϕ)
= ϕ¯(H˜k) − ϕ¯ − 1(1,2](α)Hk;i∂iϕ¯ + 1(1,2](α)(Hk;i + Fk;i)∂iϕ¯
+p1(1,2](α)(Hk;i + Fk;i)r−21 x
iϕ¯ +


rp1 (H˜
k)
rp1
− 1

 (ϕ¯(H˜k) − 1(1,2](α)ϕ¯)
+1(1,2](α)


rp1 (H˜
k)
rp1
− 1 + pHk;ir−21 x
i

 ϕ¯, (3.3.2)
where ϕ¯ := r−pϕ. By Taylor’s formula, for all ϕ ∈ Cα(Rd1; Rd2) and all k, ω, t, x, and z, we
have
|ϕ(H˜k) − ϕ − 1(1,2](α)Hk;i∂iϕ| ≤ rα1 |ϕ|α|r
−1
1 H|
α
0 . (3.3.3)
Combining (3.3.2), (3.3.3), and the estimates given in Lemma 3.4.10 (1), for all k, ω, t, x
and z, we obtain
r−α1 |ρ
k(H˜k;−1) − ρk| ≤ N |ρ|α∧1|r−11 H
k|α∧10
and
r−λn−α1 |v˜(H˜
k) − v˜ + 1(1,2](α)Fk;i∂iv˜|
≤ N |r−λn1 v˜|α
(
|r−11 H
k|α0 + |r
−1
1 H|0[H
k]1 + |r−11 H|
[α]−+1
0 + [H]
[α]−+1
1
)
, (3.3.4)
for some constant N = N(d1, λn,N0, η1, η2). Therefore, P-a.s for all t,
∫
Rd1


2∑
k=1
2|r−λn1 v˜||r
−λn−2
1 I
kv˜| +
∫
D1∪E1
|r−λ−11 Izv˜|
2π1(dz)

 r
−d′
1 dx ≤ N sup
t≤T
|r−λn1 v˜|
2
β′ ,
and ∫
Rd1
(
2|r−λn1 v˜||r
−λn−2
1 I
k
z v˜| + |r
−λn−1
1 Izv˜|
2
)2
r−d
′
1 dx ≤ N sup
t≤T
|r−λn1 v˜t|
4
β′ ,
for some constant N = N(d1, d2, λn,N0, η1, η2).
Let L2(Rd1; Rd2) be the space of square-integrable functions f : Rd1 → Rd2 with norm
∥ · ∥0 and inner product (·, ·)0. Moreover, let L2(Rd1; ℓ2(Rd2)) be the space of square-
integrable functions f : Rd1 → ℓ2(Rd2) with norm ∥ · ∥0. With the help of the above
3.3. Proof of main theorems 51
estimates and Corollary 3.4.13, denoting v¯ = r−λv˜, P-a.s. for all t, we have
d∥v¯t∥20 =
(
2(v¯lt, L¯
1
t v¯t)0 + ∥N¯
1
t v¯t∥
2
0 + 2(v¯t, I¯
1
t,zv¯t)0 +
∫
D1∪E1
∥I¯1t,zv¯t∥
2
0π
1(dz)
)
dt
+
(
2(v˜t, b
i
t∂iv˜t + c¯
l¯
tv˜
l¯
t)0 + 2(v˜t, L¯
2
t v˜t)0 + 2(v˜t, I¯
2
t,zv˜t)0
)
dt + 2(vt, N¯
1;ϱ
t v˜t)0dw
1;ϱ
t
+
∫
Z1
(
2(v˜t−, I¯1t,zv˜t−)0 + ∥I¯
1
t,zv˜t−∥
2
0
)
q1(dt, dz), τn < t ≤ τn+1,
∥v¯t∥20 = 0, t ≤ τn, l ∈ {1, . . . , d2}, (3.3.5)
where all coefficients and operators are defined as in (3.2.1) with the following changes:
(1) for each k ∈ {1, 2}, υk is replaced with
υ¯k;ll¯ := υk;ll¯ + 1{2}(α)λσk;iϱr−21 x
iδll¯;
(2) for each k ∈ {1, 2}, ρk replaced with
ρ¯k;ll¯ := ρk;ll¯ +


rλ1(H˜
k)
rλ1
− 1

 (Ill¯d2 + ρ
k;ll¯);
(3) c is replaced with
c¯ll¯ = cll¯ + λbir−2xiδll¯ +
2∑
k=1
λ2σk;iϱσk; jϱr−41 x
ix j
+
2∑
k=1
∫
Dk




rλ1
rλ1(H˜
k;−1)
− 1

 (Ill¯m + ρ
k(H˜k;−1)) − 1(1,2](α)λr−21 xiH
k;i(H˜k;−1)

 π
k(dz).
Since for all k, ω and t, |r−11 σ
k|0 + |r−11 ∇σ
k|β¯−1 + |υk|β˜ ≤ N0, for β¯ > 1 ∨ α and β˜ > α, it is
clear that |υ¯k|α ≤ N. Moreover, since for all k, ω and t, |r−11 H
k|0 + |Hk|β¯ ≤ Kk and |ρ|β˜′ ≤ lk,
applying the estimates in Lemma (3.4.10) (1), we get
|ρ¯k|α ≤ lk + Kk(1 + lk) and |c|α ≤ N0.
We will now estimate the drift terms of (3.3.5) in terms of ∥v¯t∥20. We write f ∼ g if∫
Rd1
| f (x)|dx =
∫
Rd1
|g(x)|dx and f ≪ g if
∫
Rd1
| f (x)|dx ≤
∫
Rd1
|g(x)|dx. Using the divergence
theorem, for any v : Rd1 → Rd2 , σ : Rd1 → Rd1 and υ : Rd1 → R2d2 and all x, we get
σiσ jvlvli j ∼
1
2
(σiσ j)i jv − σiσ jvliv
l
j = (σ
i
i jσ
j + σijσ
j
i )|v|
2 − σiσ jvliv
l
j,
2σijσ
jvlvli ∼ −(σ
i
jσ
j)i|v|2 = (σii jσ
j + σijσ
j
i )|v|
2,
52 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
and
σivlυll¯vl¯i + σ
ivl¯υll¯vli = σ
ivlυll¯symv
l¯
i ∼ −(σ
iυll¯sym)i|v|
2 = −(σiiυ
ll¯
sym + σ
iυll¯sym)|v|
2,
where υll¯sym = (υ
ll¯ + υl¯l)/2. Consequently, for all ω, t, and x, we have
2v¯lL¯1;lv¯ + |N¯1v¯|2 ∼
1
2
(
| divσ1|2 − ∂iσ1; jϱ∂ jσ1;iϱ
)
|v¯|2
− υ¯1;ll¯ϱsym v¯
lv¯l¯ divσ1;ϱ + |υ¯1v¯|2 ≪ N |v¯|2
and
2v¯lL¯(2);lv¯ ≪ −(1 + ϵ)|σ2;i∂iv¯|2 + N|v¯|2,
for any ϵ > 0, where in the last estimate we have also used Young’s inequality. By Lemma
3.4.10 (2) and basic properties of the determinant, there is a constant N = N(d,N0, η1, η2)
such that for all k, ω, t, x, and z,
det H˜k;−1 − 1 = det(Id1 + F
k) − 1 ≤ |∇Fk| ≤ N|∇Hk|
and
det H˜k;−1 − 1 − div Fk ≤ |∇Fk|2 ≤ N |∇Hk|2.
Thus, integrating by parts, for all ω, t, and x, we get
2v¯lI¯1;lv¯ +
∫
D1∪E1
|I¯1v¯|2π1(dz) ∼ 2
∫
D1
ρ¯1;ll¯sym(H˜
1;−1)(det∇H˜1;−1 − 1)π1(dz)v¯l¯v¯l
+
∫
D1∪E1
(
det∇H˜1;−1 − 1 + 1(1,2](α)1D1 div F
1
)
π1(dz)|v¯|2
+
∫
D1∪E1
(
1E12ρ¯
1;ll¯
sym(H˜
1;−1)v¯l¯v¯l + |ρ¯1(H˜1;−1)v¯|2
)
det∇H˜1;−1π1(dz)
≪ N
(∫
D1
(
K1(z)2 + l1(z)K1(z) + l1(z)2
)
π1(dz) +
∫
E1
(
Kk(z) + lk(z)
)
π1(dz)
)
|v¯|2.
Analogously, for all ω, t, and x, we attain
2v¯lI¯2;lv¯ ≤ −(1 + ϵ)
∫
D2∪E2
|v¯(H˜2) − v¯|2π2(dz) + N|v¯|2.
Therefore, combining the above estimates, P-a.s. for all t,
Qt ≤ N
∫ t
0
Qsds + Mt, (3.3.6)
where (Mt)t≤T is a càdlàg square-integrable martingale. Taking the expectation of (3.3.6)
3.3. Proof of main theorems 53
and applying Gronwall’s lemma, we get supt≤T EQt = 0, which implies that P-a.s. for all t
and x, v˜t(x) = 0. This completes the proof. □
3.3.2 Small jump case
Set (wϱ)ϱ∈N = (w1;ϱ)ϱ∈N, (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz) =
q1(dt, dz). Let σt(x) = (σ
iϱ
t (x))1≤i≤d1,ϱ≥1 be a ℓ2(R
d1)-valued RT ⊗B(Rd1)-measurable func-
tion defined on Ω × [0,T ] × Rd1 and Ht(x, z) = (Hit(x, z))1≤i≤d1 be a PT ⊗ B(R
d1) ⊗ Z-
measurable function defined on Ω × [0,T ] × Rd1 × Z.
We introduce the following assumption for β > 1 ∨ α.
Assumption 3.3.1 (β). (1) There is a constant N0 > 0 such that for all (ω, t) ∈ Ω× [0,T ],
|r−11 bt|0 + |r
−1
1 σt|0 + |∇bt|β−1 + |∇σt|β−1 ≤ N0.
Moreover, for all (ω, t, z) ∈ Ω × [0,T ] × Z,
|r−11 Ht(z)|0 ≤ Kt(z) and |∇Ht(z)|β−1 ≤ K¯t(z),
where K : Ω × [0,T ] × Z → R+ is a PT ⊗Z-measurable function satisfying
Kt(z) + K¯t(z) +
∫
Z
(
Kt(z)
α + K¯t(z)
2
)
π(dz) ≤ N0,
for all (ω, t, z) ∈ Ω × [0,T ] × Z.
(2) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ {(ω, t, x, z) ∈ Ω × [0,T ] ×
Rd1 × Z : |∇Ht(ω, x, z)| > η},
|
(
Id1 + ∇Ht(x, z)
)−1 | ≤ N0.
Let Assumption 3.3.1(β) hold for some β > 1 ∨ α. Let τ ≤ T be a stopping time.
Consider the system of SIDEs on [0,T ] × Rd1 given by
dvt(x) =
(
1{2}(α)
1
2
σ
iϱ
t (x)σ
jϱ
t (x)∂i jvt(x) + b
i
t(x)∂ivt(x)
)
dt + 1{2}(α)σ
iϱ
t (x)∂ivt(x)dw
ϱ
t
+
∫
Z
(vt−(x + Ht(x, z)) − vt−(x)) [1(1,2](α)q(dt, dz) + 1[0,1](α)p(dt, dz)]
+ 1(1,2](α)
∫
Z
(vt(x + Ht(x, z)) − vt(x) + Ft(x, z)∂ivt(x)) π(dz)dt, τ < t ≤ T,
vt(x) = x, t ≤ τ, (3.3.7)
54 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
where
bit(x) := 1[1,2](α)b
i
t(x) + 1{2}(α)σ
jϱ
t (x)∂ jσ
iϱ
t (x)
and
Ft(x, z) := −Ht(H˜−1t (x, z), z).
We associate with (3.3.7) the stochastic flow Yt = Yt(τ, x), (t, x) ∈ [0,T ] × Rd1 , generated
by the SDE
dYt = −1[1,2](α)bt(Yt)dt − 1{2}(α)σ
ϱ
t (Yt)dw
ϱ
t
+
∫
Z
Ft(Yt−, z)[1(1,2](z)q(dt, dz) + 1[0,1](z)p(dt, dz)], τ < t ≤ T, (3.3.8)
Yt = x, t ≤ τ.
Owing to parts (1) and (2) of Lemma 3.4.10, for all ω, t, and z, the inverse of the mapping
F˜t(x, z) := x+Ft(x, z) = x−Ht(H˜−1t (x, z), z) is H˜t(x, z) := x+Ht(x, z) and there is a constant
N = N(d1,N0, β, η) such that for all ω, t, x, y, and z,
|r−11 Ft(z)|0 ≤ NKt(z), |∇Ft(z)|β−1 ≤ Kt(z), |(Id1 + ∇Ft(x, z))
−1| ≤ N.
Thus, by Theorem 2.1 in [LM14c], there is a modification of the solution of (3.3.8), which
we still denote by Yt = Yt(τ, x), that is a C
β′
loc-diffeomorphism for any β
′ ∈ [1, β). Moreover,
P-a.s. Y·(τ, ·),Y−1· (τ, ·) ∈ D([0,T ];C
β′
loc(R
d1; Rd1)), and Y−1t− (τ, ·) coincides with the inverse
of Yt−(τ, ·) for all t. In the proof of the following proposition, we will show that the inverse
flow Y−1t (τ) solves (3.3.7).
Proposition 3.3.1. Let Assumption 3.3.1(β) hold for some β > 1 ∨ α. For any stopping
time τ ≤ T and β′ ∈ [1 ∨ α, β), vt(x) = vt(τ, x) = Y−1t (τ, x) solves (3.3.7) and for all ϵ > 0
and p ≥ 2, there is a constant N = N(d1, p,N0,T, β′, η, ϵ) such that
E
[
sup
t≤T
|r−(1+ϵ)1 vt(τ)|
p
0
]
+ E
[
sup
t≤T
|r−ϵ1 ∇vt(τ)|
p
β′−1
]
≤ N. (3.3.9)
Proof. The estimate (3.3.9) is given in Theorem 2.1 in [LM14c], so we only need to show
that Y−1t (τ, x) solves (3.3.7). Let (δn)n≥1 be a sequence such that δn ∈ (0, η) for all n and
δn → 0 as n→ ∞. It is clear that there is a constant N = N(N0) such that for all ω and t,
π({z : Kt(z) > δn}) ≤
N
δαn
. (3.3.10)
3.3. Proof of main theorems 55
For each n, consider the system of SIDEs on [0,T ] × Rd1 given by
dv(n)t (x) =
(
1{2}(α)
1
2
σ
iϱ
t (x)σ
jϱ
t (x)∂i jv
(n)
t (x) + b
i
t(x)∂iv
(n)
t (x)
)
dt
+1(1,2](α)
∫
Z
1{Kt>δn}(z)
(
v(n)t (x + Ht(x, z)) − v
(n)
t (x) + F
i
t(x, z)∂iv
(n)
t (x)
)
π(dz)dt
+
∫
Z
1{Kt>δn}(z)
(
v(n)t− (x + Ht(x, z)) − v
(n)
t− (x)
)
[1(1,2](α)q(dt, dz) + 1[0,1](α)p(dt, dz)],
+1{2}(α)σ
iϱ
t (x)∂iv
(n)
t (x)dw
ϱ
t , τ < t ≤ T, v
(n)
t (x) = x, t ≤ τ, (3.3.11)
and the stochastic flow Y (n)t = Y
(n)
t (τ, x), (t, x) ∈ [0,T ] × R
d1 , generated by the SDE
dY (n)t = −1[1,2](α)bt(Y
(n)
t )dt − 1{2}(α)σ
ϱ
t (Y
(n)
t )dw
ϱ
t
+
∫
Z
1{Kt>δn}(z)Ft(Y
(n)
t− , z)[1(1,2](α)q(dt, dz) + 1[0,1](α)p(dt, dz)], τ < t ≤ T,
Y (n)t (x) = x, t ≤ τ. (3.3.12)
Since (3.3.10) holds, we can rewrite equation (3.3.12) as
dY (n)t = −
(
1[1,2](α)bt(Y
(n)
t ) + 1(1,2](α)
∫
Z
1{Kt>δn}(z)Ft(Y
(n)
t , z)π(dz)
)
dt (3.3.13)
− 1{2}(α)σ
ϱ
t (Yn(t))dw
ϱ
t +
∫
Z
1{Kt>δn}(z)Ft(Y
(n)
t− , z)p(dt, dz), τ < t ≤ T,
and (3.3.11) as
dv(n)t (x) =
(
1{2}(α)
1
2
σ
iϱ
t (x)σ
jϱ
t (x)∂i jv
(n)
t (x) + b
i
t(x)∂ jσ
iϱ
t (x)
)
dt
+ 1{2}(α)σ
iϱ
t (x)∂iv
(n)
t (x)dw
ϱ
t + 1(1,2](α)
∫
Z
1{Kt>δn}(z)F
i
t(x, z)π(dz)∂iv
(n)
t (x)dt
+
∫
Z
1{Kt>δn}(z)
(
v(n)t− (x + Ht(x, z)) − v
(n)
t− (x)
)
p(dt, dz), τ < t ≤ T. (3.3.14)
It is well-known that the solution Y (n)t = Y
(n)
t (x) of (3.3.13) can be written as the solution of
continuous SDEs with a finite number of jumps interlaced. Indeed, for each n and stopping
time τ′ ≤ T , consider the stochastic flow Y˜ (n)t = Y˜
(n)
t (τ
′, x), (t, x) ∈ [0,T ] × Rd1 , generated
by the SDE
dY˜ (n)t = −
(
1[1,2](α)bt(Y˜
(n)
t ) + 1(1,2](α)
∫
Z
1{K>δn}(t, z)Ft(Y˜
(n)
t , z)π(dz)
)
dt
− 1{2}(α)σ
ϱ
t (Y˜
(n)
t )dw
ϱ
t , τ
′ < t ≤ T,
Y˜ (n)t = x, t ≤ τ
′.
56 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
By Theorems 2.1 and 2.4 and Remark 2.5, there is a modification of Y˜ (n)t = Y˜
(n)
t (τ
′, x), still
denoted Y˜ (n)t (τ
′, x), that is a Cβ
′
loc-diffeomorphism. Furthermore, P-a.s. we have that
Y˜ (n)· (τ
′, ·), Y˜ (n);−1· (τ
′, ·) ∈ C([0,T ];Cβ
′
loc)
and v˜(n)t = v˜
(n)
t (τ
′, x) = Y˜ (n);−1t (τ
′, x) solves the SPDE given by
dv˜(n)t (x) =
(
1{2}(α)
1
2
σ
iϱ
t (x)σ
jϱ
t (x)∂i jv
(n)
t (x) + b
i
t(x)∂iv
(n)
t (x)
)
dt
+ 1{2}(α)σ
iϱ
t (x)∂iv
(n)
t (x)dw
ϱ
t
+ 1(1,2](α)
∫
Z
1{K>δn}(t, z)F
i(t, z)π(dz)∂iv
(n)
t (x)dt, τ
′ < t ≤ T,
v˜(n)t (x) = x, t ≤ τ
′.
For each n, let
A(n)t =
∫
]0,t]
∫
Z
1{Ks>δn}(z)p(ds, dz), t ≥ 0,
and define the sequence of stopping times (τ(n)l )
∞
l=1 recursively by τ
(n)
0 = τ and
τ
(n)
l+1 = inf
{
t > τ(n)l : ∆A
(n)
t , 0
}
∧ T.
Fix some n ≥ 1. It is clear that P-a.s. for all x and t ∈ [0, τ(n)1 ),
Y (n);−1t (τ, x) = Y˜
(n);−1
t (τ, x) = v˜
(n)
t (τ, x)
satisfies (3.3.14) up to, but not including time τ(n)1 . Moreover, P-a.s. for all x,
Y (n)
τ
(n)
1
(τ, x) = Y˜ (n)
τn1−
(τ, x) +
∫
Z
F
τ
(n)
1
(Y˜ (n)
τ
(n)
1 −
(τ, x), z)p({τ(n)1 }, dz),
and hence
Y (n);−1
τ
(n)
1
(τ, x) =
∫
Z
v˜(n)
τ
(n)
1 −
(τ, x + H
τ
(n)
1
(x, z))p({τ(n)1 }, dz).
Consequently, v(n)t (τ, x) = Y
(n);−1
t (τ, x) solves (3.3.14) up to and including time τ
(n)
1 . Assume
that for some l ≥ 1, v(n)t (τ, x) = Y
(n);−1
t (τ, x) solves (3.3.14) up to and including time τ
(n)
l .
Clearly, P-a.s. for all x and t ∈ [τ(n)l , τ
(n)
l+1), Y
(n)
t (x) = Y˜
(n)
t (τ
(n)
l ,Y
(n)
τ
(n)
l −
(x)), and thus P-a.s. for
all x and t ∈ [τ(n)l , τ
(n)
l+1),
Y (n);−1t (x) = Y˜
(n)
t (τ
(n)
l ,Y
(n)
τ
(n)
l −
(x)) = v˜(n)t (τ
(n)
l ,Y
(n)
τ
(n)
l −
(x)).
3.3. Proof of main theorems 57
Moreover, P-a.s. for all x,
Y−1n (τ
n
l+1, x) =
∫
U
v˜n(τ
n
l , τ
n
l+1−, x + H(τ
n
l+1, x, z))p({τ
n
l+1}, dz),
which implies that v(n)t (τ, x) = Y
(n);−1
t (τ, x) solves (3.3.14) up to and including time τ
n
l+1.
Therefore, by induction, for each n, v(n)t (τ, x) = Y
(n);−1
t (τ, x) solves (3.3.14). It is easy to
see that for all ω, t, and z,
|r−11 1{Kt>δn}(z)Ft(z) − r
−1
1 Ft(z)|0 + |1{Kt>δn}(z)∇Ft(z) − ∇Ft(z)|β−1 ≤ 1{Kt≤δn}(z)Kt(z)
and thus
dPdt − lim
n→∞
∫
D
1{K≤δn}(t, z)Kt(z)
2π(dz) + dPdt − lim
n→∞
∫
E
1{K≤δn}(t, z)Kt(z)π(dz) = 0.
By virtue of Theorem 2.3 in [LM14c], for all ϵ > 0 and p ≥ 2 we have
lim
n→∞
(
E
[
sup
t≤T
|r−(1+ϵ)1 (Y
(n)
t (τ) − r
−(1+ϵ)
1 Yt(τ)|
p
0
]
+ E
[
sup
t≤T
|r−ϵ1 ∇Y
(n)
t (τ) − r
−ϵ
1 ∇Yt(τ)|
p
β′−1
])
= 0,
lim
n→∞
E
[
sup
t≤T
|r−(1+ϵ)1 Y
(n);−1
t (τ) − r
−(1+ϵ)
1 Y
−1
t (τ)|
p
0
]
= 0
and
lim
n→∞
E
[
sup
t≤T
|r−ϵ1 ∇Y
(n);−1
t (τ) − r
−ϵ
1 ∇Y
−1
t (τ)|
p
β′−1
]
= 0.
Then passing to the limit in both sides of (3.3.11) and making use of Assumption 3.3.1(β),
the estimate (3.3.4), and basic convergence properties of stochastic integrals, we discover
that vt(τ, x) = X−1t (τ, x) solves (3.3.7) . □
3.3.3 Adding free and zero-order terms
Set (wϱ)ϱ∈N = (w1;ϱ)ϱ∈N, (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz)
= p1(dt, dz) − π1(dz)dt. Also, set D = D1, E = E1, and assume Z = D ∪ E. Let υt(x) =
(υll¯ϱt (ω, x))1≤l,l¯≤d2, ϱ∈N be a ℓ2(R
2d2)-valued RT ⊗ B(Rd1)-measurable function defined on
Ω×[0,T ]×Rd1 and ρt(x, z) = (ρll¯t (ω, x, z))1≤l,l¯≤d2 be a PT ⊗B(R
d1)⊗Z-measurable function
defined on Ω × [0,T ] × Rd1 × Z.
We introduce the following assumptions for β > 1 ∨ α and β˜ > α.
Assumption 3.3.2 (β). (1) There is a constant N0 > 0 such that for all (ω, t) ∈ Ω× [0,T ],
|r−11 bt|0 + |r
−1
1 σt|0 + |∇bt|β−1 + |∇σt|β−1 ≤ N0.
58 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
Moreover, for all (ω, t, z) ∈ Ω × [0,T ] × Z,
|r−11 Ht(z)|0 ≤ Kt(z) and |∇Ht(z)|β−1 ≤ K¯t(z),
where K : Ω × [0,T ] × Z → R+ is a PT ⊗Z-measurable function satisfying
Kt(z) + K¯t(z) +
∫
D
(
Kt(z)
α + K¯t(z)
2
)
π(dz) +
∫
E
(
Kt(z)
α∧1 + K¯t(z)
)
π(dz) ≤ N0,
for all (ω, t, z) ∈ Ω × [0,T ] × Z.
(2) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ {(ω, t, x, z) ∈ Ω × [0,T ] ×
Rd1 × Z : |∇Ht(ω, x, z)| > η},
|
(
Id1 + ∇Ht(x, z)
)−1 | ≤ N0.
Assumption 3.3.3 (β˜). There is a constant N0 > 0 such that for all (ω, t) ∈ Ω × [0,T ],
|ct|β˜ + |υt|β˜ + |r
−θ
1 ft|β˜ + |r
−θ
1 gt|β˜ ≤ N0.
Moreover, for all (ω, t, z) ∈ Ω × [0,T ] × Z,
|ρt(z)|β˜ + |r
−θ
1 ht(z)|β˜ ≤ lt(z),
where l : Ω × [0,T ] × Z → R+ is a PT ⊗Z-measurable function satisfying
lt(z) +
∫
D
lt(z)
2π(dz) +
∫
E
lt(z)π(dz) ≤ N0.
(ω, t, z) ∈ Ω × [0,T ] × Z.
Let Assumptions 3.3.2(β¯) and 3.3.3(β˜) hold for some β¯ > 1 ∨ α and β˜ > α. Let τ ≤ T
be a stopping time and φ : Ω × Rd1 → Rd2 be a Fτ ⊗ B(Rd1)-measurable random field.
Consider the system of SIDEs on [0,T ] × Rd1 given by
dvlt =
(
Lltvt + bˆ
i
t∂iϕ
l + cˆll¯t ϕ
l¯ + fˆlt
)
dt +
(
N lϱt vt + g
lϱ
t
)
dwϱt
+
∫
Z
(
Ilt,zvt− + h
l
t(z)
)
[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,
vlt = φ
l, t ≤ τ, l ∈ {1, . . . , d2}, (3.3.15)
where for ϕ ∈ C∞c (R
d1; Rd2) and l ∈ {1, . . . , d2},
Lltϕ(x) := 1{2}(α)
1
2
σ
iϱ
t (x)σ
jϱ
t (x)∂i jϕ
l(x) + 1{2}(α)σ
iϱ
t (x)a
ll¯ϱ
t (x)∂iϕ
l¯(x)
3.3. Proof of main theorems 59
+
∫
Dk
ρll¯t (x, z)
(
ϕl¯(x + Ht(x, z)) − ϕl¯(x)
)
π(dz)
+
∫
Dk
(
ϕl(x + Ht(x, z)) − ϕl(x) − 1(1,2](α)∂iϕl(x)Hit(x, z)
)
π(dz)
N lϱt ϕ
l(x) := 1{2}(α)σ
iϱ
t (x)∂iϕ
l(x) + υll¯ϱt (x)ϕ
l¯(x),
Ilt,zϕ
l(x) := (Id2 + ρ
ll¯
t (x, z))ϕ
l¯(x + Ht(x, z)) − ϕl(x),
and where
bˆit(x) : = 1[1,2](α)b
i
t(x) + 1{2}(α)σ
jϱ
t (x)∂ jσ
iϱ
t (x)
+
∫
D
(
1(1,2](α)Hit(x, z) − H
i
t(H˜
−1
t (x, z), z)
)
π(dz),
cˆll¯t (x) : = c
ll¯
t (x) + 1{2}(α)σ
jϱ
t (x)∂ jυ
ll¯ϱ
t (x) +
∫
D
(
ρll¯t (x, z) − ρ
ll¯
t (H˜
−1
t (x, z), z)
)
π(dz),
fˆlt(x) : = f
l
t (x) + 1{2}(α)σ
jϱ
t (x)∂ jg
l
t(x) +
∫
D
(
hlt(x, z) − h
l
t(H˜
−1
t (x, z), z)
)
π(dz).
We associate with (3.3.15) the stochastic flow Xt = Xt(x) = Xt (τ, x) , (t, x) ∈ [0,T ] × Rd1 ,
given by (3.3.8). Let Γt(x) = Γt(τ, x), (t, x) ∈ [0,T ]×Rd1 , be the solution of the linear SDE
given by
dΓt(x) = (ct(Xt(x))Γt(x) + ft(Xt(x))) dt +
(
υ
ϱ
t (Xt(x))Γt(x) + g
ϱ
t (Xt(x))
)
dwϱt
+
∫
Z
ρt(H˜
−1
t (Xt−(x), z), z)Γt−(x)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)]
+
∫
Z
ht(H˜
−1
t (Xt−(x), z), z)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,
Γt(x) = 0, t ≤ τ. (3.3.16)
Let Ψt(x) = Ψt(τ, x), (t, x) ∈ [0,T ] × Rd1 , be the unique solution of the linear SDE given
by
dΨt(x) = ct(Xt(x))Ψt(x)dt + υ
ϱ
t (Xt(x))Ψt(x)dw
ϱ
t
+
∫
Z
ρt(H˜
−1
t (Xt−(x), z), z)Ψt−(x)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,
Ψt(x) = Id2 , t ≤ τ.
In the following lemma, we obtain p-th moment estimates of the weighted Hölder
norms of Γ and Ψ.
Lemma 3.3.2. Let Assumptions 3.3.2(β¯) and 3.3.3(β˜) hold for some β¯ > 1 ∨ α and β˜ > α.
For any stopping time τ ≤ T and β′ ∈ [0, β¯ ∧ β˜), there exists a D([0,T ],Cβ
′
loc(R
d1; Rd2))-
60 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
modification of Γ(τ, ) and Ψ(τ), also denoted by Γ¯(τ) and Ψ(τ), respectively. Moreover, for
all ϵ > 0 and p ≥ 2, there is a constant N = N(d1, d2, p,N0,T, β′, η, ϵ, θ) such that
E
[
sup
t≤T
|r−(θ+ϵ)1 Γt(τ)|
p
β′
]
+ E
[
sup
t≤T
|r−ϵ1 Ψt(τ)|
p
β′
]
≤ N. (3.3.17)
Proof. Let τ ≤ T be a fixed stopping time and β := β¯∧ β˜. Estimating (3.3.16) directly and
using the Burkholder-Davis-Gundy inequality and Lemma 3.4.1, we get
E
[
sup
t≤T
|Γt(x)|p
]
≤ NE
∫ (
|Γt(x)|p + r1(Xt(x))pθ|r−θ1 ft|
p
0
)
dt + E
(∫
r1(Xt(x))
2θ|r−θ1 gt|
2
0dt
)p/2
+E
(∫ ∫
D
∫
r1(H˜
−1
t (Xt(x)))
2θ|r−θ1 ht|
2
∞π(dz)dt
)p/2
+E
∫ ∫
D∪E
r1(H˜
−1
t (Xt(x)))
pθ|r−θ1 ht|
p
∞π(dz)dt
+E
(∫ ∫
E
r1(H˜
−1
t (Xt(x)))
θ|r−θ1 ht|∞π(dz)dt
)p
.
Then using multiplicative decomposition
ht(x, H˜
−1
t (Xt−(x), z), z) = r1(Xt−(x))
θ r1(H˜
−1
t (Xt−(x), z))
θ
r1(Xt(x))θ
ht(H˜−1t (Xt−(x), z), z)
r1(H˜−1t (Xt−(x), z))θ
,
Hölder’s inequality, Lemma 3.4.10 (1), Lemma 3.2 in [LM14c], and Gronwall’s inequality,
we get that for all x and y,
E
[
sup
t≤T
|Γt(x)|p
]
≤ Nr−θp1 (x),
where N = N(d1, p,N0,T, η, θ) is a positive constant. In a similar way, grouping terms in
the obvious manner and using Lemma 3.4.3 and Lemma 3.4.10 (3), we obtain
E
[
sup
t≤T
|Γt(x) − Γt(y)|p
]
≤ N(r−pθ1 (x) ∨ r
−pθ
1 (y))|x − y|
(β′∧1)p.
Now, assume that [β]− ≥ 1. As in the proof of Theorem 3.4 in [Kun04], it follows that
Ut = ∇Γt(τ, x) solves
dUt = (ct(Xt)Ut + ∇ct(Xt)∇XtΓt + ∇ ft(Xt)∇Xt) dt
+
∫
Z
ρt(H˜
−1
t (Xt−, z), z)Ut−[1D(z)q(dt, dz) + 1E(z)p(dt, dz)]
+
∫
Z
∇ρt(H˜−1t (Xt−, z), z)∇[H˜
−1
t (Xt−)]Γt−[1D(z)q(dt, dz) + 1E(z)p(dt, dz)]
+
∫
Z
∇ht(x, H˜−1t (Xt, z), z)∇[H˜
−1
t (Xt−)]][1D(z)q(dt, dz) + 1E(z)p(dt, dz)]
3.3. Proof of main theorems 61
+
(
υ
ϱ
t (Xt)Ut + ∇υ
ϱ
t (Xt)∇XtΓt + ∇g
ϱ
t (Xt)∇Xt
)
dwϱt τ < t ≤ T,
Ut = 0, t ≤ τ.
Recall that by Lemma 3.4.6, a function ϕ : Rd1 → Rn, n ≥ 1 satisfies |r−θϕ|β < ∞ if and
only if |r−θϕ|0, . . . , |r−θ∂γϕ|0, |γ| ≤ [β]−, and [r−θ∂γϕ]|{β}+ are finite. Estimating as above and
using Proposition 3.4 in [LM14c] and Lemma 3.4.10 we obtain that for all p ≥ 2, there is
a constant N = N(d1, d2, p,N0,T, θ) such that for all x and y,
E
[
sup
t≤T
|∇Γt(x)|p
]
≤ r−pθ1 (x)N
and
E
[
sup
t≤T
|∇Γt(x) − ∇Γt(y)|p
]
≤ N(r−pθ1 (x) ∨ r
−pθ
1 (y))|x − y|
((β−1)∧1)p.
Using induction we get that for all p ≥ 2 and all multi-indices γ with 0 ≤ |γ| ≤ [β]− and
all x,
E sup
t≤T
[|∂γΓt(x)|p] ≤ r
−pθ
1 (x)N,
and for all multi-indices γ with |γ| = [β]− and all x, y,
E
[
sup
t≤T
|∂γΓt(x) − ∂γΓt(y)|p
]
≤ N(r−pθ1 (x) ∨ r
−pθ
1 (y))|x − y|
(β−[β]−)p,
for a constant N = N(d1, d2, p,N0,T, β, η, θ). It is also clear that for all p ≥ 2 and all
multi-indices γ with 0 ≤ |γ| ≤ [β]− and all x,
E
[
sup
t≤T
|∂γΨt(x)|p
]
≤ N,
and for all multi-indices γ with |γ| = [β]− and all x, y,
E
[
sup
t≤T
|∂γΨt(x) − ∂γΨt(y)|p
]
≤ N|x − y|(β−[β]
−)p.
We obtain the existence of a D([0,T ],Cβ
′
loc(R
d1; Rd2))-modification of Γ(τ) and Ψ(τ) using
estimate (3.3.17) and Corollary 5.3 in [LM14c]. This completes the proof. □
Let Φ˜t(x) = Φ˜t(τ, x), (t, x) ∈ [0,T ] × Rd1 , be the solution of the linear SDE given by
dΦ˜t(x) =
(
ct(Xt(x))Φ˜t(x) + ft(Xt(x))
)
dt +
(
υ
ϱ
t (Xt(x))Φ˜t(x) + g
ϱ
t (Xt(x))
)
dwϱt
+
∫
Z
ρt(H˜
−1
t (Xt−(x), z), z)Φ˜t−(x, y)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)]
62 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
+
∫
Z
ht(H˜
−1
t (Xt−(x), z), z)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,
Φ˜t(x) = φ(x), t ≤ τ.
The following is a simple corollary of Lemma 3.3.2.
Corollary 3.3.3. Let Assumptions 3.3.2(β¯) and 3.3.3(β˜) hold for some β¯ > 1∨α and β˜ > α.
For any stopping time τ ≤ T andFτ⊗B(Rd1)-measurable random field φ such that for some
β′ ∈ [0, β¯∧ β˜), P-a.s. φ ∈ Cβ
′
loc(R
d1; Rd2), there is a D([0,T ];Cβ
′
loc(R
d1 ,Rd2))-modification of
Φ˜(τ), also denoted by Φ˜(τ), and P-a.s. for all (t, x) ∈ [0,T ] × Rd1 ,
Φ˜t(τ, x) = Ψt(x)φ(x) + Γt(x).
Moreover, if for some θ′ ≥ 0 and β′ ∈ [0, β¯ ∧ β˜), P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1; Rd2), then for all
ϵ > 0 and p ≥ 2, there is a constant N = N(d1, d2, p,N0,T, θ, θ′, β′, ϵ) such that
E
[
sup
t≤T
|r−(θ∨θ
′)−ϵ
1 Φ˜t(τ)|
p
β′
∣∣∣Fτ
]
≤ N(|r−θ
′
1 φ|
p
β′
+ 1). (3.3.18)
Let us now state our main result concerning degenerate SIDEs and their connection
with linear transformations of inverse flows of jump SDEs.
Proposition 3.3.4. Let Assumptions 3.3.2(β¯) and 3.3.3(β˜) hold for some β¯ > 1 ∨ α and
β˜ > α. For any stopping time τ ≤ T and Fτ ⊗ B(Rd1)-measurable random field φ such
that for some β′ ∈ (α, β¯ ∧ β˜) and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1; Rd2), we have that P-
a.s. Φ˜(τ, X−1(τ)) ∈ D([0,T ];Cβ
′
loc(R
d1; Rd2)) and vt(x) = vt(τ, x) = Φ˜t(τ, X−1t (τ, x)) solves
(3.3.15). Moreover, for all ϵ > 0 and p ≥ 2,
E
[
sup
t≤T
|r−(θ∨θ
′)−ϵ
1 vt(τ)|
p
β′
∣∣∣Fτ
]
≤ N(|r−θ
′
1 φ|
p
β′
+ 1), (3.3.19)
for a constant N = N(d1, d2, p,N0,T, β′, η, ϵ, θ, θ′).
Proof. Fix a stopping time τ ≤ T and random field φ such that for some β′ ∈ (α, β¯ ∧ β˜)
and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1 ; Rd2). By virtue of Corollary 3.3.3 and Theorem 2.1 in
[LM14c], P-a.s.
Φ˜(τ, X−1(τ)) ∈ D([0,T ];Cβ
′
loc(R
d1 ,Rd2)).
Then using the Ito-Wenzell formula (Proposition 3.4.16) and following a simple calcula-
tion, we obtain that vt(τ, x) := Φ˜t(τ, X−1t (τ, x)) solves (3.3.15). By Theorem 2.1 in [LM14c]
and Corollary 3.3.3, for all ϵ > 0 and p ≥ 2, there exists a constant N = N(d1, p,N0,T,
3.3. Proof of main theorems 63
β′, η, ϵ) such that
E
[
sup
t≤T
|r−(1+ϵ)1 X
−1
t (τ)|
p
β′
]
+ E
[
sup
t≤T
|r−ϵ1 ∇X
−1
t (τ)|
p
β′−1
]
≤ N. (3.3.20)
Therefore applying Lemma 3.4.9 and Hölder’s inequalty and using the estimates (3.3.20)
and (3.3.18), we procure (3.3.19), which completes the proof. □
3.3.4 Adding uncorrelated part (Proof of Theorem 3.2.2)
Proof of Theorem 3.2.2 . Fix a stopping time τ ≤ T and random field φ such that for some
β′ ∈ (α, β¯∧ β˜) and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1; Rd2). Consider the system of SIDEs given
by
dv˜lt =
(
(L1;lt +L
2;l
t )v˜t + 1[1,2](α)bˆ
i
t∂iu
l
t + cˆ
ll¯
t u
l¯
t(x) + fˆ
l
t
)
dt +
(
N1;lϱt v˜t + g
lϱ
t
)
dw1;ϱt
+N2;lϱt v˜tdw
2;ϱ
t +
∫
Z1
(
I1;lt,z v˜t− + h
l
t(z)
)
[1D1(z)q
1(dt, dz) + 1E1 p
1(dt, dz)]
+
∫
Z2
I2;lt,z v˜t−[1D2(z)q
2(dt, dz) + 1E2(z)p
2(dt, dz)] τ < t ≤ T,
v˜lt = φ
l, t ≤ τ, l ∈ {1, . . . , d2},
where for ϕ ∈ C∞c (R
d1; Rd2) and l ∈ {1, . . . , d2},
N2;lϱt ϕ(x) := 1{2}(α)σ
2;iϱ
t (x)∂iϕ
l(x) + υ2;ll¯ϱt (x)ϕ
l¯(x), ϱ ∈ N,
I2;lt,zϕ(x) := (I
ll¯
d2 + ρ
2;ll¯
t (x, z))ϕ
l¯(x + H2t (x, z)) − ϕ
l(x).
By Proposition 3.3.4, P-a.s. Φ(τ, X−1(τ)) ∈ D([0,T ];Cβ
′
loc(R
d1; Rd2)) and v˜t(τ, x) = Φt(τ,
X−1t (τ, x)) solves (3.3.15). We write vt(x) = vt(τ, x). Moreover, for all ϵ > 0 and p ≥ 2,
E
[
sup
t≤T
|r−(θ∨θ
′)−ϵ
1 v˜t(τ)|
p
β′
∣∣∣Fτ
]
≤ N(|r−θ
′
1 φ|
p
β′
+ 1), (3.3.21)
where N = N(d1, d2, p,N0,T, β′, η1, η2, ϵ, θ, θ′) is a positive constant. Without loss of gen-
erality we will assume that for all ω and t, |r−θ
′
1 φ|β′ ≤ N, since we can always multiply the
equation by indicator function. For each n ∈ N0, let Cnloc(R
d1; Rd2) be the separable Fréchet
space of n-times continuously differentiable functions f : Rd1 → Rd2 endowed with the
countable set of semi-norms given by
| f |n,k =
∑
0≤|γ|≤n
sup
|x|≤k
|∂γ f (x)|, k ∈ N.
64 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
Owing to Lemma 3.4.2, there is a the family of measures Etω(dU), (ω, t) ∈ Ω × [0,T ]
on D([0,T ]; C[β]
−
loc (R
d1; Rd2)), corresponding to A = v˜ such that for all bounded G : Ω ×
[0,T ]×[0,T ]×D([0,T ]; C[β]
−
loc (R
d1; Rd2))→ Rd2 that are OT ×B ([0,T ])×B(D([0,T ]; C
[β]−
loc
(Rd1; Rd2))) measurable, P-a.s. for all t, we have
Et[Gt(t, v˜)] =
∫
D([0,T ];C[β
′]−
loc (R
d1 ;Rd2 ))
Gt(t,U)E
t(dU) = E [Gt(t, v˜)|Ft] ,
where the right-hand-side is the càdlàg modification of the conditional expectation. Set
uˆt(x) = uˆt(τ, x) = E
t[v˜t(τ, x)] =
∫
D([0,T ];C[β
′]−
loc (R
d1 ;Rd2 ))
Ut(x)E
t(dU).
Let λ = (θ ∨ θ′) + ϵ. We claim that for all multi-indices γ with |γ| ≤ [β]−, P-a.s. for all t
and x,
∂γ[r−λ1 (x)uˆt(x)] =
∫
D([0,T ];C[β
′]−
loc (R
d1 ;Rd2 ))
∂γ[r−λ1 (x)Ut(x)]E
t(dU) = Et[∂γ[r−λ1 (x)v˜t(x)]].
Indeed, since
Mt = E
t
[
sup
s≤T
|∂γ[r−λ1 v˜s]|0
]
, t ∈ [0,T ],
is a (F,P) martingale, we have
E
[
sup
t≤T
|Mt|2
]
≤ 4E
[
|MT |2
]
≤ 4E
[
sup
t≤T
|∂γ[r−λ1 v˜t]|
2
0|
]
< ∞, (3.3.22)
which implies that P-a.s. for all t,
∫
D([0,T ];C[β
′]−
loc (R
d1 ;Rd2 ))
sup
s≤T,x∈Rd1
|∂γ[r−λ1 (x)Us(x)]|E
t(dU) = Et
[
sup
t≤T
|∂γ[r−λ1 v˜t]|0
]
< ∞.
Similarly, since E
[
supt≤T |r
−λ
1 v˜t|
2
β′
]
< ∞, P-a.s. for all x and y,
|∂γ[r−λ1 (x)uˆt(x)] − ∂
γ[r−λ1 (y)uˆt(y)]|
|x − y|{β′}+
≤ Et
[
|∂γ[r−λ1 (x)v˜t(x)] − ∂
γ[r−λ1 (y)v˜t(y)]|
|x − y|{β′}+
]
≤ Et[|r−λ1 v˜t|β′],
and thus, P-a.s.
sup
t≤T
|r−λ1 uˆt|β′ ≤ sup
t≤T
Et
[
sup
t≤T
|r−λ1 v˜t|β′
]
< ∞.
3.3. Proof of main theorems 65
Thus, P-a.s. r−λ1 (·)uˆ(τ) ∈ D([0,T ];C
β′(Rd1; Rd2)) and (3.2.4) follows from (3.3.21) (see the
argument (3.3.22)). For each l ∈ {1, . . . , d2}, letAlt(x) = A
l
t(z) be defined by
Alt = φ
l +
∫
]τ,τ∨t]
(
(L1;ls +L
2;l
s )uˆs + 1[1,2](α)bˆ
i
s∂iuˆ
l
s + cˆ
ll¯
s uˆ
l¯
s + fˆ
l
s
)
ds
+
∫
]τ,τ∨t]
(
N1;lϱs uˆs + g
lϱ
s
)
dw1;ϱs
+
∫
]τ,τ∨t]
∫
Z1
(
I1;ls,zuˆs− + h
l
s(z)
)
[1D1(z)q
1(ds, dz) + 1E1(z)p
1(ds, dz)].
By Theorem 12.21 in [Jac79], the representation property holds for (F,P), and hence every
bounded (F,P)- martingale issuing from zero can be represented as
Mt =
∫
]0,t]
oϱsdw
1;ϱ
s +
∫
]0,t]
∫
Z1
es(z)q
1(ds, dz), t ∈ [0,T ],
where
E
∫
]0,T ]
|os|2ds + E
∫
]0,T ]
∫
Z1
|es(z)|2π1(dz)ds < ∞.
Then for an arbitrary F-stopping time τ¯ ≤ T and bounded (F,P)- martingale, applying
Itô’s product rule and taking the expectation, we obtain
Ev˜τ¯(τ, x)M¯τ¯ = EAτ¯(x)M¯τ¯.
Since the optional projection is unique (see Theorem 13 in Chapter 1, Section 8 in [LS89]),
P-a.s. for all t and x, uˆt(x) = At(x). This completes the proof. □
3.3.5 Proof of Theorem 3.2.5
Proof of Theorem 3.2.5. Fix a stopping time τ ≤ T and random field φ such that for some
β′ ∈ (α, β¯ ∧ β˜) and θ′ ≥ 0, P-a.s. r−θ
′
1 φ ∈ C
β′(Rd1; Rd2). For any δ > 0, we can rewrite
(3.1.1) as
dult =
(
(L¯1;lt +L
2;l
t )ut + 1[1,2](α)b¯
i
t∂iu
l
t + c¯
ll¯
t u
l¯
t + f
l
t
)
dt +
(
N1;lϱt ut + g
lϱ
t
)
dw1;ϱt
+
∫
Z1
(
I¯1;lt,z ut− + h¯
l
t(z)
)
[1D1(z)q
1(dt, dz) + 1E1(z)p
1(dt, dz)]
+
∫
Z1
(
1(D1∪E1)∩{K1t >δ}(z) + 1V1(z)
) (
I1;lt,z ut− + h
l
t(z)
)
p1(dt, dz), τ < t ≤ T,
ult = φ
l, t ≤ τ, l ∈ {1, . . . , d2}, (3.3.23)
66 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
where for ϕ ∈ C∞c (R
d1; Rd2) and l ∈ {1, . . . , d2},
L¯1;lt ϕ(x) := 1{2}(α)
1
2
σ
1;iϱ
t (x)σ
1; jϱ
t (x)∂i jϕ
l(x) + 1{2}(α)σ
k;iϱ
t (x)υ
1;ll¯ϱ
t (x)∂iϕ
l¯(x)
+
∫
D1
ρ¯
1;ll¯
t (x, z)
(
ϕl¯(x + H¯1t (x, z)) − ϕ
l¯(x)
)
π1(dz)
+
∫
D1
(
ϕl(x + H¯1t (x, z)) − ϕ
l(x) − 1(1,2](α)H¯
1;i
t (x, z)∂iϕ
l(x)
)
π1(dz),
I¯1t,zϕ
l(x) = (Ill¯d2 + 1{K1t ≤δ}(z)ρ
1;ll¯
t (x, z))ϕ
l¯(x + 1{K1t ≤δ}(z)H
1
t (x, z)) − ϕ
l(x),
H¯1 := 1{K1t ≤δ}H
1, ρ¯1 := 1{K1t ≤δ}ρ
1, h¯ := 1{K1t ≤δ}h,
b¯it(x) := b
i
t(x) −
∫
D1∩{K1t >δ}
1(1,2](α)H
1;i
t (x, z)π
1(dz),
c¯ll¯t (x) := c
ll¯
t (x) −
∫
D1∩{K1t >δ}
ρ
1;ll¯
t (x, z)π
1(dz).
For an arbitrary stopping time τ′ ≤ T and Fτ′ ⊗ B(Rd1)-measurable random field φτ
′
:
Ω × Rd1 → Rd2 satisfying for some θ(τ′) > 0, P-a.s. r−θ(τ
′)
1 φ
τ′ ∈ Cβ
′
(Rd1; Rd2), consider the
system of SIDEs on [0,T ] × Rd1 given by
dvlt =
(
(L¯t
1;l
+L2;lt )vt + 1[1,2](α)b¯
i
t∂iv
l
t + c¯
ll¯
t v
l¯
t + f
l
t
)
dt +
(
N1;lϱt vt + g
lϱ
t
)
dw1;ϱt
+
∫
Z1
(
I¯1;lt,z ut− + h¯
l
t(z)
)
[1D1(z)q
1(dt, dz) + 1E1(z)p
1(dt, dz)], τ′ < t ≤ T,
vlt = φ
τ′;l, t ≤ τ′, l ∈ {1, . . . , d2}. (3.3.24)
Set H¯2 = H2 and ρ¯2 = ρ2. In order to invoke Theorem 3.2.2 and obtain a unique solution
vt = vt(τ′, x) = vt(τ′, φτ
′
, x) of (3.3.24), we will show that for all ω and t,
|r−11 b˜t|0 + |∇b˜t|β¯−1 + |c˜t|β˜ + |r
−θ f˜ |β˜ ≤ N0, (3.3.25)
where
b˜it(x) : = 1[1,2](α)b¯
i
t(x) −
2∑
k=1
1{2}(α)σ
k; jϱ
t (x)∂ jσ
k;iϱ
t (x)
−
2∑
k=1
∫
Dk
(
1(1,2](α)H¯
k;i
t (x, z) − H¯
k;i
t (
˜¯Hk;−1t (x, z), z)
)
πk(dz),
c˜ll¯t (x) : = c¯
ll¯
t (x) −
2∑
k=1
1{2}(α)σ
k;iϱ
t (x)∂iυ
k;ll¯ϱ
t (x)
3.3. Proof of main theorems 67
−
2∑
k=1
∫
Dk
(
ρ¯
k;ll¯
t (x, z) − ρ¯
k;ll¯
t (
˜¯Hk;−1t (x, z), z)
)
πk(dz),
f˜ lt (x) : = f
l
t (x) − σ
1; jϱ
t (x)∂ jg
l
t(x) −
∫
D1
(
h¯lt(x, z) − h¯
l
t(
˜¯H1;−1t (x, z), z)
)
π1(dz).
Owing to Assumption 3.2.3(β¯, δ1), we easily deduce that there is a constant N = N(d1, N0,
β¯) such that for each k ∈ {1, 2} and all ω and t,
|σk; jϱt ∂ jσ
k;ϱ
t |β¯ + |σ
k; jϱ
t ∂ ja
k;ϱ
t (x)|β¯ + |σ
1; jϱ
t ∂ jg
ϱ
t |β¯ ≤ N, if α = 2.
Since |∇H¯1t |0 ≤ δ, for any fixed η
1 < 1, for all (ω, t, x, z) ∈ {(ω, t, x, z) ∈ Ω × [0,T ] × Rd1 ×
(D1 ∪ E1) : |∇H¯1t (ω, x, z)| > η
1},
∣∣∣∣
(
Id1 + ∇H
1
t (ω, x, z)
)−1∣∣∣∣ ≤
1
1 − δ
.
Appealing to Assumption 3.2.3(β¯, δ1) and applying Lemma 3.4.10, we obtain that there is
a constant N = N(d1, d2,N0) such that for each k ∈ {1, 2} and all ω, t, and z,
|H¯k;it (z) − H¯
k;i
t (
˜¯Hk;−1t (z), z)|β¯ ≤ N(K
k
t (z) + K¯
k
t (z))
2 + N1(0,1]({β¯}+ + δk)K˜kt (z)K
k
t (z)
δk
+ N1(1,2]({β¯}+ + δk)
(
K˜kt (z)K
k
t (z)
δk + K¯kt (z)
2
)
,
|ρ¯kt (z) − ρ¯
k
t (
˜¯Hk;−1t (z), z)|β¯ ≤ Nl
k
t (z)(K
k
t (z) + K¯
k
t (z)) + N1(0,1]({β¯}
+ + µk)l˜kt (z)K
k
t (z)
µk
+ N1(1,2]({β¯}+ + µk)
(
l˜kt (z)K
k
t (z)
µk + lkt (z)K¯
k
t (z)
)
,
and
|r−θ1 h¯t(z) − r
−θ
1 h¯t(
˜¯H1;−1t (z), z)|β¯ ≤ Nl
1
t (z)(K
1
t (z) + K¯
k
t (z)) + N1(0,1]({β¯}
+ + µ1)l˜kt (z)K
1
t (z)
µ1
+ N1(1,2]({β¯}+ + µ1)
(
l˜kt (z)K
1
t (z)
µ1 + lkt (z)K¯
1
t (z)
)
.
Moreover, using Lemma 3.4.10, we find that there is a constant N = N(d1, d2,N0) such
that for each k ∈ {1, 2} and all ω, t, and z,
|r−11 H¯
k
t (
˜¯Hk;−1t (z), z)|0 ≤ |r
−1
1 H
k|0, |∇[H¯
k;i
t (
˜¯Hk;−1t (z), z)]|β¯ ≤ |∇H
k|β¯−1.
Combining the above estimates and using Hölder’s inequality and the integrability prop-
erties of lkt (z) and K
k
t (z), we obtain (3.3.25). Therefore, by Theorem 3.2.2, for each stop-
ping time τ′ ≤ T and and Fτ′ ⊗ B(Rd1)-measurable random field φτ
′
satisfying for some
θ(τ′) > 0, P-a.s. r−θ(τ
′)
1 φ
τ′ ∈ Cβ
′
(Rd1; Rd2), there exists a unique solution vt(x) = vt(τ′, φτ
′
, x)
68 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
of (3.3.24) such that
E
[
sup
t≤T
|r−θ(τ
′)∨θ−ϵ
1 vt(τ
′)|p
β′
∣∣∣Fτ′
]
≤ N(|r−θ(τ
′)
1 φ
τ′ |p
β′
+ 1), (3.3.26)
where N = N(d1, d2, p,N0,T, β′, η1, η2, ϵ, θ, θ(τ′)) is a positive constant. Let
At =
∫
]0,t]
∫
Z1
(
1(D1∪E1)∩{K1s>η1}(z) + 1V1(z)
)
p1(ds, dz), t ≤ T.
Define a sequence of stopping times (τn)n≥0 recursively by τ1 = τ and
τn+1 = inf(t > τn : ∆At , 0) ∧ T.
We construct the unique solution u = u(τ) of (3.3.23) in Cβ
′
(Rd1; Rd2) by interlacing solu-
tions of (3.3.24) along the sequence of stopping times (τn). For (ω, t) ∈ [[0, τ1)), we set
ut(τ, x) = vt(τ, φ, x) and note that
E
[
sup
t≤τ1
|r−θ
′∨θ−ϵ
1 ut(τ)|
p
β′
∣∣∣Fτ
]
≤ N(|r−θ
′
1 φ|
p
β′
+ 1).
For each ω and x, we set
uτ1(x) = uτ1−(x)
+
∫
Z1
(
1(D1∪E1)∩{K1>η1}(τ1, z) + 1V1(z)
) (
I1τ1,zuτ1−(x) + h
l
τ1
(x, z)
)
p1({τ1}, dz).
By virtue of Lemma 3.4.9, there is a constant N = N(d1, d2, θ, θ′, ζτ1(z), β
′)
|uτ1− ◦ H˜
1
τ1
(z) · r
−ξτ1 (z)(θ∨θ
′+ϵ+β′)
1 |β′ ≤ N|r
−θ∨θ′−ϵ
1 u
l
τ1−|β′ ,
and hence
|r−λ11 uτ1(x)|β′ ≤ N |r
−θ∨θ′−ϵ
1 u
l
τ1−|β′ + ζτ1(z),
where
λ1 = (ξτ1(z)(θ ∨ θ
′ + 1 + ϵ + β′)) ∨ θ ∨ (θ ∨ θ′ + ϵ).
We then proceed inductively, each time making use of the estimate (3.3.26), to obtain a
unique solution u = u(τ) of (3.3.23), and hence (3.1.1), in Cβ
′
(Rd1; Rd2). This completes
the proof of Theorem 3.2.5. □
3.4 Appendix
3.4. Appendix 69
3.4.1 Martingale and point measure moment estimates
Set (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz) = q1(dt, dz). The follow-
ing moment estimates are used to derive the estimates of Γt and Ψt in Lemma 3.3.2. The
notation a ∼
p,T
b is used to indicate that the quantity a is bounded above and below by a
constant depending only on p and T times b.
Lemma 3.4.1. Let h : Ω × [0,T ] × Z → Rd1 be PT ⊗Z-measurable
(1) For any stopping time τ ≤ T and p ≥ 2,
E
[
sup
t≤τ
∣∣∣∣∣∣
∫
]0,τ]
∫
Z
hs(z)q(ds, dz)
∣∣∣∣∣∣
p]
∼
p,T
E
[∫
]0,τ]
∫
Z
|hs(z)|p π(dz)ds
]
+ E


(∫
]0,τ]
∫
Z
|hs(z)|2 π(dz)ds
)p/2 .x
(2) For any stopping time τ ≤ T and p¯ ≥ 1,
E

sup
t≤τ
(∫
]0,τ]
∫
Z
|hs(z)|p(ds, dz)
)p¯ ∼
p,T
E
[∫
]0,τ]
∫
Z
|hs(z)| p¯ π(dz)ds
]
+ E


(∫
]0,τ]
∫
Z
|hs(z)|π(dz)ds
) p¯ ,
Proof. We will only prove part (2), since part (1) is well-known (see, e.g., [Nov75] or
[Kun04]). Assume that ht(ω, z) > 0 for all ω, t and z. Let
At =
∫
]0,t]
∫
Z
hs(z)p(ds, dz) and Lt =
∫
]0,t]
∫
Z
hs(z)π(dz)ds, t ≤ T.
It suffices to prove (2) for p > 1, since the case p = 1 is obvious. Fix an arbitrary stopping
time τ ≤ T and p > 1. For all t, we have
Apt =
∑
s≤t
[
(As− + ∆As)
p − Aps−
]
.
Thus, by the inequality
bp ≤ (a + b)p − ap ≤ p(a + b)p−1b ≤ p2p−1[ap−1b + bp], a, b ≥ 0,
we get
Apt ≥
∫
]0,t]
∫
Z
hs(z)
p p(ds, dz)
and
Apt ≤ p2
p−2
[∫
]0,t]
∫
h
hs(z)
p p(ds, dz) +
∫ t
0
∫
Z
Ap−1s− hs(z)p(ds, dz)
]
.
70 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
Since At is increasing, we obtain
E
∫
]0,τ]
∫
Z
hs(z)
p p(ds, dz) ≤ EApτ ≤ p2
p−2E
[∫
]0,τ]
∫
Z
hs(z)
p p(ds, dz) + Ap−1τ Lτ
]
.
It is easy to see that
E[Lpτ ] = pE
∫
]0,τ]
Lp−1s dLs = pE
∫
]0,τ]
Lp−1s dAs ≤ pE[L
p−1
τ Aτ].
Applying Young’s inequality, for any ε > 0, we get
Ap−1τ Lτ ≤ εA
p
τ +
(p − 1)p−1
εp−1 pp
Lpτ and L
p−1
τ Aτ ≤ εL
p
τ +
(p − 1)p−1
εp−1 pp
Apτ .
Combining the estimates for any ε1 ∈ (0, 1p ), we have


ε
p−1
1 p
p(1 − pε1)
p(p − 1)p−1
ELpτ

 ∨ E
∫
]0,τ]
∫
Z
hs(z)
p p(ds, dz) ≤ E[Apτ ].
and for any ε2 ∈ (0, 1p2p−2 )
E[Apτ ] ≤
p2p−2
(1 − p2p−2ε2)
E


∫
]0,τ]
∫
Z
hs(z)
p p(ds, dz) +
(p − 1)p−1
ε
p−1
2 p
p
Lpτ

 ,
which completes the proof. □
3.4.2 Optional projection
The following lemma concerning the optional projection plays an integral role in Sec-
tion 3.3.4 and the proof of Theorem 3.2.2. For more information on the Skorokhod J1-
topology, we refer the reader to Chapter 6, Section 1 of [LS89]. Also, we refer the reader
to Theorem 5.3 [Kal97] for the construction of regular conditional probability measures
on Borel spaces.
Lemma 3.4.2. (cf. Theorem 1 in [Mey76]) Let X be a Polish space and D ([0,T ];X)
be the space of X-valued càdlàg trajectories with the Skorokhod J1-topology. If A is a
random variable taking values in D ([0,T ];X), then there exists a family of B([0,T ])×F -
measurable non-negative measures Et(dU), (ω, t) ∈ Ω × [0,T ], on D ([0,T ];X) and a
random-variable ζ satisfying P (ζ < T ) = 0 such that Et(D ([0,T ];X)) = 1 for t < ζ
and Et(D([0,T ];X)) = 0 for t ≥ ζ. In addition, Et is càdlàg in the topology of weak
convergence, Et = Et+ for all t ∈ [0,T ], and for each continuous and bounded functional
F on D ([0,T ];X) , the process Et (F) is the càdlàg version of E[F (A) |Ft]. If G : Ω ×
3.4. Appendix 71
[0,T ] × [0,T ] × D ([0,T ];X) → Rd2 is bounded and O × B ([0,T ]) × B (D ([0,T ];X))-
measurable, then ∫
D([0,T ];X)
Gt(ω, t,U)E
t(dU) = Et(Gt)
is the optional projection of Gt(A) = Gt(ω, t,A). Furthermore, if G = Gt(ω, t,U) is
bounded and P × B([0,T ]) × B(D([0,T ];X))-measurable, then Et−(Gt) is the predictable
projection of Gt(A) = Gt(ω, t,A).
Proof. We follow the proof of Theorem 1 in [Mey76]. Since D([0,T ];X) is a Polish space,
for each t ∈ [0,T ], there is family of probability measures E˜tω(dw), ω ∈ Ω, on D([0,T ];X)
such that for each A ∈ B(D([0,T ];X)), E˜t(A) is Ft-measurable and P-a.s.
P (A ∈ A|Ft) = E˜t (A) .
For each ω ∈ Ω, let I (ω) be the set of all t ∈ (0,T ] such that for any bounded continuous
function F on D(([0,T ];X), the function
r 7→ E˜rω(F) =
∫
D([0,T ];X)
F(w)E˜r(dw)
has a right-hand limit on [0, s) ∩ Q and a left-hand limit on (0, s] ∩ Q for every rational
s ∈ [0,T ] ∩ Q. Let ζ (ω) = sup (t : t ∈ I(ω)) ∧ T. It is easy to see that P (ξ < T ) = 0.
We set E˜tω = 0 if ξ(ω) < t ≤ T . The function E˜
t
ω has left-hand and right-hand limits for
all t ∈ Q ∩ [0,T ]. We define Etω = E˜
t+
ω for each t ∈ [0,T ) (the limit is taken along the
rationals), and ETω is the left-hand limit at T along the rationals. The statement follows by
repeating the proof of Theorem 1 in [Mey76] in an obvious way. □
3.4.3 Estimates of Hölder continuous functions
In the coming lemmas, we establish some properties of weighted Hölder spaces that are
used Section 3.3.5 and the proof of Theorem 3.2.5.
Lemma 3.4.3. Let β ∈ (0, 1] and θ1, θ2 ∈ R with θ1 − θ2 ≤ β.
(1) There is a constant c1 = c1 (θ2, β) such that for all ϕ : Rd1 → R with |r
−θ1
1 ϕ|0 +
[r−θ21 ϕ]β =: N1 < ∞,
|ϕ(x) − ϕ(y)| ≤ c1N1(r1(x)θ2 ∨ r1(y)θ2)|x − y|β,
for all x, y ∈ Rd1 .
72 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
(2) Conversely, if ϕ : Rd1 → R satisfies |r−θ11 ϕ|0 < ∞ and there is a constant N2 such that
for all x, y ∈ Rd1 ,
|ϕ(x) − ϕ(y)| ≤ N2(r1(x)θ2 ∨ r1(y)θ2)|x − y|β,
then
[r−θ21 ϕ]β ≤ c1|r
−θ1
1 ϕ|0 + N2.
Proof. (1) For all x, y with r1 (x)
θ2 ≥ r1 (y)θ2 , we have
|ϕ(x) − ϕ(y)| ≤ r1(x)θ2[r
−θ2
1 ϕ]β|x − y|
β + r1(y)
θ1−θ2 |r−θ11 ϕ|0|r
θ2
1 (x) − r1(y)
θ2 |
≤ ([r−θ21 ϕ]β + c1|r
−θ
1 ϕ|0)r1(x)
θ2 |x − y|β,
where c1 := 1 + supt∈(0,1)
1−tθ2
(1−t)β if θ2 ≥ 0 and c1 := 1 + supt∈(1,∞)
(tθ2−1)tβ
(t−1)β if θ2 < 0, which
proves the first claim. (2) For all x and y with r1(x)θ2 > r1(y)θ2 , we have
|r1(x)−θ2ϕ(x) − r1(y)−θ2ϕ(y)|
≤ r1(x)−θ2 |ϕ(x) − ϕ(y)| + r1(y)θ1−θ2 |r
−θ1
1 (y)ϕ(y)||r1(y)
θ2r1(x)
−θ2 − 1|
≤ (c1|r−θ1ϕ|0 + N2)|x − y|β,
which proves the second claim. □
Lemma 3.4.4. Let β, µ ∈ (0, 1] and θ1, θ2, θ3, θ4 ∈ R with θ1 − θ2 ≤ β, θ3 − θ4 ≤ µ, and
θ3 ≥ 0. If ϕ : Rd1 → R and H : Rd1 → Rd1 are such that
|r−θ11 ϕ|0 + [r
−θ2
1 ϕ]β =: N1 < ∞ and |r
−θ3
1 H|0 + [r
−θ4
1 H]µ =: N2 < ∞,
then
|ϕ ◦ H · r−θ1θ31 |0 ≤ |r
−θ1
1 ϕ|0(1 + |r
−θ3
1 H|0) ≤ N1 (1 + N2)
θ1
and there is a constant N = N(β, µ, θ1, θ2) such that
[ϕ ◦ H · r−θ2θ3−βθ41 ]βµ ≤ NN1(1 + N2)
θ2+β.
Proof. For all x, we have
r1(H(x)) ≤ (1 + |r
−θ3
1 H|0)r1(x)
θ3 ≤ (1 + N2)r1(x)θ3 ,
and hence
|ϕ ◦ H · r−θ1θ31 |0 ≤ |r
−θ1
1 ϕ|0|r
θ1
1 ◦ H · r
−θ1θ3
1 |0 ≤ N1(1 + N2)
θ1 .
3.4. Appendix 73
Using Lemma 3.4.3, for all x and y, we get
|ϕ(H(x)) − ϕ(H(y))| ≤ NN1(r1(H(x)) ∨ r1(H(y)))θ2 |H(x) − H(y)|β
≤ NN1(1 + N2)θ2(r1(x) ∨ r1(y))θ2θ3 N
β
2 (r1(x) ∨ r1(y))
βθ4 |x − y|βµ
≤ NN1(1 + N2)θ2+β(r1(x) ∨ r1(y))θ2θ3+βθ4 |x − y|βµ,
for some constant N = N(β, µ, θ1, θ2). Noting that
θ1θ3 − θ2θ3 − βθ4 = (θ1 − θ2)θ3 − βθ4 ≤ β(θ3 − θ4) ≤ βµ,
we apply Lemma 3.4.3 to complete the proof. □
Remark 3.4.5. Let β ∈ (0, 1] and θ1, θ2 ∈ R. Then there is a constant N = N(β, θ1, θ2) such
that for all ϕ : Rd1 → R with |r−θ11 ϕ|0 + [r
−θ2
1 ϕ]β =: N1 < ∞, we have |r
−θϕ|β ≤ NN1, where
θ = max {θ1, θ2} . In particular, if in Lemma 3.4.4, θ1 = θ2 and θ4 ≥ 0, then
|ϕ ◦ H · r−θ1θ3−βθ4 |βµ ≤ NN1(1 + N2)θ1+β.
Proof. If θ2 ≥ θ1, then the claim is obvious and if θ1 > θ2, for all x and y, we find
|r1(x)−θ1ϕ(x) − r1(y)−θ1ϕ(y)| ≤ r1(x)θ2−θ1 |r1(x)−θ2ϕ(x) − r1(y)−θ2ϕ(y)|
+
∣∣∣∣∣∣
r(y)θ1−θ2
r(x)θ1−θ2
− 1
∣∣∣∣∣∣
|r−θ11 ϕ|0 ≤ N1(1 + c1)|x − y|
β,
where c1 := supt∈(0,1)
1−tθ1−θ2
(1−t)β
. □
Lemma 3.4.6. For any θ ≥ 0 and β > 1 , there are constants N1 = N1(d1, θ, β) and
N2(d1, θ, β) such that for all ϕ : Rd1 → R,
N1|r−θ1 ϕ|β ≤
∑
|γ|≤[β]−
|r−θ1 ∂
γϕ|0 +
∑
|γ|=[β]−
|r−θ1 ∂
γϕ|{β}+ ≤ N2|r−θ1 ϕ|β. (3.4.1)
Proof. For any multi-index γ with |γ| ≤ [β]− and x, we have
∂γ(r−θ1 ϕ)(x) =
∑
γ1+γ2+=γ
|γ1 |≥1
r1(x)
θ∂γ1(r−θ1 )(x)r1(x)
−θ∂γ2ϕ(x) + r1(x)
−θ∂γϕ(x).
It is easy to show by induction that for all multi-indices γ, |rθ1∂
γ(r−θ1 )|1 < ∞. Moreover, for
all multi-indices γ with |γ| < [β]−,
|r−θ1 ∂
γϕ|1 ≤ |∇(r−θ1 ∂
γϕ)| ≤ |r−θ1 ∇(r
−θ
1 )|0|r
−θ
1 ∂
γ∇ϕ|0.
74 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
Thus, for any multi-index γ with |γ| ≤ [β]−,
|∂γ(r−θ1 ϕ)|0 ≤
∑
γ1+γ2+=γ
|γ1 |≥1
|rθ1∂
γ1(r−θ1 )|0|r
−θ
1 ∂
γ2ϕ|0 + |r−θ1 ∂
γϕ|0
and for any multi-index γ with |γ| = [β]−,
|∂γ(r−θ1 ϕ)|{β}+ ≤
∑
γ1+γ2+=γ
|γ1 |≥1
|rθ1∂
γ1(r−θ1 )|1|r
−θ
1 ∇(r
−θ
1 )|0|r
−θ
1 ∂
γ2∇ϕ|0 + |r−θ1 ∂
γϕ|0.
This proves the leftmost inequality in (3.4.1). For all i ∈ {1, . . . , d} and x,
r−θ1 ∂iϕ(x) = ∂i(r
−θ
1 ϕ)(x) − r1(x)
−θϕ(x)r1(x)
θ∂i(r
−θ
1 )(x).
It follows by induction that for all multi-indices γ with |γ| ≤ [β]− and x, r−θ1 ∂
γϕ(x) is
a sum of ∂γ(r−θ1 ϕ)(x), a finite sum of terms, each of which is a product of one term of
the form ∂γ˜(r−θ1 ϕ)(x), |γ˜| < |γ|, and a finite number of terms of the form ∂
γ1(rθ1)∂
γ2(r−θ1 ),
|γ1|, |γ2| ≤ |γ|. Since for all multi-indices γ1 and γ2, we have |∂γ1(rθ1)∂
γ2(r−θ1 )|1 < ∞, the
rightmost inequality in (3.4.1) follows. □
Corollary 3.4.7. For any θ ≥ 0 and β > 1 , there are constants N1 = N1(d1, θ, β) and
N2(d1, θ, β) such that for all ϕ : Rd1 → R,
N1|r−θ1 ϕ|β ≤ |r
−θ
1 ϕ|0 +
∑
|γ|=[β]−
|r−θ1 ∂
γϕ|{β}+ ≤ N2|r−θ1 ϕ|β.
Proof. It is well known that for an arbitrary unit ball B ⊂ Rd1 and any 1 ≤ k < [β]−, there
is a constant N such that for any ε > 0,
sup
x∈B,|γ|=k
|∂γϕ| ≤ N(ε sup
x∈B,|γ|=[β]−
|∂γϕ(x)| + ε−k sup
x∈B
|ϕ(x)|).
Let U0 = {x ∈ Rd1 : |x| ≤ 1} and U j = {x ∈ Rd1 : 2 j−1 ≤ |x| ≤ 2 j}, j ≥ 1. For all j, we have
sup
x∈U j,|γ|=k
|∂γϕ(x)| = sup
B⊆U j
sup
x∈B,|γ|=k
|∂γϕ(x)|
≤ N(ε sup
B⊆U j
sup
x∈B,|γ|=[β]−
|∂γϕ(x)| + ε−k sup
B⊆U j
sup
x∈B
|ϕ(x)|)
≤ N(ε sup
x∈U j,|γ|=[β]−
|∂γϕ(x)| + ε−k sup
x∈U j
|ϕ(x)|).
3.4. Appendix 75
Since for every j,
2−θ/22− jθ sup
x∈U j,|γ|=k
|∂γϕ(x)| ≤ sup
x∈U j,|γ|=k
|r−θ∂γϕ(x)| ≤ 2
θ
2−( j−1)θ sup
x∈U j,|γ|=k
|∂γϕ(x)|,
we see that
2−θ/2 sup
j
2− jθ sup
x∈U j,|γ|=k
|∂γϕ(x)| ≤ sup
j
sup
x∈U j,|γ|=k
|r−θ∂γϕ(x)| = |r−θ∂γϕ|0
≤ 2θ sup
j
2− jθ sup
x∈U j,|γ|=k
|∂γϕ(x)|,
and the statement follows. □
Remark 3.4.8. If ϕ : Rd1 → R is such that |r−θ1ϕ|0 + |r−θ2∇ϕ|0 < ∞ for θ1, θ2 ∈ R with
θ1 − θ2 ≤ 1, then
[r−θ2ϕ]1 ≤ N(|r−θ1ϕ|0 + |r−θ2∇ϕ|0)
Proof. Indeed, for all x and y, we have
|ϕ(x) − ϕ(y)| ≤ |r−θ2∇ϕ|0
∫ 1
0
rθ2(x + s(y − x))ds|y − x|
≤ |r−θ2∇ϕ|0(r(y)θ2 ∨ r(x)θ2)|y − x|,
and hence the claim follows from Lemma 3.4.3. □
Lemma 3.4.9. Let n ∈ N, β, µ ∈ (0, 1], θ3, θ4 ≥ 0 be such that θ3 − θ4 ≤ 1. There is a
constant N = N(d1, θ1, θ3, θ4, n, β) such that for all ϕ : Rd1 → R with r
−θ1
1 ϕ ∈ C
n+β(Rd1 ,Rd1)
and H : Rd1 → Rd1 with
|r−θ31 H|0 + |r
−θ4
1 ∇H|n−1+µ =: N2 < ∞,
we have
|ϕ ◦ H · r−θ1θ3 |0 ≤ |r
−θ1
1 ϕ|0(1 + |r
−θ3
1 H|0)
θ1
and
|r−θ1θ3−θ4(n+µ∧β)1 ∇(ϕ ◦ H)|n−1+µ∧β ≤ N|r
−θ1
1 ϕ|n+β(1 + N2)
θ1+µ∧β+n.
Proof. It follows immediately from Lemma 3.4.4 and Remark 3.4.8 that
|ϕ ◦ H · r−θ1θ3 |0 ≤ |r
−θ1
1 ϕ|0(1 + |r
−θ3
1 H|0)
θ1 .
Using induction, we get that for all x and |γ| = n,
∂γ(ϕ(H(x))) = Iγ1(x) + I
γ
2(x) + I
γ
3 (x),
76 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
where
Iγ1(x) =
d1∑
i=1
∂iϕ(H(x))∂
γHi(x)
Iγ2(x) is a finite sum of terms of the form
∂i1 · · · ∂i|γ|ϕ(H(x))∂
γ˜1 Hi1 · · · ∂γ˜|γ|Hi|γ|
with i1, . . . , i|γ| ∈ {1, 2, . . . , d}, |γ˜1| = · · · = |γ˜|γ|| = 1, and
∑|γ|
k=1 γ˜k = γ, if n ≥ 2 and zero
otherwise, and where Iγ3(x) is a finite sum of terms of the form
∂i1 · · · ∂ikϕ(H(x))∂
γ˜1 Hi1(x) · · · ∂γ˜k Hik(x)
with 2 ≤ k < n, i1, i2, . . . , ik ∈ {1, . . . , d}, and
∑k
j=1 γ˜ j = γ, 1 ≤ |γ˜ j| < |γ|, if n ≥ 3, and zero
otherwise. Thus, owing to Lemmas 3.4.4 and 3.4.6, for any multi-index γ with |γ| = n, we
have
|r−θ3θ1−θ41 I
γ
1 |0 ≤ N |r
−θ1
1 ∇ϕ|0(1 + |r
−θ3
1 H|0)
θ1 |r−θ41 ∂
γH|0,
|r−θ3θ1−nθ41 I
γ
2 |0 ≤ N|r
−θ1
1 ∂
γϕ|0(1 + |r
−θ3
1 H|0)
θ1 |r−θ41 ∇H|
n
0,
and
|r−θ3θ1−(n−1)θ41 I
γ
3 |0 ≤ N |r
−θ1
1 ϕ|n−1(1 + |r
−θ3
1 H|0 + |r
−θ4
1 ∇H|)
θ1+n−1,
and hence
|r−θ1θ3−nθ4∂γ(ϕ ◦ H)|0 ≤ N |r
−θ1
1 ϕ|n(1 + |r
−θ3
1 H|0 + |r
−θ4
1 ∇H|)
θ1+n.
Appealing again to Lemmas 3.4.4 and 3.4.6, for all multi-indices γ with |γ| = n, we get
|r−θ1θ3−(1+µ∧β)θ41 I
γ
1 |µ∧β ≤ N|r
−θ1
1 ϕ|1+µ∧β (1 + N2)
θ1+µ∧β+1 ,
|r−θ1θ3−(n+µ∧β)θ41 I
γ
2 |µ∧β + |r
−θ1θ3−(n−1+µ∧β)θ4
1 I
γ
3 |µ∧β ≤ N|r
−θ1
1 ϕ|n+µ∧β (1 + N2)
θ1+n+µ∧β .
Then applying Lemmas 3.4.4 and 3.4.6, we complete the proof. □
We shall now provide some useful estimates of composite functions of diffeomor-
phisms.
Lemma 3.4.10. Let H : Rd1 → Rd1 be continuously differentiable and assume that for all
x ∈ Rd1 ,
|H(x)| ≤ L0 + L1|x| and |∇H(x)| ≤ L2.
Assume that for all x ∈ Rd1 , κ(x) = (Id1 + ∇H(x))
−1 exists and |κ(x)| ≤ Nκ.
(1) Then the mapping H˜(x) := x + H(x) is a diffeomorphism with H˜−1(x) = x−H(H˜−1(x))
3.4. Appendix 77
=: x + F(x) and for all x ∈ Rd1 ,
|F(x)| ≤ L0 + L1L0Nκ + L1Nκ|x|, |∇F(x)| ≤ NκL2, |
(
Id1 + ∇F(x)
)−1 | ≤ 1 + L2.
For all p ∈ R, there is a constant N = N(L0, L1,Nκ, p) such that for all x ∈ Rd1 ,
rp1 (H˜(x))
rp1 (x)
+
rp1 (H˜
−1(x))
rp1 (x)
≤ N, r−11 (x)|H
i(x) + Fk;i(x)| ≤ N[H]1|r−11 H|0.
Moreover, there is a constant N = N(L0, L1,Nκ, p) such that
∣∣∣∣∣∣
rp1 (H˜)
rp1
− 1 + 1(1,2](α)pHir−21 x
i
∣∣∣∣∣∣
α
+
∣∣∣∣∣∣
rp1 (H˜
−1)
rp1
− 1 − 1(1,2](α)pF ir−21 x
i
∣∣∣∣∣∣
α
≤ N(|r−11 H|
[α]−+1
0 + [H]
[α]−+1
1 ).
(2) If for some β > 1, |∇H|β−1 ≤ L3, then there is a constant N = N(d1, β,Nκ, L3) such that
|∇F|β−1 ≤ N |∇H|β−1. (3.4.2)
(3) If for some β ≥ 1, |∇H|β−1 ≤ L3, then for all θ ≥ 0, there is a constant N = N(d1, β,Nκ,
L1, L3, θ) such that
∣∣∣∣∣∣
rθ1 ◦ H˜
−1
rθ1
− 1
∣∣∣∣∣∣
β
≤ N(|r−11 H|0 + |∇H|β−1).
(4) If |H|0 ≤ L4, and for some β > 0, |∇H|β∨1−1 ≤ L5 and ϕ : Rd1 → R is such that
for some µ ∈ (0, 1] and θ ≥ 0, r−θ1 ϕ ∈ C
β+µ(Rd1; R), then there is a constant N =
N(d1, β, µ,Nκ, L4, L5, θ) such that
|r−θ1 (ϕ ◦ H˜
−1 − ϕ)|β ≤ N |r−θ1 ϕ|β(|H|0 + |∇H|β∨1−1)
+N1(0,1]({β}+ + µ)
∑
|γ|=[β]−
[∂γ(r−θ1 ϕ)]{β}++µL
µ
4
+N1(1,2]({β}+ + µ)
∑
|γ|=[β]−
(
[∇∂γ(r−θ1 ϕ)]{β}++µ−1L
µ
4 + |∇∂
γ(r−θ1 ϕ)|0|∇H|0
)
.
Proof. (1) Since (Id1 +∇H(x))
−1 exists for all x, it follows from Theorem 0.2 in [DMGZ94]
that the mapping H˜ is a global diffeomorphism. For all x, we easily verify H˜−1(x) =
x − H(H˜−1(x)) by substituting H˜(x) into the expression. Simple computations show that
for all x, we have
|∇H˜(x)| ≤ 1 + L2, |∇H˜−1(x)| = |κ(H˜−1(x))| ≤ Nκ,
78 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
|∇F(x)| = |∇H(H˜−1(x))∇H˜−1(x)| ≤ NκL2,
|(Id1 + ∇F(x))
−1| = |∇H˜−1(x)−1| = |κ(H˜−1(x))−1| = |Id1 + ∇H(H˜
−1(x))| ≤ 1 + L2.
For all x and y, we easily obtain
|H˜(x) − H˜(y)| ≤ (1 + L2)|x − y|, |H˜−1(x) − H˜−1(y)| ≤ Nκ|x − y|,
and hence
N−1κ |x − y| ≤ |H˜(x) − H˜(y)|, (1 + L2)
−1|x − y| ≤ |H˜−1(x) − H˜−1(y)|. (3.4.3)
Making use of (3.4.3), for all x, we get
N−1κ |x| ≤ L0 + |H˜(x)|, |H˜
−1(x)| ≤ NκL0 + Nκ|x|, |x| ≤ L0 + L1|H˜−1(x)|,
and thus
|F(x)| ≤ L0 + L1NκL0 + L1Nκ|x|.
The rest of the estimates then follow easily from the above estimates and Taylor’s theorem.
(2) Using the chain rule, for all x, we obtain
∇F(x) = −∇H(H˜−1(x))∇H˜−1(x) = −∇H(H˜−1(x))κ(H˜−1(x)), (3.4.4)
and hence |∇F|0 ≤ Nκ|∇H|0. For all x and y, we have
κ(H˜−1(y)) − κ(H˜−1(x)) = κ(y)[∇H(H˜−1(x)) − ∇H(H˜−1(y))]κ(x),
and thus since [H˜−1]1 ≤ (1 + NκL3) by part (1), we have for all δ ∈ (0, 1 ∧ β],
[κ(H˜−1)]δ ≤ N2κ (1 + NκL3)
δ[∇H]δ.
It follows that there is a constant N = N(Nκ, L3) such that for all δ ∈ (0, 1 ∧ β],
|∇F|δ ≤ N|H|δ.
It is well-known that the inverse map I on the set of invertible d1×d1 matrices is infinitely
differentiable and for each n, there exists a constant N = N(n, d1) such that for all invertible
matrices A, the n-th derivative of I evaluated at A, denoted I(n)(A), satisfies
|I(n)(A)| ≤ N |A−n−1| ≤ N |A−1|n+1.
3.4. Appendix 79
Using induction we find that for all multi-indices γ with |γ| ≤ [β]− and for all x, ∂γF(x) is
a finite sum of terms, each of which is a finite product of
∂γ¯H(H˜−1(x)), κ(H˜−1(x))n¯, I(n¯−1)(I + ∇H(H˜−1(x))), |γ¯| ≤ |γ|, n¯ ∈ {1, . . . , |γ|}.
Therefore, differentiating (3.4.4) and estimating directly we easily obtain (3.4.2).
(3) For each x, we have
r1(H˜−1(x))θ
r1(x)θ
− 1 = r1(x)−θ
∫ 1
0
r1(Gs(x))
θ−2Gs(x)
∗F(x)ds
=
∫ 1
0
rθ−11 (Gs(x))
r1(x)θ−1
K(Gs(x))
∗dsr1(x)
−1F(x),
where Gs(x) := x + sF(x), s ∈ [0, 1], and J(x) := r1(x)−1x. According to part (1) and (2),
we have |r−11 F|0 ≤ N|r
−1
1 H|0 and |∇F|β−1 ≤ N|∇H|β−1, and hence
|r−11 Gs|0 ≤ N(1 + |r
−1
1 H|0), |∇Gs(x)|β−1 ≤ N(1 + |∇H|β−1).
and
|J ◦Gs|β ≤ N(1 + |r−11 H|0 + |∇H|β−1),
for some constant N independent of s. Moreover, using Lemma 3.4.9 we find
|rθ−11 ◦Gs · r
1−θ
1 |β ≤ N
(
1 + |r−11 H|0 + |∇H|β−1
)θ+β
.
The statement then follows.
(4) First, we will consider the case θ = 0. By part (1), we have that for all µ¯ ∈ (0, (β+µ)∧1],
|ϕ ◦ H˜−1 − ϕ|0 ≤ [ϕ]µ¯|H ◦ H˜−1|
µ
0 ≤ [ϕ]µ¯|H|
µ¯
0.
First, let us consider the case β ≤ 1. For each x, let J(x) = ϕ(H˜−1(x))−ϕ(x). For all x and
y, it is clear that
|J(x) − J(y)| ≤ A(x, y) + B(x, y) + C(x, y),
where
A(x, y) := |J(x)|1[L4,∞)(|x − y|), B(x, y) := |J(y)|1[L4,∞)(|x − y|),
and
C(x, y) := |J(x) − J(y)|1[0,L4)(|x − y|).
Moreover, owing to part (1), if β + µ ≤ 1, then for all x and y, we have
A(x, y) ≤ [ϕ]β+µL
β+µ
4 1[L4,∞)(|x − y|) ≤ [ϕ]β+µL
µ
4 |x − y|
{β}+ ,
80 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
B(x, y) ≤ [ϕ]β+µL
µ
4 |x − y|
β,
and
C(x, y) ≤ [ϕ]β+µ|[H˜−1]
β+µ
1 |x − y|
β+µ1[0,L4)(|x − y|) + [ϕ]β+µ|x − y|
β+µ1[0,L4)(|x − y|)
≤ N[ϕ]β+µL
µ
4 |x − y|
β
for some constant N = N(µ,Nκ, L4). Using the identity
J(x) − J(y)
= −
∫ 1
0
(
∇ϕ
(
x − θH(H˜−1(x))
)
− ∇ϕ
(
y − θH(H˜−1(y))
))
H(H˜−1(x))dθ
−
∫ 1
0
∇ϕ
(
y − θH(H˜−1(y))
)
(H(H˜−1(y)) − H(H˜−1(x))),
and part (1), if β + µ > 1, then there is a constant N = N(µ,Nκ, L4) such that for all x and
y,
|J(x) − J(y)|1[L4,∞)(|x − y|) ≤ N([∇ϕ]β+µ−1|x − y|
β+µ−1L4
+ |∇ϕ|0|x − y|[H]1)1[L4,∞)(|x − y|)
≤ N[∇ϕ]β+µ−1L
µ
4 |x − y|
β + N|∇ϕ|0|∇H|0|x − y|.
Moreover, since
J(x) − J(y)
=
∫ 1
0
∇ϕ
(
H˜−1(x + θ(y − x))
) (
∇H˜−1(x + θ(y − x)) − Id1
)
(x − y)dθ
+
∫ 1
0
(
∇ϕ
(
H˜−1 (x + θ(y − x))
)
− ∇ϕ (x + θ(y − x))
)
(x − y)dθ,
by part (1) and (3.4.2), if β + µ > 1, we attain that there is a constant N = N(µ,Nκ, L4)
such that for all x and y,
|J(x) − J(y)|1[0,L4)(|x − y|) ≤ (|∇ϕ|0|∇H|0 + [∇ϕ]β+µ−1L
β+µ−1
4 )|x − y|1[0,L4)(|x − y|)
≤ |∇ϕ|0|∇H|0|x − y| + [∇ϕ]β+µ−1L
µ
4 |x − y|
β.
Combining the above estimates, we get that for all β ≤ 1 and µ ∈ (0, 1], there is a constant
N = N(µ,Nκ, L4) such that
[ϕ ◦ H˜−1 − ϕ]β ≤ N1[0,1](β+ µ)[ϕ]β+µL
µ
4 + N1(1,2](β+ µ)
(
[∇ϕ]β+µ−1 + |∇ϕ|0|∇H|0
)
. (3.4.5)
3.4. Appendix 81
This proves the desired estimate for β ≤ 1 and θ = 0. We now consider the case β > 1.
For β > 1, it is straightforward to prove by induction that for all multi-indices γ with
1 ≤ |γ| ≤ [β]− and for all x,
∂γ(ϕ(H˜−1))(x) = Jγ1 (x) +J
γ
2 (x) +J
γ
3 (x) +J
γ
4 (x),
where
Jγ1 (x) := ∂
γϕ(H˜−1(x)),
Jγ2 (x) = ∂
γϕ(H˜−1)(∂1H˜
−1;1)γ1 · · · (∂dH˜−1;d)γd − 1,
Jγ3 (x) is a finite sum of terms of the form
∂ j1 · · · ∂ jkϕ(H˜
−1(x))∂γ˜1 H˜−1; j1(x) · · · ∂γ˜k H˜−1; jk(x)
with 1 ≤ k < [β]−, j1, . . . , jk ∈ {1, . . . , d}, and
∑k
j=1 γ˜ j = γ, and J4(x) is a finite sum of
terms of the form
∂ j1 . . . ∂ j[β]−ϕ(H˜
−1(x))∂i1 H˜
−1; j1(x) · · · ∂i[β]− H˜
−1; j[β]− (x)
with i1, j1, . . . , i[β]− , j[β]− ∈ {1, . . . , d} and at least one pair ik , jk. Since for all x,
∇H˜−1(x) = I + ∇F(x)
and (3.4.2) holds, there is a constant N = N(d1, β) such that
∑
1≤|γ|≤β
4∑
i=2
|Jγi |0 +
∑
|γ|=β
4∑
i=2
|Jγi |{β}+ ≤ N |∇ϕ|β−1|∇F|β−1 ≤ N |∇ϕ|β−1|∇H|β−1.
If β > 2, then for all multi-indices γ with 1 ≤ |γ| < [β]−, we get
|Jγ1 − ∂
γϕ|0 = |∂γϕ ◦ H˜−1 − ∂γϕ|0 ≤ [∂γϕ]1|H|0.
It is easy to see that there is a constant N = N(L4,Nκ) such that for all γ with |γ| = [β]−
and all µ¯ ∈ (0, ({β}+ + µ) ∧ 1],
|Jγ1 − ∂
γϕ|0 = |∂γϕ ◦ H˜−1 − ∂γϕ|0 ≤ [∂γϕ]µ¯|H|
µ¯
0.
Moreover, appealing to the estimate (3.4.5) we obtain
[Jγ1 − ∂
γϕ]{β}+
≤ N1[0,1]({β}+ + µ)[∂γϕ]{β}++µL
µ
4 + N1(1,2]({β}
+ + µ)
(
[∇∂γϕ]{β}++µ−1 + |∇∂γϕ|0|∇H|0
)
.
82 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
Let us now consider the case θ > 0. The following decomposition obviously holds for all
x:
r1(x)
−θϕ(H˜−1(x)) − r1(x)−θϕ(x) = ϕˆ(H˜−1) − ϕˆ(x) +
(
r1(H˜−1(x))θ
r1(x)θ
− 1
)
ϕˆ(H˜−1(x)),
where ϕˆ = r−θ1 ϕ ∈ C
β(Rd1; Rd1). Thus, to complete the proof we require
|ϕˆ ◦ H˜−1|β ≤ N |ϕˆ|β and
∣∣∣∣∣∣
rθ1 ◦ H˜
−1
rθ1
− 1
∣∣∣∣∣∣
β
≤ N(|H|0 + |∇H|β∨1−1).
The latter inequality was proved in part (3) and the first inequality follows from part (2)
and Lemma 3.4.9. □
Remark 3.4.11. Let H : Rd1 → Rd1 be continuously differentiable and assume that for all
x,
|∇H(x)| ≤ η < 1.
Then for all x ∈ Rd1 ,
|(Id1 + ∇H(x))
−1| ≤ |Id1 +
∞∑
k=1
(−1)k∇H(x)k| ≤
1
1 − η
.
3.4.4 Stochastic Fubini theorem
Let m = (mϱ)t≤T , ϱ ∈ N, be a sequence of F-adapted locally square integrable continuous
martingales issuing from zero such that P-a.s. for all t ∈ [0,T ], ⟨mϱ1 ,mϱ2⟩t = 0 for ϱ1 , ϱ2
and ⟨mϱ⟩t = Nt for ϱ ∈ N, where Nt is a PT -measurable continuous increasing processes
issuing from zero. Let η(dt, dz) be a F-adapted integer-valued random measure on ([0,T ]×
E,B([0,T ])⊗E), where (U,U) is a Blackwell space. We assume that η(dt, dz) is optional,
P˜T -sigma-finite, and quasi-left continuous. Thus, there exists a unique (up to a P-null set)
dual predictable projection (or compensator) ηp(dt, dz) of η(dt, dz) such that µ(ω, {t}×U) =
0 for all ω and t. We refer the reader to Chapter II, Section 1, in [JS03] for any unexplained
concepts relating to random measures.
Let (X,Σ, µ) be a sigma-finite measure space; that is, there is an increasing sequence
of Σ-measurable sets Xn, n ∈ N, such that X = ∪∞n=1Xn and µ(Xn) < ∞ for each n. Let
f : Ω × [0,T ] × X → Rd2 be RT ⊗ Σ-measurable, g : Ω × [0,T ] × X → ℓ2(Rd2) be
RT ⊗ Σ/B(ℓ2(Rd2))-measurable, and h : Ω × [0,T ] × X × U → Rd2 be PT ⊗ Σ ⊗ U-
measurable. Moreover, assume that for all t ∈ [0,T ] and x ∈ X, P-a.s.
∫
]0,T ]
|gt(x)|2dNt +
∫
]0,T ]
∫
U
|ht(x, z)|2ηp(dt, dz) < ∞.
3.4. Appendix 83
Let F = Ft(x) : Ω × [0,T ] × X → Rd2 be OT ⊗ B(X)-measurable and assume that for
dPµ-almost all (t, x) ∈ [0,T ] × X,
Ft(x) =
∫
]0,t]
gϱs(x)dm
ϱ
s +
∫
]0,t]
∫
U
hs(x, z)η˜(dt, dz),
where η˜(dt, dz) = η(dt, dz) − ηp(dt, dz).
The following version of the stochastic Fubini theorem is a straightforward extension
of Lemma 2.6 [Kry11] and Corollary 1 in [Mik83]. See also Proposition 3.1 in [Zho13],
Theorem 2.2 in [Ver12], and Theorem 1.4.8 in [Roz90]. Indeed, to prove it for a bounded
measure we can use a monotone class argument as in Theorem 64 in [Pro05]. To handle
the general setting with possibly infinite µ, we use assumptions (ii) and (iii) below and
take limits on the sets Xn using the Lenglart domination lemma (Theorem 1.4.5 in [LS89])
and the following well-known inequalities:
E sup
t≤T
∣∣∣∣∣∣
∫
]0,t]
gϱsdm
ϱ
s
∣∣∣∣∣∣
≤ NE
(∫
]0,T ]
|gt(x)|2dm
ϱ
t
)1/2
E sup
t≤T
∣∣∣∣∣∣
∫
]0,t]
∫
U
ht(x, z)η˜(dt, dz)
∣∣∣∣∣∣
≤ NE
(∫
]0,T ]
∫
U
|ht(x, z)|2ηp(dt, dz)
)1/2
,
where τ ≤ T is an arbitrary stopping time and N = N(T ) is a constant independent of g
and h.
Proposition 3.4.12 (c.f. Corollary 1 in [Mik83] and Lemma 2.6 in [Kry11]). Assume that
(1) P-a.s. for all n ≥ 1,
∫
Xn
(∫
]0,T ]
|gt(x)|2dNt
)1/2
µ(dx) +
∫
Xn
(∫
]0,T ]
∫
U1
|ht(x, z)|2ηp(dt, dz)
)1/2
µ(dx) < ∞;
(2) P-a.s.
∫
]0,T ]
(∫
X
|gt(x)|µ(dx)
)2
dt +
∫
]0,T ]
∫
U
(∫
X
|ht(x, z)|µ(dx)
)2
ηp(dt, dz);
(3) P-a.s. for al t ∈ [0,T ], ∫
X
|Ft(x)|µ(dx) < ∞.
Then P-a.s. for all t ∈ [0,T ],
∫
X
Ft(x)µ(dx) =
∫
]0,t]
∫
X
gϱs(x)µ(dx)dm
ϱ
s +
∫
]0,t]
∫
U
∫
X
hs(x, z)µ(dx)η˜(dr, dz).
We obtain the following corollary by applying Minkowski’s integral inequaility.
84 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
Corollary 3.4.13. Assume that P-a.s.
∫
X
(∫
]0,T ]
|gt(x)|2dNt
)1/2
µ(dx) +
∫
X
(∫
]0,T ]
∫
U1
|ht(x, z)|2ηp(dt, dz)
)1/2
µ(dx) < ∞. (3.4.6)
Then P-a.s. for all t ∈ [0,T ],
∫
X
Ft(x)µ(dx) =
∫
]0,t]
∫
X
gϱs(x)µ(dx)dm
ϱ
s +
∫
]0,t]
∫
U
∫
X
hs(x, z)µ(dx)η˜(dr, dz).
Remark 3.4.14. If µ is a finite-measure and P-a.s.
∫
X
∫
]0,T ]
|gt(x)|2dNtµ(dx) +
∫
X
∫
]0,T ]
∫
U1
|ht(x, z)|2ηp(dt, dz)µ(dx) < ∞,
then (3.4.6) holds by Hölder’s inequality.
3.4.5 Itô-Wentzell formula
Definition 3.4.15. We say that an Rd1-valued F-adapted quasi-left continuous semimartin-
gale Lt = (Lkt )1≤k≤d1 , t ≥ 0, is of α-order for α ∈ (0, 2], if P-a.s. for all t ≥ 0,
∑
s≤t
|∆Ls|α < ∞
and
Lt = L0 +
∫
]0,t]
∫
R
d1
0
zpL(ds, dz), if α ∈ (0, 1),
Lt = L0 + At +
∫
]0,t]
∫
|z|≤1
zqL(ds, dz) +
∫
]0,t]
∫
|z|>1
zpL(ds, dz), if α ∈ [1, 2),
Lt = L0 + At + L
c
t +
∫
]0,t]
∫
|z|≤1
zqL(ds, dz) +
∫
]0,t]
∫
|z|>1
zpL(ds, dz), if α = 2,
where pL(dt, dz) is the jump measure of L with dual predictable projection πL(dt, dz), qL
(dt, dz) = pL(dt, dz) − πL(dt, dz) is a martingale measure, At = (Ait)1≤i≤d1 is a continuous
process of finite variation with A0 = 0, and Lct = (L
c;i
t )1≤i≤d1 is a continuous local martingale
issuing from zero.
Set (wϱ)ϱ∈N = (w1;ϱ)ϱ∈N, (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz)
= q1(dt, dz). Also, set D = D1, E = E1, and assume Z = D ∪ E.
Let f : Ω × [0,T ] × Rd1 → Rd2 be RT ⊗ B(Rd1)-measurable, g : Ω × [0,T ] × Rd1 →
ℓ2(Rd2) be RT ⊗ B(Rd1)/B(ℓ2(Rd2))-measurable, and h : Ω × [0,T ] × Rd1 × Z → Rd2 be
3.4. Appendix 85
PT ⊗ B(Rd1) ⊗Z-measurable. Moreover, assume that, P-a.s. for all x ∈ Rd1 ,
∫
]0,T ]
| ft(x)|dt +
∫
]0,T ]
|gt(x)|2dt < ∞
+
∫
]0,T ]
∫
D
|ht(x, z)|2π(dz)dt +
∫
]0,T ]
∫
E
|ht(x, z)|π(dz)dt < ∞.
Let F = Ft(x) : Ω × [0,T ] × Rd1 → Rd2 be OT ⊗ B(Rd1)-measurable and assume that for
all x, P-a.s. for all t,
Ft(x) = F0(x) +
∫
]0,t]
fs(x)ds +
∫
]0,t]
gϱs(x)dw
ϱ
s
+
∫
]0,t]
∫
Z
hs(x, z)[1D(z)q(ds, dz) + 1E(z)p(ds, dz)].
For each n ∈ {1, 2}, let Cnloc(R
d1; Rd2) be space of n-times continuously differentiable func-
tions f : Rd1 → Rd2 . We now state our version of the Itô-Wentzell formula. For each ω, t
and x, we denote ∆F(x) = Ft(x) − Ft−(x).
Proposition 3.4.16 (cf. Proposition 1 in [Mik83] ). Let (Lt)t≥0 be an Rd1-valued quasi-left
continuous semimartingale of order α ∈ (0, 2]. Assume that:
(1) (a) P-a.s. F ∈ D([0,T ];Cαloc(R
d; Rm) if α is fractional and F ∈ D([0,T ]; Cαloc(R
d; Rm)
if α = 1, 2 ;
(b) for dPdt-almost-all (ω, t) ∈ Ω × [0,T ], ft(x) and gt(x) = (g
iϱ
t (x))ϱ∈N ∈ ℓ2(R
d2) are
continuous in x and
dPdt − lim
y→x
[∫
D
|ht(y, z) − ht(x, z)|2π(dz) +
∫
E
|ht(y, z) − ht(x, z)|π(dz)
]
= 0;
(c) for all ϱ ∈ N and i ∈ {1, . . . , d1} and for dPd|⟨Lc;i,wϱ⟩|t-almost-all (ω, t) ∈ Ω ×
[0,T ], giϱt ∈ C
1
loc(R
d; R), if α = 2, ;
(2) for all compact subsets K of Rd1 , P-a.s.
∫
]0,T ]
sup
x∈K
(
| ft(x)| + |gt(x)|2 +
∫
D
|ht(x, z)|2π(dz) +
∫
E
|ht(x, z)|π(dz)
)
dt < ∞,
∑
ϱ∈N
∫
]0,T ]
sup
x∈K
|∇giϱt (x)|d|⟨L
c;i,wϱ⟩|t +
∑
t≤T
|∆Ft|α∧1;K |∆Lt|α∧1 < ∞.
Then P-a.s for all t ∈ [0,T ],
Ft(Lt) = F0(L0) +
∫
]0,t]
fs(Ls)ds +
∫
]0,t]
gϱs(Ls)dw
ϱ
s
86 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
+
∫
]0,t]
∫
Z
hs(Ls−, z)[1D(z)q(dr, dz) + 1E(z)p(dr, dz)]
+
∫
]0,t]
∂iFs−(Ls−)[1[1,2](α)dAis + 1{2}(α)dL
c;i
s ]
+
∑
s≤t
(
Fs−(Ls) − Fs−(Ls−) − 1[1,2](α)∇Fs−(Ls−)∆Ls
)
+ 1{2}(α)
1
2
∫
]0,t]
∂i jFs(Ls)d⟨Lc;i, Lc; j⟩s (3.4.7)
+ 1{2}(α)
∫
]0,t]
∂ig
ϱ
s(Ls)d⟨w
ϱ, Lc;i⟩s +
∑
s≤t
(∆Fs(Ls) − ∆Fs(Ls−)) .
Proof. Since both sides have identical jumps and we can always interlace a finite set of
jumps, we may assume that |∆Lt| ≤ 1 for all t ∈ [0,T ]; that is, it is enough to prove the
statement for L˜t = Lt −
∑
s≤t 1[1,∞)(|∆Ls|)∆Ls, t ∈ [0,T ]. It suffices to assume that for some
K and all ω, |L0| ≤ K. For each R > K, let
τR = inf

t ∈ [0,T ] : |A|t + |⟨L
c⟩|t +
∑
s≤t
|∆Ls|α + |Lt| > R

 ∧ T
and note that P-a.s. τR ↑ T as R tends to infinity. If instead of L, f , g, h, and F, we take
L·∧τR , f 1(0,τR], g
ϱ1(0,τR], h1(0,τR], F1(0,τR], then the assumptions of the proposition hold for
this new set of processes. Moreover, if we can prove (3.4.7) for this new set of processes,
then by taking the limit as R tends to infinity, we obtain (3.4.7). Therefore, we may assume
that for some R > 0, P-a.s. for all t ∈ [0,T ],
|A|t + |⟨Lc⟩|t +
∑
s≤t
|∆Ls|α + |Lt| ≤ R. (3.4.8)
Let ϕ ∈ C∞c (R
d1 ,R) have support in the unit ball in Rd1 and satisfy
∫
Rd1
ϕ(x)dx =
1, ϕ(x) = ϕ(−x), and ϕ(x) ≥ 0, for all x ∈ Rd1 . For each ε ∈ (0, 1), let ϕε(x) =
ε−dϕ (x/ε) , x ∈ Rd1 . By Itô’s formula, for all x ∈ Rd1 , P-a.s. for all t ∈ [0,T ],
Ft(x)ϕε(x − Lt) = F0(x)ϕε(x − L0) −
∫
]0,t]
Fs−(x)∂iϕε(x − Ls−)dLis
+
∫
]0,t]
ϕε (x − Ls) fs(x)ds +
∫
]0,t]
ϕε (x − Ls) gϱs(x)dw
ϱ
s
+ 1{2}(α)
1
2
∫
]0,t]
Fs(x)∂i jϕε(x − Ls)d⟨Lc;i, Lc; j⟩s
+ 1{2}(α)
∫
]0,t]
gϱs(x)∂iϕε(x − Ls)d⟨w
ϱ, Lc;i⟩s
+
∫
]0,t]
∫
Z
ϕε (x − Ls−) hs(x, z)[1D(z)q(dr, dz) + 1E(z)p(dr, dz)]
3.4. Appendix 87
+
∑
s≤t
∆Fs(x) (ϕε(x − Ls) − ϕε (x − Ls−))
+
∑
s≤t
Fs−(x) (ϕε(x − Ls) − ϕε(x − Ls−) + ∂iϕε (x − Ls−) ∆Ls) .
Appealing to assumption (2) and (3.4.8) (i.e. for the integrals against F), we integrate
both sides of the above in x, apply Corollary 3.4.13 (see, also, Remark 3.4.14) and the
deterministic Fubini theorem, and then integrate by parts to get that P-a.s. for all t ∈ [0,T ],
F(ε)t (Lt) = F
(ε)
0 (L0) +
∫
]0,t]
∇F(ε)s− (Ls−)[1[1,2](α)dA
i
s + 1{2}(α)dL
c;i
s ] +
∫
]0,t]
f (ε)s (Ls)dr
+
∫
]0,t]
g(ε)s (Ls)dw
ϱ
s +
∫
]0,t]
∫
Z
h(ε)s (Ls−, z)[1D(z)q(dr, dz) + 1E(z)p(dr, dz)]
+ 1{2}(α)
1
2
∫
]0,t]
∂i jF
(ε)
s (Ls)d⟨L
c;i, Lc; j⟩s + 1{2}(α)
∫
]0,t]
∂ig
(ε);ϱ
s (Ls)d⟨w
ϱ, Lc;i⟩s
+
∑
s≤t
(
∆F(ε)s (Ls) − ∆F
(ε)
s (Ls−)
)
+
∑
s≤t
(
F(ε)s− (Ls) − F
(ε)
s− (Ls−) − 1[1,2](α)∇F
(ε)
s− (Ls−)∆Ls
)
(3.4.9)
where for all ω, t, x, and z,
F(ε)t (x) := ϕε ∗ Ft(x), f
(ε)
t = ϕε ∗ ft(x), g
(ε);ϱ
t (x) = ϕε ∗ g
ϱ
t (x), h
(ε)
t (x, z) = ϕε ∗ ht(x, z),
and ∗ denotes the convolution operator on Rd1 . Let BR+1 = {x ∈ Rd1 : |x| ≤ R+1}. Owing to
assumption (1)(a) and standard properties of mollifiers, for any multi-index γ with |γ| ≤ α,
P-a.s. for all t,
|∂γF(ε)t (Lt)| ≤ sup
t≤T
sup
x∈BR+1
|∂γFt(x)| < ∞
and for all x,
dPdt − lim
ε↓0
|∂γF(ε)t (x) − ∂
γF(ε)t (x)| = 0.
Similarly, by assumption 1(b), dPdt-almost-all (ω, t) ∈ Ω × [0,T ],
| f (ε)t (Lt)| ≤ sup
x∈BR+1
| ft(x)| < ∞, |g
(ε)
t (Lt)| ≤ sup
x∈BR+1
|gt(x)| < ∞,
∫
D
|hεt (Lt, z)|
2π(dz) ≤ sup
x∈BR+1
∫
D
|ht(x, z)|2π(dz),
∫
E
|hεt (Lt, z)|π(dz) ≤ sup
x∈BR+1
∫
E
|ht(x, z)|π(dz)
88 Chapter 3. The method of stochastic characteristics for parabolic SIDEs
and for all x,
dPdt − lim
ε↓0
| f (ε)t (x) − ft(x)| = 0, dPdt − lim
ε→0
|g(ε)t (x) − gt(x)| = 0
and
dPdt − lim
ε↓0
∫
Z
[1D(z)|h
(ε)
t (x, z) − ht(x, z)|
2 + 1E(z)|h
(ε)
t (x, z) − ht(x, z)|]π(dz) = 0,
where in the last-line we have also used Minkowski’s integral inequality and a standard
mollifying convergence argument. Using assumption 1(d), for all ϱ ∈ N and i ∈ {1, . . . , d1}
and for dPd|⟨Lc;i,wϱ⟩|t-almost-all (ω, t) ∈ Ω × [0,T ]
|∇g(ε);iϱt (Lt)| ≤ sup
x∈BR+1
|∇giϱt (x)|
and for all x,
dPd|⟨Lc;i,wϱ⟩|t − lim
ε→0
|∇g(ε);iϱt (x) − ∇g
iϱ
t (x)| = 0, if α = 2.
Owing to assumption 1(a) and (3.4.8), P-a.s.
∑
s≤t
|F(ε)s− (Ls) − F
(ε)
s− (Ls−) − 1[1,2](α)∇F
(ε)
s− (Ls−)∆Ls|
≤ sup
t≤T
|Ft|α;BR+1
∑
s≤t
|∆Ls|α ≤ R sup
t≤T
|Ft|α;BR+1 .
Since P-a.s. F ∈ D([0,T ];Cα(Rd; Rm), it follows that for all x, P-a.s. for all t,
lim
ε↓0
|∆Fεt (x) − ∆Ft(x)| = 0.
By assumption (2), P-a.s for all t, we have
∑
s≤t
(
∆F(ε)s (Ls− + ∆Ls) − ∆F
(ε)
s (Ls−)
)
≤
∑
s≤t
|∆Ft|α∧1;BR+1 |∆Ls|
α∧1.
Combining the above and using assumptions (1)(a) and (2) and the bounds given in (3.4.8)
and the deterministic and stochastic dominated convergence theorem, we obtain conver-
gence of all the terms in (3.4.9), which complete the proof. □
Chapter 4
The L2-Sobolev theory for parabolic
SIDEs
4.1 Introduction
Let (Ω,F ,P) be a probability space with the filtration F = (Ft)0≤t≤T of sigma-algebras
satisfying usual conditions. In a triple of Hilbert spaces (Hα+µ,Hα,Hα−µ) with parameters
µ ∈ (0, 1] and α ≥ µ, we consider a linear stochastic evolution equation given by
dut = (Ltut + ft) dVt + (Mtut− + gt) dMt, t ≤ T, (4.1.1)
u0 = φ,
where Vt is a continuous non-decreasing process, Mt is a cylindrical square integrable mar-
tingale, L andM are linear adapted operators, and ϕ, f , and g are adapted input functions.
By virtue of Theorems 2.9 and 2.10 in [Gyö82], under some suitable conditions on
the data φ, f and g, if L satisfies a growth assumption and L andM satisfy a coercivity
condition in the triple (Hα+µ,Hα,Hα−µ), then there exists a unique solution (ut)t≤T of (4.1.1)
that is strongly càdlàg in Hα and belongs to L2(Ω × [0,T ],OT , dVtdP; Hα+µ), where OT is
the optional sigma-algebra on Ω × [0,T ]. In this chapter, under a weaker assumption than
coercivity (see Assumption 4.2.1(α, µ)) and using the method of vanishing viscosity, we
prove that there exists a unique solution (ut)t≤T of (4.1.1) that is strongly càdlàg in Hα
′
for
all α′ < α and belongs to L2(Ω × [0,T ], dVtdP; Hα). Furthermore, under some additional
assumptions on the operators L andM we can show that the solution u is weakly càdlàg
in Hα.
The variational theory of deterministic degenerate linear elliptic and parabolic PDEs
was established by O.A. Oleinik and E.V. Radkevich in [Ole65] and [OR71]. In [Par75],
É. Pardoux developed the variational theory of monotone stochastic evolution equations,
which was extended in [KR77, KR79, GK81, Gyö82] by N.V. Krylov, B.L Rozovskiı˘, and
I. Gyöngy. Degenerate parabolic stochastic partial differential equations (SPDEs) driven
by continuous noise were first investigated by N.V. Krylov and B.L. Rozovskiı˘ in [KR82].
These types of equations arise in the theory of non-linear filtering of continuous diffusion
processes as the Zakai equation and as equations governing the inverse flow of continuous
89
90 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
diffusions. In [GGK14], the solvability of systems of linear SPDEs in Sobolev spaces was
proved by M. Gerencsér, I. Gyöngy, and N.V. Krylov, and a small gap in the proof of
the main result of [KR82] was fixed. In Chapters 2, 3, and 4 of [Roz90], B.L. Rozovskiı˘
offers a unified presentation and extension of earlier results on the variational framework
of linear stochastic evolution systems and SPDEs driven by continuous martingales (e.g.
[Par75, KR77, KR79, KR82]). Our existence and uniqueness result on degenerate linear
stochastic evolution equations driven by jump processes (Theorem 4.3.2) extends Theorem
2 in Chapter-3-Section 2.2 of [Roz90] to include the important case of equations driven
by jump processes. It is also worth mentioning that the semigroup approach for non-
degenerate SPDEs driven by Lévy processes is well-studied (see, e.g. [PZ07, PZ13]).
As a special case of (4.1.1), we will consider a system of SIDEs. Before introducing
the equation, let us describe our driving processes. Let η(dt, dz) be an integer-valued
random measure on (R+ × Z,B(R+) ⊗ Z) with predictable compensator πt(dz)dVt. Let
η˜(dt, dz) = η(dt, dz) − πt(dz)dt be the martingale measure corresponding to η(dt, dz). Let
(Z2,Z2) be a measurable space with RT -measurable family π2t (dz) of sigma-finite random
measures on Z. Let wϱt , t ≥ 0, ϱ ∈ N, be a sequence of continuous local uncorrelated
martingales such that d⟨wϱ⟩t = dVt, for all ρ ∈ N. Let d1, d2 ∈ N. For convenience, we
set (Z1,Z1) = (Z,Z) and π1t = πt. We consider the d2-dimensional system of SIDEs on
[0,T ] × Rd1 given by
dult =
(
(L1;lt +L
2;l
t )ut + b
i
t∂iu
l
t + c
ll¯
t u
l¯
t(x) + f
l
t
)
dVt + (N
lϱ
t ut + g
lϱ
t )dw
ϱ
t (4.1.2)
+
∫
Z1
(
Ilt,zu
l¯
t− + h
l
t(z)
)
η˜(dt, dz),
ul0 = φ
l, l ∈ {1, . . . , d2},
where for k ∈ {1, 2}, l ∈ {1, . . . , d2}, and ϕ ∈ C∞c (R
d1; Rd2),
Lk;lt ϕ(x) :=
1
2
σ
k;iϱ
t (x)σ
k; jϱ
t (x)∂i jϕ
l(x) + σk;iϱt (x)υ
k;ll¯ϱ
t (x)∂iϕ
l¯(x)
∫
Zk
((
δll¯ + ρ
k;ll¯
t (x, z)
) (
ϕl¯(x + ζk(x, z)) − ϕl¯(x)
)
− ζk;it (x, z)∂iϕ
l(x)
)
πkt (dz)
N lϱt ϕ(x) := σ
1;iϱ
t (x)∂iϕ
l(x) + υ1;ll¯ϱt (x)ϕ
l¯(x), ϱ ∈ N,
Ilt,zϕ(x) := (δll¯ + ρ
1;ll¯
t (x, z))ϕ
l¯(x + ζ1t (x, z)) − ϕ
l(x),
and where δll¯ is the Kronecker delta (i.e. δll¯ = 1 if l = l¯ and δll¯ = 0 otherwise). The
summation convention with respect to repeated indices is used here and below; summation
over i is performed over the set {1, . . . , d1} and the summation over l, l¯ is performed over
the set {1, . . . , d2}. Without the noise term η˜(dt, dz) and integro-differential operators in L1
and L2, equation (4.1.2) has been well-studied (see, e.g. [KR82], [Roz90] (Chapter 3),
4.2. Degenerate linear stochastic evolution equations 91
and the recent paper [GGK14]). The choice to use the notation ζk(x, z) rather than the
traditional Hk(x, z) (as is used above) in this chapter due to the conflict of notation with
our spaces.
Let (Hα(Rd1; Rd2))α∈R be the L2-Sobolev-scale. For each m ∈ N, using our theorem on
degenerate stochastic evolution equations discussed above, under suitable measurability
and regularity conditions on the coefficients, initial condition, and free terms, we derive
the existence of a unique solution (ut)t≤T of (4.1.2) that is weakly càdlàg in Hm(Rd1; Rd2),
strongly càdlàg in Hα(Rd1; Rd2) for all α < m, and belongs to L2(Ω × [0,T ],OT , dVtdP;
Hm(Rd1; Rd2)).
Degenerate SIDEs of type (4.1.2) arise in the theory of non-linear filtering of semi-
martingales as the Zakai equation and as the equations governing the inverse flow of jump
diffusion processes. We constructed solutions of the above equation (with πt(dz) determin-
istic and independent of time) using the method
This chapter is organized as follows. We derive our existence and uniqueness result
for (4.1.1) in Section 4.2 and for (4.1.2) in Section 4.3.
4.2 Degenerate linear stochastic evolution equations
4.2.1 Basic notation and definitions
All vector spaces considered in this paper are assumed to have base field R. We also
assume that all Hilbert spaces are separable. For a Hilbert space H, we denote by H∗ the
dual of H and by B(H) the Borel sigma-algebra of H. Unless otherwise stated, the norm
and inner product of a Hilbert space H are denoted by | · |H and (·, ·)H, respectively. For
Hilbert spaces H and U and a bounded linear map L : H → U, we denote by L∗ the Hilbert
adjoint of L. Whenever we say that a map F from a sigma-finite measure space (S ,S, µ) to
a Hilbert space H is S-measurable without specifying the sigma-algebra on H, we always
mean that F is S/B(H)-measurable. For any Hilbert space H and sigma-finite measure
space (S ,S, ν), we denote by L2(S ,S, µ; H) the linear space of all S-measurable functions
F : S → H such that
|F|L2(S ,S,ν;H) =
∫
S
|F(s)|2Hν(ds) < ∞,
where we identify functions F,G : S → H that are equal µ-almost-everywhere (ν-a.e.).
The linear space L2(S ,S, ν; H) is a Hilbert space when endowed with the inner product
(F,G)L2(S ,S,ν;H) :=
∫
S
(F(s),G(s))Hν(ds).
We use the notation N = N(·, · · · , ·) below to denote a positive constant depending only
on the quantities appearing in the parentheses. In a given context, the same letter is often
92 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
used to denote different constants depending on the same parameter. All the stochastic
processes considered below are (at least) F-adapted unless explicitly stated otherwise.
Furthermore, we will often drop the dependence on ω ∈ Ω for random quantities.
In this section, we consider a scale of Hilbert spaces (Hα)α∈R and a family of operators
(Λα)α∈R satisfying the following properties:
• for all α, β ∈ R with β > α, Hβ is densely embedded in Hα;
• for all α, β, µ ∈ R with α < β < µ and all ε > 0, there is a constant N = N(α, β, µ, ε)
such that
|v|β ≤ ε|v|µ + N|v|α, ∀v ∈ Hµ; (4.2.1)
• Λ0 = I; for all α, µ ∈ R, Λα : Hµ → Hµ−α is an isomorphism; for all α, β ∈ R,
Λα+β = ΛαΛβ;
• for all α ∈ R, the inner product in Hα is given by (·, ·)α = (Λα·,Λα·)0;
• for all α > 0, the dual (Hα)∗ can be identified with H−α through the duality product
given by
⟨u, v⟩α = ⟨u, v⟩Hα,H−α =
(
Λαu,Λ−αv
)
0 , u ∈ H
α, v ∈ H−α;
• We assume that for every α ≥ 0, Λα is selfadjoint as an unbounded operator in H0
with domain Hα ⊆ H0: i.e. (Λαu, v)0 = (u,Λαv)0 for all u, v ∈ Hα.
Remark 4.2.1. It follows from the above properties that for all α ∈ R, the Hα norm is given
by |v|α = |Λαv|0, Λα is defined and linear on ∪β∈RHβ, Λ−α = (Λα)
−1, and ΛαΛβ = ΛβΛα,
for all β ∈ R. Moreover, for each α ≥ 0, if u ∈ Hα and v ∈ H0, then ⟨u, v⟩α = (u, v)0.
We will now describe our driving cylindrical martingale (Mt)t≥0 in (4.1.1) ) and the
associated stochastic integral. For a more thorough exposition, we refer to [MR99]. Let E
be a locally convex quasi-complete topological vector space; all bounded closed subsets
of E are complete. Let E∗ be its topological dual. Denote by ⟨·, ·⟩E∗,E the canonical bilinear
form (duality product) on E∗ × E. Assume that E∗ is weakly separable. Denote by L+(E)
the space of symmetric non-negative definite forms Q from E∗ to E; that is, for all Q ∈
L+(E), we have
⟨x,Qy⟩E∗,E = ⟨y,Qx⟩E∗,E, and ⟨x,Qx⟩E∗,E ≥ 0, ∀x, y ∈ E∗.
Recall that PT is the predictable sigma-algebra on Ω × [0,T ]. We say that a process
Q : Ω×[0,T ]→ L+(E) isPT -measurable if ⟨y,Qtx⟩E∗,E isPT -measurable for all x, y ∈ E∗.
4.2. Degenerate linear stochastic evolution equations 93
Assume that we are given a family of real-valued locally square integrable martingales
M = (Mt(y))y∈E∗)t≥0 indexed by E∗ and an increasing PT -measurable process Q : Ω ×
[0,T ]→ L+(E) such that for all x, y ∈ E∗,
Mt(x)Mt(y) −
∫ t
0
⟨x,Qsy⟩E∗,EdVs, t ≥ 0,
is a local martingale.
For each (ω, t) ∈ Ω × [0,T ], letHt = Hω,t be the Hilbert subspace of E defined as the
completion of Qω,tE∗ with respect to the inner product
(
Qω,tx,Qω,ty
)
Hω,t := ⟨x,Qω,ty⟩, x, y ∈ E
∗.
It can be shown that for all (ω, t) ∈ Ω × [0,T ], E∗ is densely embedded into H∗t , the map
Qt : E∗ → E can be extended to the Riesz isometry Qt : H∗t → Ht (still denoted Qt), and
the bilinear form ⟨x,Qty⟩E∗,E, x, y ∈ E∗, can be extended to ⟨x,Qty⟩H∗t ,Ht , x, y ∈ H
∗
t . Note
that for all x, y ∈ H∗t , we have (x, y)H∗t = ⟨x,Qty⟩H∗t ,Ht .
Let Lˆ2loc (Q) the space of all processes f such that ft ∈ H
∗
t , dVtdP-a.e., ⟨ ft,Qty⟩H∗t ,Ht is
PT -measurable for all y ∈ E∗, and P-a.s.
∫ T
0
| ft|2H∗t dVt =
∫ T
0
⟨ ft,Qt ft⟩H∗t ,HtdVt < ∞.
In [MR99], the stochastic integral of f ∈ Lˆ2loc(Q) against M, denoted It( f ) =
∫ t
0
fsdMs,
t ≥ 0, was constructed and has the following properties: (It( f ))t≥0 is a locally square
integrable martingale and P-a.s. for all t ∈ [0,T ]:
• for all y ∈ E∗, It(y) =
∫ t
0
ydMs = Mt(y) (recall that E∗ is embedded into allH∗s );
• for all g ∈ Lˆ2loc(Q).
⟨I( f ),I(g)⟩t =
∫ t
0
( fs, gs)H∗s dVs =
∫ t
0
⟨ fs,Qsgs⟩H∗s ,HsdVs;
• for all bounded PT -measurable processes ϕ : Ω × [0,T ]→ R,
∫ t
0
ϕsdIs( f ) = It(ϕ f ) =
∫ t
0
ϕs fsdMs.
For a Hilbert space H and (ω, t) ∈ Ω × [0,T ], denote by L2(H,H∗t ) the space of all
94 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
Hilbert-Schmidt operators Ψ : H → H∗t with norm and inner product given by
|Ψ|2L2(H,H∗t ) :=
∞∑
n=1
|Ψhn|2H∗t , (Ψ, Ψ˜)L2(H,H∗t ) =
∞∑
n=1
(Ψhn, Ψ˜hn)H∗t , Ψ˜ ∈ L2(H,H
∗
t ),
where (hn)n∈N is a complete orthogonal system in H. Denote by L
2
loc(H,Q) the space of
all processes Ψ such that Ψt ∈ L2(H,H∗t ), dVtdP-a.e., Ψth ∈ Lˆ
2(Q), for each h ∈ H,, and
P-a.s. ∫ T
0
|Ψt|2L2(H,H∗t )dVt < ∞.
For each Ψ ∈ L2loc (H,Q), we define the stochastic integral It(Ψ) =
∫ t
0
ΨsdMs as the unique
H-valued càdlàg locally square integrable martingale such that P-a.s. for all t ∈ [0,T ] and
h ∈ H,
(It(Ψ), h)H =
∫ t
0
ΨshdMs.
For all Ψ, Ψ˜ ∈ L2loc (H,Q) , we have that
|I· (Ψ) |2H −
∫ ·
0
|Ψs|2L2(H,H∗s )dVs and (I·(Ψ),I·(Ψ˜))H −
∫ ·
0
(Ψs, Ψ˜s)L2(H,H∗s )dVs
are real-valued local martingales. Moreover, for all bounded PT -measurable H-valued
processes u : Ω × [0,T ]→ H, P-a.s. for all t ∈ [0,T ],
∫ t
0
usdIs(Ψ) =
∫ t
0
{usΨs}HdMs,
where for a complete orthogonal system
(
e˜ns
)
n∈N inH
∗
s ,
{usΨs}H :=
∞∑
n=1
(
Ψsus, e˜
n
s
)
H∗s
e˜ns .
If H and Y are Hilbert spaces and L : H → Y is a bounded linear operator and Ψ ∈
L2loc(H,Q), then it follows that LIt(Ψ) = It(ΨL
∗); indeed, for all y ∈ Y , we have
(LIt(Ψ), y)Y = (It(Ψ), L
∗y)H =
∫ t
0
ΨsL
∗ydMs
and ΨL∗ ∈ L2loc(Y,Q).
4.2. Degenerate linear stochastic evolution equations 95
4.2.2 Statement of main results
In this section, for µ ∈ (0, 1], we consider the linear stochastic evolution equation in the
triple (H−µ,H0,Hµ) given by
dut = (Ltut + ft) dVt + (Mtut− + gt) dMt, t ≤ T, (4.2.2)
u0 = φ,
where φ is an F0-measurable H0-valued random variable and Vt is a continuous non-
decreasing process such that Vt ≤ C for all (ω, t) ∈ Ω × [0,T ], for some positive constant
C. Let α ≥ µ be given. We assume that:
(1) the mapping L : Ω × [0,T ] × Hµ → H−µ is linear in Hµ, and for all v ∈ Hµ, Lv is
RT/B(H−µ)-measurable; in addition, dVtdP-a.e., Ltv ∈ Hα−µ for all v ∈ Hα+µ;
(2) for dVtdP-almost-all (ω, t) ∈ Ω × [0,T ],Mω,t : Hµ → L2(H0,H∗ω,t) is linear, and for
all v ∈ Hµ, ϕ ∈ H0, y′ ∈ E∗, ⟨(Mv)ϕ,Qty′⟩H∗t ,Ht is PT -measurable for all y
′ ∈ E∗; in
addition, dVtdP-a.e,Mtv ∈ L2(Hα,H∗t ) for all v ∈ H
α+µ;
(3) the process f : Ω × [0,T ] → Hα−µ is RT/B(Hα−µ)-measurable and g ∈ L2loc (H
α,Q) ∩
L2loc(H
0,Q),
Let us introduce the following assumption for λ ∈ {0, α}. Recall that (u, v)λ =
(
Λλu,Λλv
)
0
.
Assumption 4.2.1 (λ, µ). There are positive constants L and K and an RT -measurable
function f¯ : Ω × [0,T ]→ R such that the following conditions hold dVtdP-a.e.:
(1) for all v ∈ Hλ+µ,
2(Λµv,Λ−µLtv)λ + |Mtv|2L2(Hλ,H∗t ) ≤ L|v|
2
λ;
2(Λµv,Λ−µt Ltv + ft)λ + |Mtv + gt|
2
L2(Hλ,H∗t )
≤ L|v|2λ + f¯t;
(2) for all v ∈ Hλ+µ,
|Ltv|λ−µ ≤ K|v|λ+µ, |Mtv|L2(Hλ,H∗t ) ≤ K|v|λ+µ;
(3)
| ft|2λ−µ + |gt|
2
L2(Hλ,H∗t )
≤ f¯t, E
∫ T
0
f¯tdVt < ∞.
Let OT be the optional sigma-algebra on Ω × [0,T ]. For µ ∈ (0, 1]) and λ ∈ R+
with λ ∈ {0, α}, we denote byWλ,µ the space of all Hλ-valued strongly càdlàg processes
v : Ω × [0,T ] → Hλ that belong to L2(Ω × [0,T ],OT , dVtdP; Hλ+µ). The following is our
definition of the solution of (4.2.2) and is standard in the variational theory or L2-theory
of stochastic evolution equations.
96 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
Definition 4.2.2. A process u ∈ W0,µ is said to be a solution of the stochastic evolution
equation (4.2.2) if P-a.s. for all t ∈ [0,T ]
ut
H−µ
= u0 +
∫ t
0
(Lsus + fs)dVs +
∫ t
0
(Msus− + gs)dMs,
where
H−µ
= indicates that the equality holds in the H−µ. That is, P-a.s. for all t ∈ [0,T ] and
v ∈ Hµ,
(v, ut)0 = (v, u0) +
∫ t
0
⟨v,Lsus + fs⟩µdVs +
∫ t
0
{v(Msus− + gs)}H0dMs.
Remark 4.2.3. In Definition 4.2.2, it is implicitly assumed that the integrals in (4.2.2) are
well-defined. Moreover, it is easy to check that if Assumption 4.2.1(0, µ) holds, then the
integrals in Definition 4.2.2 are well-defined.
In order to obtain estimates of the second moments of the supremum in t of the solution
of (4.2.2), in the Hα norm, we will need to impose the upcoming assumption. Before
introducing this assumption, we describe a few notational conventions. For two real-
valued semimartingales Xt and Yt, we write P-a.s. dXt ≤ dYt if with probability 1, Xt−Xs ≤
Yt − Ys for any 0 ≤ s ≤ t ≤ T . For v ∈ Wλ,µ, we define
Mt (v) :=
∫ t
0
MsvsdMs, t ∈ [0,T ],
and denote by [M(v)]λ;t the quadratic variation process of Mt (v) in H
λ.
Assumption 4.2.2 (λ, µ). There is a positive constant L, a PT -measurable function g¯ : Ω×
[0,T ]→ R, and increasing adapted processes A, B : Ω×[0,T ]→ R with dAtdP ≤LdVtdP,
dBtdP ≤g¯tdVtdP on PT such that the following conditions hold P-a.s.:
(1) for all v ∈ Wλ,µ,
(Λµvt,Λ
−µLtvt)λdVt + d [M(v)]λ;t + 2{vt−Mtvt−}HλdMt ≤ |vt−|
2
λdAt + Gt(v)dMt,
where G(v) ∈ Lˆ2loc (Q) satisfies |Gt(v)|H∗t dVt ≤ L|vt−|
2
λ
dVt;
(2) for all v ∈ Wλ,µ,
2d
[
M (v) ,I (g)
]
λ,t + 2{vt−gt}HλdMt ≤ |vt−|λdBt + G˜t (v) dMt,
where G¯(v) ∈ Lˆ2loc (Q) satisfies |G¯t(v)|H∗t dVt ≤ L|vt−|λg¯tdVt, and
E
∫ T
0
g¯2t dVt < ∞.
4.2. Degenerate linear stochastic evolution equations 97
Although Assumption 4.2.2(λ, µ) looks rather technical, it is satisfied for a large class
of parabolic stochastic integro-differential equations (see Section 4.3) under what we con-
sider to be reasonable assumptions.
Let T be the set of all stopping times τ ≤ T and T p be the set of all predictable
stopping times τ ≤ T .
Theorem 4.2.4. Let µ ∈ (0, 1] and α ≥ µ. Let Assumption 4.2.1(λ, µ) hold for λ ∈ {0, α}
and assume that E
[
|φ|2α
]
< ∞.
(1) Then there exists a unique solution u = (ut)t≤T of (4.2.2) such that for any α′ < α, u
is an Hα
′
-valued strongly càdlàg process and there is a constant N = N(L,K,C) such
that
E
[
sup
t≤T
|ut|2α−µ
]
+ sup
τ∈T
E
[
|uτ|2α
]
+ E
∫ T
0
|us|2α dVs ≤ N
(
E
[
|φ|2α
]
+ E
∫ T
0
f¯tdVt
)
.
Moreover, for all p ∈ (0, 2) and α′ < α, there is a constant N = N(L,K,C, p, α′) such
that
E
[
sup
t≤T
|ut|
p
α′
]
≤ NE


(
|φ|2α +
∫ T
0
f¯tdVt
) p
2

 .
(2) If, in addition, Assumption 4.2.2(λ, µ) holds for λ ∈ {0, α}, then u is an Hα-valued
weakly càdlàg process and there is a constant N = N(L,K,C) such that
E
[
sup
t≤T
|ut|2α
]
≤ NE
[
|φ|2α +
∫ T
0
( f¯t + g¯
2
t )dVt
]
.
Remark 4.2.5. If V is an arbitrary continuous increasing adapted process, then Theo-
rem 4.2.4 can be applied locally by considering VCt = Vt∧τC , t ∈ [0,T ], with τC = inf
(t ∈ [0,T ] : Vt ≥ C) ∧ T.
4.2.3 Proof of Theorem 4.2.4
We will construct a sequence of approximations in Wα,µT of the solution of (4.2.2) by
solving in the triple (H−µ,H0,Hµ) the equation
dut =
(
Lnt ut + ft
)
dVt + (Mtut− + gt) dMt, t ≤ T, (4.2.3)
u0 = φ,
where Lnt = Lt −
1
n (Λ
µ)2. In order to apply the foundational theorems on stochastic evolu-
tion equations with jumps established in [GK81] and [Gyö82], it is convenient for us first
98 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
to consider the following equation in the triple (H−µ,H0,Hµ) :
dvt =
(
ΛαLtΛ−αvt −
1
n
(Λµ)2vt + Λ
α ft
)
dVt +
(
MtΛ−αvt−(Λα)∗ + gt(Λα)∗
)
dMt, t ≤ T,(4.2.4)
v0 = Λ
αφ.
The solutions of (4.2.3) and (4.2.4) are to be understood following Definition 4.2.2.
Lemma 4.2.6. Let µ ∈ (0, 1] and α ≥ µ. Let Assumption 4.2.1(α, µ) hold and assume that
E
[
|φ|2α
]
< ∞.
(1) For each n ∈ N, there is a unique solution vn = (vnt )t≤T of (4.2.3), and there is a
constant N = N(L,K,C) independent of n such that
sup
τ∈T
E
[
|vnτ|
2
0
]
+ E
∫ T
0
|vnt |
2
0dVt +
1
n
E
∫ T
0
|vnt |
2
µdVt ≤ NE
[
|φ|2α +
∫ T
0
f¯tdVt
]
. (4.2.5)
Moreover, for all p ∈ (0, 2), there is a constant N = N(L,K,T, p)
E
[
sup
t≤T
|vnt |
p
0
]
≤ NE


(
|φ|2α +
∫ T
0
f¯tdVt
) p
2

 . (4.2.6)
(2) If, in addition, Assumption 4.2.2(α, µ) holds, then there is a constant N = N(L,K,C)
such that
E
[
sup
t≤T
|vnt |
2
0
]
≤ NE
[
|φ|2α +
∫ T
0
( f¯t + g¯
2
t )dVt
]
. (4.2.7)
Proof. (1) For each (ω, t) ∈ Ω × [0,T ] and n ∈ N, let
Lαt v = Λ
αLtΛ−αv, L
α,n
t v = L
α
t v −
1
n
(Λµ)2 v, Mαt v =MtΛ
−αv(Λα)∗.
Using basic properties of the spaces (Hα)α∈R and the operators (Λα)α∈R, dVtdP-a.e. for all
v ∈ Hµ, we have
2⟨v,Lαt v⟩µ = 2(Λ
µv,Λ−µΛαLtΛ−αv)0 = 2(ΛµΛ−αv,Λ−µLtΛ−αv)α,
2⟨v, (Λµ)2v⟩µ = 2(Λµv,Λµv)0 = |v|2µ,
and
∣∣∣Mαt v
∣∣∣
2
L2(H0,H∗t )
=
∞∑
k=1
|Λα(MtΛ−αv)∗e˜nt |
2
H0 =
∞∑
k=1
|(MtΛ−αv)∗e˜nt |
2
Hα
=
∞∑
k=1
|MtΛ−αvh¯k|2H∗t =
∣∣∣Mαt v
∣∣∣
2
L2(Hα,H∗t )
,
4.2. Degenerate linear stochastic evolution equations 99
where (e˜kt )k∈N,and (h¯
k)k∈N are orthonormal basis ofHt and Hα, respectively. It follows from
Assumption 4.2.1(α, µ) that dVtdP-a.e. for all v ∈ Hµ, we have
|Lα,nt v|−µ ≤
(
K +
1
n
)
|v|µ,
∣∣∣Mαt v
∣∣∣
L2(H0,H∗t )
≤ K|v|µ,
and
2⟨v,Lα,nt v + Λ
α ft⟩µ +
∣∣∣Mαt v + gt(Λ
α)∗
∣∣∣
2
L2(H0,H∗t )
≤ −
2
n
|v|2µ + L|v|
2
0 + f¯t. (4.2.8)
In [Gyö82], the variational theory for monotone stochastic evolution equations driven
by locally square integrable Hilbert-space-valued martingales was derived; it is worth
mentioning that the càdlàg version of the variational solution in the pivot space and the
uniqueness of the solution was obtained using Theorem 2 in [GK81]. The theorems and
proofs given in [Gyö82] continue to hold for equations driven by the cylindrical martin-
gales we consider in this paper. Therefore, by Theorems 2.9 and 2.10 in [Gyö82], for every
n ∈ N, there exists a unique solution vn = (vnt )t≤T of the stochastic evolution equation given
by
vnt = Λ
αφ +
∫ t
0
(Lα,ns v
n
s + Λ
α fs)dVs +
∫ t
0
(Mαs v
n
s− + Λ
αgs)dMs, t ≤ T.
Furthermore, by virtue of Theorem 4.1 in [Gyö82], there is a constant N(n) = N(n, L,
K,C) such that
E
[
sup
t≤T
|vnt |
2
0
]
+ E
∫ T
0
|vnt |
2
µdVt ≤ N(n)E
[
|φ|2α +
∫ T
0
f¯tdVt
]
. (4.2.9)
We will now use our assumptions to obtain bounds of the solutions vnt , n ∈ N, in the
H0-norm independent of n. Applying Theorem 2 in [GK81], P-a.s. for all t ∈ [0,T ], we
have
|vnt |
2
0 = |φ|
2
α +
∫ t
0
2⟨vns ,L
α,n
t v
n
s + Λ
α fs⟩µdVs +
[
M˜
]
0,t
+ mt, (4.2.10)
where (M˜t)t≤T and (mt)t≤T are local martingales given by
M˜t :=
∫ t
0
(Mαs v
n
s− + Λ
αgs)dMs, mt := 2
∫ t
0
{vns−(M
α
s v
n
s− + Λ
αgs)}H0dMs.
Thus, taking the expectation of (4.2.10) and making use of Assumption 4.2.1(α, µ), (4.2.8),
and (4.2.9), we find that for all τ ∈ T ,
E
[
|vnτ|
2
0
]
≤ E
[
|φ|2α
]
+ E
∫ τ
0
(
2⟨vnt ,L
α,n
t v
n
t + Λ
α ft⟩µ + |Mαt v + Λ
αgt|2L2(H0,Ht)
)
dVt
≤ E
[
|φ|2α +
∫ τ
0
(
L|vnt |
2
0 −
2
n
|vnt |
2
µ + f¯t
)
dVt
]
.
100 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
This implies that for any τ ∈ T p,
E
[
|vnτ−|
2
0
]
+
2
n
E
∫ τ
0
|vnt |
2
µdVt ≤ E
[
|φ|2α +
∫ τ
0
(
L|vns |
2
0 + f¯s
)
dVs
]
.
By virtue of Lemmas 2 and 3 in [GM83], we deduce that there is a constant N = N(L,K,C)
such that for any τ ∈ T ,
E
[
|vnτ|
2
0
]
+
1
n
E
∫ τ
0
|vnt |
2
µdVt ≤ NE
[
|φ|2α +
∫ τ
0
f¯sdVs
]
,
which implies that (4.2.5) holds since Vt is uniformly bounded by the constant C. Finally,
owing to Corollary II in [Len77], we have (4.2.6).
(2) Using Assumption 4.2.2(α, µ) and estimating (4.2.10), we get that P-a.s.
d|vnt |
2
0 ≤ |φ|
2
α + |v
n
t−|
2
λdAt + |v
n
t−|λdBt + (Gt(v
n) + G¯t (v
n))dMt.
Moreover, for any τ ∈ T , we obtain
E
[
sup
t≤τ
|vnt |
2
0
]
≤ NE
[
|φ|2α +
∫ τ
0
|vnt |
2
0dVt +
∫ τ
0
( f¯t + g¯
2
t )dVt + sup
t≤τ
|lnt |
]
,
where lnt :=
∫ t
0
(Gs(vn)+G¯s(vn))dMs.Moreover, by the Burkholder-Davis-Gundy inequality
and Young’s inequality, we have
E sup
t≤τ
|lnt | ≤ NE


(∫ τ
0
(|vnt−|
4
α + |v
n
t−|
2
αg¯
2
t )dVt
) 1
2

 ≤ NE

sup
t≤τ
|vnt |0
(∫ τ
0
(|vnt−|
2
α + g¯
2
t )dVt
) 1
2


≤
1
4N
E
[
sup
t≤τ
|vnt |
2
0
]
+ NE
∫ τ
0
(|vnt |
2
α + g¯
2
t )dVt,
from which estimate (4.2.7) follows. □
Proof of Theorem 4.2.4 . (1) Let vn = (vnt )t≤T be the unique solution of (4.2.4) constructed
in Lemma 4.2.6. Since vn ∈ W0,µ, it follows that un := Λ−αvn ∈ Wα,µ ⊆ W0,µ is a solution
of (4.2.3) in the triple (H−µ,H0,Hµ).
We will first show that (un)n∈N is Cauchy in H0. Letting un,m = un−um, for all n,m ∈ N,
we have
un,mt =
∫ t
0
(Lnt u
n
t − L
m
t u
m
t )dVt +
∫ t
0
Mtun,mdMt, t ≤ T.
Applying Theorem 2 in [GK81], we obtain that P-a.s. for all t ∈ [0,T ],
|unt − u
m
t |
2
0 =
∫ t
0
2⟨uns − u
m
s ,L
n
su
n
s − L
m
s u
m
s ⟩µ,0dVs + [M
n,m]t + η
n,m
t (4.2.11)
4.2. Degenerate linear stochastic evolution equations 101
where (Mn,mt )t≤T and (η
n,m
t )t≤T are local martingales given by
Mn,mt :=
∫ t
0
Ms(uns− − u
m
s−)dMs, η
m,n
t := 2
∫ t
0
{(uns− − u
m
s−)Ms(u
n
s− − u
m
s−)}H0dMs.
Assumption 4.2.1(0, µ) (1) implies that for any τ ∈ T ,
E
[
|un,mτ |
2
0
]
≤ E
∫ τ
0
(
2⟨un,ms ,L
n
su
n,m
s ⟩µ + |Mtu
n,m
s |
2
L2(H0,H∗t )
)
dVs
≤ E
∫ τ
0
(
L|un,ms |
2
0 +
1
n
|uns |
2
µ +
1
m
|ums |
2
µ
)
dVs,
and hence for any τ ∈ T p, we have
E
[
|un.mτ− |
2
0
]
≤ E
∫ τ
0
(
L|un.ms |
2
0 +
1
n
|uns |
2
µ +
1
m
|ums |
2
µ
)
dVs.
By virtue of Lemmas 2 and 4 in [GM83] and (4.2.5) (noting that |unt |0 = |Λ
−αvnt |0 ≤ N|v
n
t |0),
there is a constant N such that for any τ ∈ T ,
E
[
|un,mτ |
2
0
]
≤ NE
∫ τ
0
(
1
n
|uns |
2
µ +
1
m
|ums |
2
µ
)
dVs ≤ N
(
1
n
+
1
m
)
E
[
|φ|2α +
∫ T
0
f¯tdVt
]
.
Using Corollary II in [Len77], we have that for all p ∈ (0, 2), there is a constant N such
that
E
[
sup
t≤T
|un,mt |
p
0
]
≤ N
(
1
n
+
1
m
) p
2
[
E
[
|φ|2α +
∫ T
0
f¯tdVt
]] p
2
.
Therefore,
lim
n,m→∞
[
sup
τ∈T
E
[
|un,mτ |
2
0
]
+ E
[
sup
t≤T
|un,mt |
p
0
]]
= 0, (4.2.12)
and hence there exists a strongly càdlàg H0-valued process u = (ut)t≤T such that
dP − lim
n→∞
sup
t≤T
|ut − unt |0 = 0. (4.2.13)
Since for all n, un is solution of (4.2.3), we have that P-a.s. for all t ∈ [0,T ] and
ϕ ∈ Hµ,
(ϕ, unt )0 = (ϕ, φ)0 +
∫ t
0
⟨ϕ,Lnsu
n
s + fs⟩µ,0dVs +
∫ t
0
{ϕ(Msuns− + gs)}H0dMs. (4.2.14)
Owing to (4.2.13), we know that the left-hand-side of (4.2.14) converges P-a.s. for all
t ∈ [0,T ] to (ϕ, ut)0 as n tends to infinity. Our aim, of course, is to pass to the limit as n
tends to infinity on the right-hand-side.
102 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
This can be done quite simply when α > µ. Indeed, owing to the interpolation inequal-
ity (4.2.1), for all ε > 0, α′ < α, and p ∈ (0, 2), there is a constant N = N(α, α′, ε, p) such
that
sup
t≤T
|un,mt |
p
α′
≤ ε sup
t≤T
|un,mt |
p
α + N sup
t≤T
|un,mt |
p
0 . (4.2.15)
Since |unt |α = |Λ
−αvnt |α ≤ N |v
n
t |0, by (4.2.7) and (4.2.15), we have that for all α
′ < α and
p ∈ (0, 2),
E
[
sup
t≤T
|un,mt |
p
α′
]
≤ εNE


(
|φ|2α +
∫ T
0
f¯tdVt
) p
2

 + NE
[
sup
t≤T
|un,mt |
p
0
]
. (4.2.16)
Using (4.2.12) and passing to the limit as n and m tend to infinity on both sides of (4.2.16),
and then taking ε ↓ 0, we get that for all α′ < α and p ∈ (0, 2),
lim
n,m→∞
E
[
sup
t≤T
|un,mt |
p
α′
]
= 0.
Combining the above results, we conclude that for any α′ < α, u is an Hα
′
-valued strongly
càdlàg process and
dP − lim
n→∞
sup
t≤T
|ut − unt |α′ = 0. (4.2.17)
In particular, if α > µ, then taking α′ > µ in (4.2.17) and appealing to Assumption
4.2.1(0, µ) (2), (4.2.5), and the identity,
⟨ϕ,Λ2µuns⟩µ = (Λ
µϕ,Λµuns)0,
we can take the limit as n tends to infinity on the right-hand-side of (4.2.14) by the domi-
nated convergence theorem to conclude that u is a solution of (4.2.2).
The case α = µ must be handled with weak convergence. Let
S (OT ) = (Ω × [0,T ],OT , dV¯tdP) and S (PT ) = (Ω × [0,T ],PT , dV¯tdP),
where V¯t =: Vt +t. It follows from (4.2.5) that there exists a subsequence (unk)k∈N of (un)n∈N
that converges weakly in L2(S (OT ); Hµ) to some u¯ ∈ L2(S (OT ); Hµ) which satisfies
E
∫
]0,T ]
|u¯t|2µdV¯t ≤ NE
[
|φ|2µ +
∫ T
0
f¯tdVt
]
.
For any ϕ ∈ H0 and bounded predictable process ξt, we have
lim
k→∞
E
∫ T
0
ξt⟨ϕ, u
nk
t ⟩µdV¯t = lim
k→∞
E
∫ T
0
ξt(u
nk
t , ϕ)0dV¯t = E
∫ T
0
ξt(ut, ϕ)0dV¯t,
4.2. Degenerate linear stochastic evolution equations 103
and hence u = u¯ in L2(S (OT ); Hµ) and unk converges to u weakly in L2(S (OT ); Hµ) as k
tends to infinity. Define unk− = (u
nk
t−)t≤T and u− = (ut−)t≤T , where the limits are taken in
H0. By repeating the argument above, we conclude that unk− converges to u− weakly in
L2(S (OT ); Hµ) as k tends to infinity.
Fix ϕ ∈ Hµ and a PT -measurable process (ξt)t≤T bounded by the constant K. Define
the linear functionals ΦL : L2(S (OT ); Hµ)→ R and ΦM : L2(S (PT ); Hµ)→ R by
ΦL(v) = E
∫
]0,T ]
ξt
∫
]0,t]
⟨ϕ,Lsvs⟩µ,0dVsdV¯t, ∀v ∈ L2(S (OT ); Hµ)
and
ΦM(v) = E
∫
]0,T ]
ξt
∫
]0,t]
{ϕMsvs}H0dMsdV¯t, ∀v ∈ L
2(S (PT ); Hµ),
respectively. Owing to Assumption 4.2.1(0, µ) (2), the Burkholder-Davis-Gundy inequal-
ity, and the fact that (V¯t)t≤T is uniformly bounded by the constant C, there is a constant
N = N(K,C) such that
|ΦL(v)| ≤ N |ϕ|µ
(
E
∫ T
0
|vt|2µdV¯t
) 1
2
, ∀v ∈ L2(S (OT ); Hµ),
and
|ΦM(v)| ≤ N|ϕ|µ
(
E
∫
]0,T ]
|vs|2µdV¯s
) 1
2
, ∀v ∈ L2(S (PT ); Hµ).
This implies that ΦL is a continuous linear functional on L2(S (OT ); Hµ) and ΦM is a con-
tinuous linear functional on L2(S (PT ); Hµ), and hence that
lim
k→∞
ΦL(unk) = ΦL(u), lim
k→∞
ΦL(unk− ) = Φ
M(u−). (4.2.18)
For each k, we have that
E
∫ T
0
ξt(ϕ, u
nk
t )0dV¯t = E
∫ T
0
ξt(ϕ, φ)0dV¯t + E
∫ T
0
ξt
∫ t
0
⟨ϕ,Lnks u
nk
s + fs⟩µ,0dVsdV¯t(4.2.19)
+ E
∫ T
0
ξt
∫ t
0
{ϕ(Msuns− + gs)}H0dMsdV¯t.
Passing to the limit as k tends to infinity on both sides of (4.2.19) using (4.2.18) and
⟨ϕ,Λ2µunks ⟩µ = (Λ
µϕ,Λµunks )0,
104 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
we obtain that dV¯tdP-a.e.
(ϕ, ut)0 = (ϕ, φ)0 +
∫
]0,t]
⟨ϕ, (Lsus + fs)⟩µdVs +
∫
]0,t]
{ϕ(Msus− + gs)}H0dMs, t ≤ T.
Therefore, for all α ≥ µ, u is a solution of (4.2.2).
We will now show that
sup
τ∈T
E
[
|uτ|2α
]
≤ NE
[
|φ|2α +
∫ T
0
f¯tdVt
]
. (4.2.20)
Let (hk)k∈N be a complete orthonormal basis in Hα such that the linear subspace spanned
by (hk)k∈N is dense in H2α. Owing to (4.2.5), for all m ≥ 1 and τ ∈ T ,
E


m∑
k=1
|(unτ,Λ
2αhk)0|2

 = E


m∑
k=1
|(unτ, h
k)α|2

 ≤ E
[
|unτ|
2
α
]
≤ NE
[
|φ|2α +
∫ T
0
f¯tdVt
]
.
Applying Fatou’s lemma first in n and then in m, we have that for all τ ∈ T ,
E


∞∑
k=1
|(uτ,Λ2αhk)0|2

 ≤ NE
[
|φ|2α +
∫ T
0
f¯tdVt
]
.
Hence, for all t ∈ [0,T ], P-a.s. vt =
∑
k(ut,Λ
2αhk)0hk ∈ Hα. Since the linear subspace
spanned by (Λ2αhk)k∈N is dense in H0 and for all t ∈ [0,T ], P-a.s., (ut − vt,Λ2αhk)0 = 0, for
all k ∈ N, it follows that P-a.s. for all τ ∈ T , uτ = v and
E
[
|uτ|2α
]
= E
∞∑
k=1
∣∣∣(uτ,Λ
2αhk)0
∣∣∣
2
≤ NE
[
|φ|2α +
∫ T
0
f¯tdVt
]
,
which proves (4.2.20).
Estimating (4.2.2) directly in the Hα−µ-norm, we easily derive that
E
[
sup
t≤T
|ut|2α−µ
]
≤ NE
[
|φ|2α +
∫ T
0
f¯sdVt
]
.
If (vt)t≤T be another solution of (4.2.2), then by Theorem 2 in [GK81] and Assumption
4.2.1(0, µ)(1), P-a.s. for all t ∈ [0,T ], we have
|ut − vt|20 ≤ L
∫ t
0
|us − vs|20dVs + mt,
where (mt)t≤T is a local martingale with m0 = 0, and hence applying Lemmas 2 and 4 in
4.2. Degenerate linear stochastic evolution equations 105
[GM83], we get
P
(
sup
t≤T
|ut − vt|0 > 0
)
= 0,
which implies that (ut)t≤T is the unique solution of (4.2.2). This completes the proof of
part (1).
(2) Estimating (4.2.11) using Assumption 4.2.2(0, µ), we get that P-a.s.
d|un,mt |
2
0 ≤ |φ|
2
0 +
(
1
n
+
1
m
)
|un,m|20dVt + |u
n,m
t− |
2
λdAt + |u
n,m
t− |λdBt + (Gt(u
n,m) + G¯t (u
n,m))dMt.
Then estimating the stochastic integrand as in the proof of part (2) of Lemma 4.2.6, for
any τ ∈ T , we get
E
[
sup
t≤τ
|un.mt |
2
0
]
≤ NE
∫ τ
0
(
|un.ms |
2
0 +
1
n
|uns |
2
µ +
1
m
|ums |
2
µ)
)
dVs,
and hence by Gronwall’s lemma and Lemma 4.2.6(2),
E
[
sup
t≤τ
|un.mt |
2
0
]
≤ N
(
1
n
+
1
m
)
E
[
|φ|2α +
∫
]0,T ]
(
f¯t + g¯
2
t
)
dVt
]
Thus,
lim
n,m→∞
E
[
sup
t≤τ
|un.mt |
2
0
]
= 0.
Let (hk)k∈N be a complete orthonormal basis Hα such that the linear subspace spanned by
(hk)k∈N is dense in H2α. Owing to (4.2.7), for all m ≥ 1 and τ ∈ T ,
E

sup
t≤T
m∑
k=1
(unt , h
k)α

 = E

sup
t≤T
m∑
k=1
|(unt ,Λ
2αhk)0|2

 ≤ E
[
sup
t≤T
|unt |
2
α
]
≤ NE
[
|φ|2α +
∫
]0,T ]
( f¯t + g¯
2
t )dVt
]
.
Applying Fatou’s lemma first in n and then in m, we have that
E

sup
t≤T
∞∑
k=1
|(ut,Λ2αhk)0|2

 ≤ NE
[
|φ|2α +
∫
]0,T ]
( f¯t + g¯
2
t )dVt
]
.
Thus, v =
∑
k(u,Λ
2αhk)0hk is an Hα-valued weakly càdlàg process. Since the linear sub-
space spanned on (Λ2αhk)k∈N is dense in H0 and
(
ut − vt,Λ2αhk
)
0
= 0, for all k ∈ N, it
follows that P-a.s. for all t ∈ [0,T ], ut = vt and
E
[
sup
t≤T
|ut|2α
]
= E

sup
t≤T
∞∑
k=1
|(ut,Λ2αhk)0|2

 ≤ NE
[
|φ|2α +
∫
]0,T ]
( f¯t + g¯
2
t )dVt
]
.
106 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
□
4.3 The L2-Sobolev theory for degenerate SIDEs
4.3.1 Statement of main results
In this section, we consider the d2-dimensional system of SIDEs on [0,T ] × Rd1 given by
dult =
(
(L1;lt +L
2;l
t )ut + b
i
t∂iu
l
t + c
ll¯
t u
l¯
t + f
l
t
)
dVt +
(
N lϱt ut + g
lϱ
t
)
dwϱt (4.3.1)
+
∫
Z1
(
Ilt,zu
l¯
t− + h
l
t(z)
)
η˜(dt, dz),
ul0 = φ
l, l ∈ {1, . . . , d2},
where for k ∈ {1, 2}, l ∈ {1, . . . , d2}, and ϕ ∈ C∞c (R
d1; Rd2),
Lk;lt ϕ(x) :=
1
2
σ
k;iϱ
t (x)σ
k; jϱ
t (x)∂i jϕ
l(x) + σk;iϱt (x)υ
k;ll¯ϱ
t (x)∂iϕ
l¯(x)
+
∫
Zk
((
δll¯ + ρ
k;ll¯
t (x, z)
) (
ϕl¯(x + ζkt (x, z)) − ϕ
l¯(x)
)
− ζk;it (x, z)∂iϕ
l(x)
)
πkt (dz)
N lϱt ϕ(x) := σ
1;iϱ
t (x)∂iϕ
l(x) + υ1;ll¯ϱt (x)ϕ
l¯(x), ϱ ∈ N,
Ilt,zϕ(x) :=
(
δll¯ + ρ
1;ll¯
t (x, z)
)
ϕl¯(x + ζ1t (x, z)) − ϕ
l(x).
We assume that
σkt (x) = (σ
k;iϱ
ω,t (x))1≤i≤d1, ϱ∈N, bt(x) = (b
i
ω,t(x))1≤i≤d1 , ct(x) = (c
ll¯
ω,t(x))1≤l,l¯≤d2 ,
υkt (x) = (υ
k;ll¯ϱ
ω,t (x))1≤l,l¯≤d2, ϱ∈N, ft(x) = ( f
i
ω,t(x))1≤i≤d2 , gt(x) = (g
iϱ
ω,t(x))1≤i≤d2, ϱ∈N
are random fields on Ω × [0,T ] × Rd1 that are RT ⊗ B(Rd1)-measurable. Moreover, we
assume that
ζ1t (x, z) = (ζ
1;i
ω,t(x, z))1≤i≤d1 , ρ
1
t (x, z) = (ρ
1;ll¯
ω,t (x, z))1≤l,l¯≤d2 , h
i
ω,t(x, z))1≤i≤d2 ,
are random fields on Ω × [0,T ] × Rd1 × Z1 that are PT ⊗ B(Rd1) ⊗Z1-measurable and
ζ2t (x, z) = (ζ
2;i
ω,t(x, z))1≤i≤d1 , ρ
2
t (x, z) = (ρ
2;ll¯
ω,t (x, z))1≤l,l¯≤d2 ,
are random fields on Ω× [0,T ]×Rd1 ×Z2 that are RT ⊗B(Rd1)⊗Z2-measurable. We also
assume that there is a constant C such that Vt ≤ C for all (ω, t) ∈ Ω × [0,T ].
Let us introduce the following assumption for an integer m ≥ 1 and a real number
α1 ∈ (1, 2). We remind the reader that the norms and seminorms | · |0 = [·]0 and | · |β,
4.3. The L2-Sobolev theory for degenerate SIDEs 107
β ∈ (0, 1], were defined in the beginning of Section 2.2.
Let us introduce the following assumption for m ∈ N and a real number β ∈ [0, 2].
Assumption 4.3.1 (m, d2). Let N0 be a positive constant.
(1) For all (ω, t, x) ∈ Ω × [0,T ] × Rd1 , the derivatives in x of the random fields bt, ct, σ2t ,
and υ2t up to order m and σ
k
t and υ
k
t , k ∈ {1, 2}, up to order m + 1 exist, and for all
x ∈ Rd1 ,
max
|γ|≤m
(
|∂γbt(x)| + |∂γct(x)| + |∂γ∇σ1t (x)| + |∂
γσ2t (x)| + |∂
γ∇υ1t (x)| + |∂
γυ2t (x)|
)
≤ N0.
(2) For each k ∈ {1, 2} and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, the derivatives in x of
the random field ζkt (z) up to order m exist, and for all x ∈ R
d1 ,
max
|γ|≤m
|∂γζ1t (x, z) | + max
|γ|=m
[
∂γζ1t (·, z)
]
β
2
≤ K1t (z) , ∀z ∈ Z
1,
max
|γ|≤m
|∂γζ2t (x, z) | ≤ K
2
t (z), ∀z ∈ Z
2,
where K1t (resp. K
2
t ) are PT ⊗Z
1 (resp. PT ⊗Z1) -measurable processes satisfying
sup
z∈Zk
Kkt (z) +
∫
Z1
K1t (z)
βπ1t (dz) +
∫
Z2
K2t (z)
2π2(dz) ≤ N0.
(3) There is a constant η < 1 such that for each k ∈ {1, 2} on the set all (ω, t, x, z) ∈ Ω ×
[0,T ] × Rd1 × Zk for which |∇ζkt (x, z)| > η,
∣∣∣∣
(
Id1 + ∇ζ
k
t (x, z)
)−1∣∣∣∣ ≤ N0.
(4) For each k ∈ {1, 2} and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, the derivatives in x of
the random field ρkt (z) up to order m exist, and for all x ∈ R
d1 ,
max
|γ|≤m
|∂γρ1t (x, z)| + max
|γ|=m
[
ρ1t (·, z)
]
β
2
≤ l1t (z),
max
|γ|≤m
|∂γρ2t (x, z)| ≤ l
2
t (z) ,
where l1 (resp. l2) is PT ⊗Z1 (resp. PT ⊗Z2) -measurable function satisfying
∫
Z1
l1t (z)
2π1t (dz) +
∫
Z2
l2t (z)
2π2(dz) ≤ N0.
Let L2 = L2(Rd1 ,B(Rd1), ν; R), where ν (differential is denoted by dx) is the Lebesgue
measure. Let S = S(Rd1) be the Schwartz space of rapidly decreasing functions on Rd1 .
108 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
The Fourier transform of an element v ∈ S is defined by
vˆ(ξ) = F v(ξ) =
∫
Rd1
v(x)e−i2πξ·xdx, ξ ∈ Rd1 .
We denote by F −1 its inverse. Denote the space of tempered distributions by S′, the dual
of S.
Let ∆ :=
∑d1
i=1 ∂
2
i be the Laplace operator on R
d1 . For α ∈ R, we define the Sobolev
scale
Hα(Rd1; Rd)
=
{
v = (vi)1≤i≤d : v
i ∈ S′ and
(
1 + 4π2 |ξ|2
)α/2
vˆi ∈ L2(Rd1), ∀i ∈ {1, . . . , d}
}
=
{
v = (vi)1≤i≤d : v
i ∈ S′ and (I − ∆)
α
2 vi ∈ L2(Rd1), ∀i ∈ {1, . . . , d}
}
with the norm and inner product given by
∥v∥α,d =


d∑
i=1
∣∣∣∣
(
1 + 4π2 |ξ|2
)α/2
vˆi
∣∣∣∣
2
L2


1/2
=


d∑
i=1
∣∣∣(I − ∆)α/2 vi
∣∣∣
2
L2


1/2
and
(v, u)α,d =
d∑
i=1
(
(I − ∆)α/2 vi, (I − ∆)α/2 ui
)
L2
, ∀u, v ∈ Hα(Rd1; Rd),
where
(I − ∆)α/2 vi = F −1
((
1 + 4π2 |ξ|2
)α/2
vˆi
)
.
It is well-known that C∞c (R
d1; Rd) is dense in Hα(Rd1; Rd) for all α ∈ R. For v ∈ H1(Rd1; Rd)
and u ∈ H−1(Rd1; Rd), we let
⟨v, u⟩1,d = (Λ1v,Λ−1u)0,d,
and identify the dual of H1(Rd1; Rd) with H−1(Rd1; Rd) through this bilinear form. More-
over, all of the properties imposed in Section 4.2 for the abstract family of spaces (Hα)α∈R
and operators (Λα)α∈R hold for the Sobolev scale. We refer the reader to [Tri10] for more
details about the Sobolev scale (see the references therein as well).
For each α ∈ R, let Hα(Rd1; Rd;F0) be the space of all F0-measurable Hα(Rd1; Rd)-
valued random variables φ˜ satisfying E
[
∥φ˜∥2α
]
< ∞.
Let Hα(Rd1; Rd2) be the space of all Hα(Rd1; Rd2)-valued RT -measurable processes f :
4.3. The L2-Sobolev theory for degenerate SIDEs 109
Ω × [0,T ]→ Hα(Rd1; Rd2) such that
E
∫ T
0
∥ ft∥2αdVt < ∞.
Let Hα(Rd1; ℓ2(Rd)) be the space of all sequences of Hα(Rd1; Rd)-valuedPT -measurable
processes g˜ = (g˜ϱ)ϱ∈N, g˜ϱ : Ω × [0,T ]→ Hα(Rd1; Rd), satisfying
E
∫ T
0
∥g˜t∥2αdVt = E
∫ T
0
∑
ρ∈N
∥g˜ϱt ∥
2
αdVt < ∞.
Let Hα(Rd1; Rd; π1) be the space of all Hα(Rd1; Rd)-valued PT ⊗Z1-measurable processes
h˜ : Ω × [0,T ] ×Z1 → Hα(Rd1; Rd) such that
E
∫ T
0
∫
Z1
∥h˜t (z) ∥2απ
1
t (dz)dVt < ∞..
For each α ∈ R, we set Hα = Hα(Rd1; Rd2), Hα(F0), Hα = Hα(Rd1; Rd2), Hα(ℓ2) =
Hα(Rd1; ℓ2(Rd2)), Hα(π1) = Hα(Rd1; Rd2; π1), and ∥ · ∥α = ∥ · ∥α,d2 , (·, ·)α = (·, ·)α,d2 , ⟨·, ·⟩1 =
⟨·, ·⟩1,d2 . We also set C
∞
c = C
∞
c (R
d1; Rd).
Definition 4.3.1. Let φ ∈ H0(F0), f ∈ H−1, g ∈ H0(ℓ2), and h ∈ H0(π1). An H0-valued
strongly càdlàg process u = (ut)t≤T is said to be a solution of the SIDE (4.3.1) if u ∈
L2(Ω × [0,T ],OT , dVtdP; H1) and P-a.s. for all t ∈ [0,T ],
ut
H−1
= φ +
∫ t
0
(
(L1;ls +L
2;l
s )us + b
i
s∂iu
l
s + c
ll¯
s u
l¯
s + f
l
s
)
dVs +
∫ t
0
(
N lϱs us + g
lϱ
s
)
dwϱs
+
∫ t
0
∫
Z1
(
Ils,zu
l¯
s− + h
l
s(z)
)
η˜(ds, dz),
where
H−1
= indicates that the equality holds in the H−1. That is, P-a.s. for all t ∈ [0,T ] and
v ∈ H1,
(v, ut)0 = (v, u0) +
∫ t
0
⟨v, (L1s +L
2
s)us + b
i
s∂ius + csus + fs⟩1dVs
+
∫ t
0
(
v,
(
N lϱs us + g
lϱ
s
))
0
dwϱs +
∫ t
0
∫
Z1
(
v,
(
Ils,zu
l¯
s− + h
l
s(z)
))
η˜(ds, dz).
The coming theorem is our existence result for equation (4.3.1). In the next section,
we prove this theorem by appealing to Theorem 4.2.4.
Theorem 4.3.2. Let Assumption 4.3.1(m, d2) hold for m ∈ N and a real number β ∈ [0, 2].
Then for every φ ∈ Hm(F0), f ∈ Hm, g ∈ Hm+1(ℓ2), h ∈ Hm+
β
2 (π1), and there exists a
unique solution u = (ut)t≤T of (4.3.1) that is weakly càdlàg as an Hm-valued process and
110 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
strongly càdlàg as an Hα
′
-valued process for any α′ < m. Moreover, there is a constant
N = N(d1, d2,N0,m, η, β) such that
E
[
sup
t≤T
∥ut∥2m
]
≤ NE
[
∥φ∥2m +
∫ T
0
(
∥ ft∥2m + ∥gt∥
2
m+1 +
∫
Z1
∥ht(z)∥2m+ β2
π1t (dz)
)
dVt
]
.
The following corollary can be proved in the same way as Corollary 1 (with p = 2) in
Chapter 4, Section 2.2 of [Roz90] by making use of the Sobolev embedding theorem.
Corollary 4.3.3. Let Assumption 4.3.1(m, d2) hold for an integer m >
d1
2 and a real number
β ∈ [0, 2]. Then the solution (ut)t≤T of (4.3.1) has a version with the following properties:
(1) for every x ∈ Rd1 , ut(x) is a OT -measurable Rd2-valued process;
(2) for β = (2m − d1)/2 and for all ω ∈ Ω, u ∈ D([0,T ];C
β
loc(R
d1 ; Rd2));
(3) it possess all the properties mentioned in Theorem 4.3.2;
(4) for each bounded subset Q ⊆ Rd,
E
[
sup
t≤T
|ut|2β;Q;Rd2
]
+ E
∫
]0,T ]
∥ut∥2β;Q;Rd2 dVt
≤ NE
[
∥φ∥2m +
∫ T
0
(
∥ ft∥2m + ∥gt∥
2
m+1 +
∫
Z1
∥ht(z)∥2m+ β2
π1t (dz)
)
dVt
]
,
where N = N(d1, d2,N0,m, η, β) is a constant;
(5) if (u1t )t≤T and (u
2
t )t≤T are solutions of (4.3.1) possessing properties (1) and (2), then
P

 sup
t≤T,x∈Rd1
|u1t (x) − u
2
t (x)| > 0

 = 0.
Remark 4.3.4. If m > α1∨α2 + d2 , then u ∈ D([0,T ];C
α1∨α2
loc (R
d1; Rd2)) and we can obtain a
representation of the solution of (4.3.1) as in Theorem 3.2.2 by applying the Ito-Wentzell
formula given in Proposition 3.4.16. However, there are two important points to notice.
First, since we have only established an integer regularity theory for our equation, then
the best we can hope for is a theory for classical solutions with integer assumptions on
the coefficients, initial condition, and free terms. This is in stark contrast with Chapter 3.
Moreover, the restriction to p = 2 has dire consequences, since the d12 term gets larger as
d1 grows, and hence one must impose more regularity as the dimension grows. This is a
strong motivation to consider the Lp-Sobolev theory for (4.3.1) in weighted spaces, so that
the d12 can be replaced with
d1
p . This way, by taking p large, the term
d1
p can be made as
small as one likes. It is worth mentioning that we could have considered weighted scale of
Sobolev spaces here (see, e.g. [GK92]), but we will leave this for a future project. At the
time of writing, the integer scale Lp-Sobolev theory for (degenerate) (4.3.1) is currently
underway and will be available soon.
4.3. The L2-Sobolev theory for degenerate SIDEs 111
4.3.2 Proof of Theorem 4.3.2
By [MR99] (see Examples 2.3-2.4), the stochastic integrals in (4.3.1) can be written as
stochastic integrals with respect to a cylindrical martingale. We will apply Theorem 4.2.4
to (4.3.1) with α = m and µ = 1 by checking that Assumptions 4.2.1(λ, 1) and 4.2.2(λ, 1)
for λ ∈ {0,m} are implied by Assumption 4.3.1(m, d2). We start with λ = 0 as our base
case and show that λ = m can be reduced to it.
We introduce our base assumption for β ∈ [0, 2] .
Assumption 4.3.2 (d2). Let N0 be a positive constant.
(1) For all (ω, t, x) ∈ Ω × [0,T ] × Rd1 , the derivatives in x of the random fields bt, σ1t , σ
2
t ,
and divσ1t exist, and for all x ∈ R
d1 ,
|∇ divσ1t (x) | + |σ
k
t (x) | + |∇σ
k
t (x)| + |divbt(x)| + |ct(x)| + |υ
2
t,sym (x) | + |∇υ
1
t (x) | ≤ N0.
(2) For each k ∈ {1, 2} and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, the derivatives in x of
the random fields ζkt (z) exist, and for all x ∈ R
d1 ,
|ζ1t (x, z)| ≤ K
1
t (z) , |∇ζ
1
t (x, z) | ≤ K¯
1
t (z) ,
[
div ζ1t (·, z)
]
β
2
≤ K˜1t (z) , ∀z ∈ Z
1,
|ζ2t (x, z)| ≤ K
2
t (z) , |∇ζ
2
t (x, z) | ≤ K¯
2
t (z) , ∀z ∈ Z
2,
where K1t , K¯
1
t , K˜
1
t (resp. K
2
t , K¯
2
t ) are PT ⊗ Z
1 (resp. PT ⊗ Z2) -measurable processes
satisfying
sup
z∈Z1
(
K1t (z) + K¯
1
t (z) + K˜
1
t (z)
)
+
∫
Z1
(
K1t (z)
β + K¯1t (z)
2 + K˜1t (z)
2
)
π1t (dz) ≤ N0,
sup
z∈Z2
(
K2t (z) + K¯
2
t (z)
)
+
∫
Z2
K¯2t (z)
2π2(dz) ≤ N0.
(3) There is a constant η < 1 such that for each k ∈ {1, 2} on the set all (ω, t, x, z) ∈ Ω ×
[0,T ] × Rd1 × Zk for which |∇ζkt (x, z)| > η,
∣∣∣∣
(
Id1 + ∇ζ
k
t (x, z)
)−1∣∣∣∣ ≤ N0.
(4) For each k ∈ {1, 2} and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, |ρkt (x, z)| ≤ l
k
t (z), and for
all (ω, t, z) ∈ Ω × [0,T ] × Z1,
[
ρ1t,sym(·, z)
]
β
2
≤ l˜1t (z) , where l
k (resp., l˜1) is a PT ⊗ Zk-
measurable (resp. PT ⊗Z1-measurable) functions satisfying
∫
Z1
(
l1t (z)
2 + l˜1t (z)
2
)
π1(dz) +
∫
Z2
l2t (z)
2π2(dz) ≤ N0.
112 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
Note that Assumption 4.3.2(d2) is weaker than Assumption 4.3.1(0, d2).
Let us make the following convention for the remainder of this section. If we do
not specify to which space the parameters ω, t, x, y, and z belong, then we mean ω ∈ Ω,
t ∈ [0,T ], x ∈ Rd1 , and z ∈ Zk. Moreover, unless otherwise specified, all statements hold
for all ω, t, x, y, and z independent of any constant N introduced is independent of ω, t, x, y,
and z. We will also drop the dependence of processes t, x, and z when we feel it will not
obscure our argument. Lastly, all derivatives and Hölder norms are taken with respect to
x ∈ Rd1 .
Remark 4.3.5. Let Assumption 4.3.2(d2) hold. For each k and θ ∈ [0, 1], on the set all ω, t,
and z in which |Kkt (z)| ≤ η, we have
∣∣∣(Id1 + θ∇ζ
k
t (x, z))
−1
∣∣∣ ≤
1
1 − θη
.
Moreover, for each k and all ω, t, and z, we have
∣∣∣(Id1 + ∇ζ
k
t (x, z))
−1
∣∣∣ ≤ max
(
1
1 − θη
,N0
)
.
Therefore, by Hadamard’s theorem (see, e.g., Theorem 0.2 in [DMGZ94] or 51.5 in
[Ber77]):
• for each k and θ ∈ [0, 1], on the set all ω, t, and z in which |Kkt (z)| ≤ η, the mapping
ζ˜kt,θ(x, z) := x + θζ
k
t (x, z)
is a global diffeomorphism in x;
• for each k and all ω, t, and z, the mapping
ζ˜kt (x, z) = ζ˜
k
t,1(x, z) = x + ζ
k
t (x, z)
is a global diffeomorphism in x.
When inverse of the mapping x 7→ ζ˜kt,θ(x, z) exists, we denote it by
ζ˜
k;−1
t,θ (x, z) =
(
ζ˜
k;−1; j
t,θ (x, z)
)
1≤ j≤d1
and note that
ζ˜
k;−1
t,θ (x, z) = x − θζ
k
t (ζ˜
k;−1
t,θ (x, z), z).
Furthermore, for each k and θ ∈ [0, 1], on the set all ω, t, and z in which |Kkt (z)| ≤ η, there
4.3. The L2-Sobolev theory for degenerate SIDEs 113
is a constant N = N(d1,N0, η) such that
|∇ζ˜k;−1t,θ (x, z)| ≤ N (4.3.2)
and for each k and all ω, t, and z,
|∇ζ˜k;−1t (x, z)| ≤ N. (4.3.3)
Using simple properties of the determinant, we can easily show that there is a constant
N = N(d1) such that for an arbitrary real-valued d1 × d1 matrix A,
| det(Id1 + A) − 1| ≤ N|A| and | det(Id1 + A) − 1 − tr A| ≤ N|A|
2.
Thus, there is a constant N = N(d1,N0, η) such that
| det∇ζ˜k;−1 − 1| =
∣∣∣∣det
(
Id − ζkt (ζ˜
k;−1
t )
)
− 1
∣∣∣∣ ≤ N |∇ζk(ζ˜k;−1)| (4.3.4)
and ∣∣∣∣det∇ζ˜k;−1 − 1 + div
(
ζk(ζ˜k;−1)
)∣∣∣∣ ≤ N |∇ζk(ζ˜k;−1)|2.
Since ∂lζ˜k;−1; j = δl j − ∂mζk; j(ζ˜k;−1)∂lζ˜k;−1;m), we have
| div(ζk(ζ˜k;−1)) − div ζk(ζ˜k;−1)| = |∂ jζk;l(ζ˜
k;−1
t )(∂lζ˜
k;−1; j − δl j)| ≤ N|∇ζk(ζ˜k;−1)|2,
and thus
∣∣∣det∇ζ˜k;−1 − 1 + div ζk(ζ˜k;−1)
∣∣∣ ≤ N |∇ζk(ζ˜k;−1)|2. (4.3.5)
In the following three lemmas, we will show that Assumptions 4.2.1(λ, 1) and 4.2.2(λ, 1)
for λ ∈ {0,m} hold under Assumption 4.3.2(β) for any β ∈ [0, 2]. For each l ∈ {1, . . . , d2}
and all ϕ ∈ C∞c , let
Lltϕ = (L
1;l
t +L
2;l
t )ϕ + b
i
t∂iϕ
l + cll¯t ϕ
l¯,
A1;lt ϕ =
1
2
σ
1;iϱ
t σ
1; jϱ
t ∂i jϕ
l + σ
1;iϱ
t υ
1;ll¯ϱ
t ∂iϕ
l¯, and J1t ϕ = L
1
t ϕ −A
1
t ϕ.
Lemma 4.3.6. Let Assumption 4.3.2(d2) hold. Then there is a constant N = N(d1, d2,N0, η)
such that for all (ω, t) ∈ Ω × [0,T ] and v ∈ H1,
∥Ltv∥−1 ≤ N∥v∥1, ∥Atv∥−1 ≤ N∥v∥1, ∥J1t v∥−1 ≤ N∥v∥1,
∥Ntv∥0 ≤ N∥ϕ∥1, and
∫
Z1
∥It,zv∥20π
1
t (dz) ≤ N∥v∥
2
1.
114 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
Proof. First we will show that there is a constant N such that
(ψ,Ltϕ)0 ≤ N∥ψ∥1∥ϕ∥1, ∀ϕ ∈ C∞c .
Once this is established, we know that L extends to a linear operator from H1 to H−1
(still denoted by L) and ∥Ltv∥−1 ≤ N∥v∥1, for all v ∈ H1. Using Taylor’s formula and the
divergence theorem, we get that for all and all ϕ, ψ ∈ C∞c
(ψ,Lϕ)0 =
2∑
k=1
(
(ψ,Lktϕ)0 + (∂iψ,Y
k;i
t ϕ)0 + (ψ, bt∂iϕ)0 + (ψ, ctϕ)0
)
,
where for each k ∈ {1, 2}, l ∈ {1, . . . , d2}, and i ∈ {1, 2, . . . , d1},
Lk;lϕ : = −
∫
K¯k<η
∫ 1
0
(
ϕl(ζ˜kθ ) − ϕ
l
)
∂iζ
k;idθπk(dz)
−
∫
K¯k<η
∫ 1
0
θ∂ jϕ
l(ζ˜kθ )∂iζ
k; jζk;idθπk(dz)
+
∫
Zk
ρk;ll¯
(
ϕl(ζ˜k) − ϕl
)
πk(dz) + σk;iϱυk;ll¯ϱ∂iϕ
l¯
+
∫
K¯k>η
(
ϕl(ζ˜k) − ϕl − ζk;i∂iϕl
)
πk(dz),
Yk;liϕ : = −
∫
K¯k<η
∫ 1
0
(
ϕl(ζ˜kθ ) − ϕ
l
)
ζk;idθπk(dz) −
1
2
∂i
(
σk;iϱσk; jϱ
)
∂ jϕ
l.
For the remainder of the proof, we make the convention that statements hold for all ϕ, ψ ∈
C∞c and that all constants N are independent of ϕ. By Minkowski’s integral inequality and
Hölder’s inequality, we have (using the notation of Remark 4.3.5)
||Lkϕ||0 ≤
(∫
K¯k<η
(Kk)2πk(dz)
) 1
2
∫ 1
0
(∫
Rd1
∫
K¯k<η
|ϕ(ζ˜kθ (z)) − ϕ|
2πk(dz)dx
) 1
2
dθ
+
∫
K¯k<η
Kk(z)2
∫ 1
0
(∫
Rd1
|∇ϕ(ζ˜kθ )|
2dx
) 1
2
πk(dz)θdθ
+
(∫
Z1
(lk)2πk(dz)
) 1
2
(∫
Rd1
∫
Z1
|ϕ(ζ˜k) − ϕ|2πk(dz)dx
) 1
2
+N∥∇ϕ∥0 +
∫
K¯k≥η
∫ 1
0
(∫
Rd1
|ϕ(ζ˜k) − ϕ − ζk;i∂iϕ|2dx
) 1
2
dθπk(dz),
and for all i ∈ {1, . . . , d1},
||Yk;iϕ||0 ≤
(∫
K¯k<η
(Kk)2πk(dz)
) 1
2
∫ 1
0
(∫
Rd1
∫
K¯k<η
|ϕ(ζ˜kθ ) − ϕ|
2πk(dz)dx
) 1
2
dθ + N∥∇ϕ∥0.
4.3. The L2-Sobolev theory for degenerate SIDEs 115
Applying the change of variable formula and appealing to (4.3.2), we find that
∫
Rd1
∫
K¯k<η
|ϕ(ζ˜kθ ) − ϕ|
2πk(dz)dx ≤ θ
∫
K¯k<η
∫ 1
0
∫
Rd1
|∇ϕ(ζ˜k
θθ¯
)|2|ζk|2dxdθ¯
≤ θ
∫
K¯k<η
(Kk)2
∫ 1
0
∫
Rd1
|∇ϕ|2
∣∣∣det∇ζ˜k;−1
θθ¯
∣∣∣ dxdθ¯ ≤ Nθ∥∇ϕ∥20.
Similarly, since πk({z ∈ Zk : K¯k ≥ η}) ≤ N0, we have
∫
Rd1
∫
K¯k≥η
|ϕ(ζ˜k) − ϕ|2πk(dz)dx ≤ 2
∫
Rd1
∫
K¯k≥η
(
|ϕ(ζ˜k)|2 + |ϕ|2
)
πk(dz)dx
≤
∫
K¯k≥η
∫
Rd1
|ϕ|2
(
1 +
∣∣∣det∇ζ˜k;−1
∣∣∣
)
dxπk(dz) ≤ N∥ϕ∥20
and
∫
K¯k≥η
(∫
Rd1
|ϕ(ζ˜k) − ϕ − ζk;i∂iϕ|2dx
) 1
2
πk(dz)
≤ N
∫
K¯k≥η
(∫
Rd1
(
|ϕ|2
(
1 +
∣∣∣det∇ζ˜k;−1
∣∣∣
)
+ (Kk)2|∇ϕ|2
)
dx
) 1
2
πk(dz) ≤ N∥ϕ∥1,
where in the last inequality we used (4.3.3). Moreover,
∫
K¯k<η
(Kk)2
(∫
Rd1
|∇ϕ(ζ˜kθ )|
2dx
) 1
2
πk(dz)
≤
∫
K¯k<η
(Kk)2
(∫
Rd1
|∇ϕ|2 det∇ζ˜k;−1
θ
dx
) 1
2
πk(dz) ≤ N∥∇ϕ∥0.
Combining the above estimates, we get that (ψ,Lϕ)0 ≤ N∥ψ∥1∥ϕ∥1. It is clear from the
above computation that
∥A1t ϕ∥−1 ≤ N∥ϕ∥1, ∥J
1
t ϕ∥−1 ≤ N∥ϕ∥1,
where actually A1 and J1 are actually extensions of the operators defined above. The
inequality ∥Nϕ∥0 ≤ N∥ϕ∥1 can easily be obtained. Following similar calculations to ones
we derived above (using (4.3.3) and (4.3.2)), we obtain
∫
Z1
||Iϕ||20π
1(dz) ≤ N(A1 + A2),
116 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
where
A1 :=
∫
Rd1
∫
K¯1≤η
∫ 1
0
|∇ϕ(ζ˜1θ )|
2|ξ1|2π1(dz)dθdx
+
∫
Rd1
∫
K1>η
(
|ϕ(ζ˜1)|2 + |ϕ|2
)
π1(dz)dx ≤ N∥ϕ∥21
and
A2 :=
∫
Rd1
∫
Z1
ρ1;ll¯ϕl¯(ζ˜1)π1 (dz) dx ≤ N∥ϕ∥20.
□
Lemma 4.3.7. Let Assumption 4.3.2(d2) hold. Then there is a constant N = N(d1, d2,N0, η, β)
such that for all (ω, t) ∈ Ω × [0,T ] and all v ∈ H1,
2⟨v,L2t v + b
i
t∂ivt + c
·l¯
t v
l¯
t⟩1 +
1
4
(σ2;iϱt ∂iv, σ
2; jϱ
t ∂ jv)0
+
1
4
∫
Z2
∥v(ζ˜2t (z)) − v∥
2
0π
2
t (dz) ≤ N||v||
2
0, (4.3.6)
2⟨v,A1t v⟩1 + ||Ntv||
2
0 ≤ N||v||
2
0, 2⟨v,J
1
t v⟩1 +
∫
Z1
||It,zv||20π
1
t (dz) ≤ N ||v||
2
0, (4.3.7)
and
2⟨v,Ltv + ft⟩1 + ||Ntv + gt||20 +
∫
Z1
||It,zv + ht (z) ||20π
1 (dz)
+
1
4
(σ2;iϱt ∂iv, σ
2; jϱ
t ∂ jv)0 +
1
4
∫
Z2
∥v(ζ˜2t (z)) − v∥
2
0π
2
t (dz)
≤ N
(
∥v∥20 + || ft||
2
0 + ||gt||
2
1 +
∫
Z1
||ht (z) ||2β
2
π1t (dz)
)
.
Proof. For the remainder of the proof, we make the convention that statements hold for
all ϕ ∈ C∞c and that all constants N are independent of ϕ. Using the divergence theorem,
we get
2⟨ϕ,A1ϕ⟩1 + ||Nϕ||20 =
1
2
∫
Rd1
(
| divσ1|2 + 2σ1;i∂i divσ1 + ∂ jσ1;i∂iσ1; j
)
|ϕ|2dx
+
∫
Rd1
(
|υ1ϕ|2 − 2ϕl
(
υ1;ll¯sym divσ
1 + σ1;i∂iυ
1;ll¯
sym
)
ϕl¯
)
dx ≤ N∥ϕ∥20.
Rearranging terms and using the identity 2a(b − a) = −|b − a|2 + |b|2 − |a|2, a, b ∈ R, we
obtain
2⟨ϕ,J1ϕ⟩1 +
∫
Z1
||Iϕ||20π
1
t (dz) = A1 + A2,
4.3. The L2-Sobolev theory for degenerate SIDEs 117
where
A1 :=
∫
Rd1
∫
Z1
(
|ϕ(ζ˜1)|2 − |ϕ|2 − 2ϕζ1;i∂iϕ
)
π1(dz)dx
A2 := 2
∫
Rd1
∫
Z1
(
ϕl(ζ˜1)ρ1;ll¯ϕl¯(ζ˜1) − ϕlρ1;l¯ϕl
)
π1(dz)dx +
∫
Rd1
∫
Z1
|ρ1ϕ(ζ˜1)|2π1(dz)dx,
Since
| div ζ1(ζ˜1;−1) − div ζ1| ≤ [div ζ1] β
2
(K1)
β
2 ≤ (K˜1)2 + (K1)β,
changing the variable of integration and making use of the estimate (4.3.5), we obtain
A1 ≤
∫
Rd1
|ϕ|2
∫
Z1
| det∇ζ˜1;−1 − 1 + div ζ1|π1(dz)dx ≤ N∥ϕ∥20
and
A2 = 2
∫
Rd1
∫
Z1
ϕl
(
ρ1;ll¯(ζ˜1;−1) − ρ1;ll¯
)
ϕl¯π1(dz)dx
+
∫
Rd1
∫
Z1
2ϕlρ1;ll¯(ζ˜1;−1)ϕl¯
(
det∇ζ˜1;−1 − 1
)
π1(dz)dx
+
∫
Rd1
∫
Z1
|ρ1(ζ˜1;−1)ϕ|2 det∇ζ˜1;−1π1(dz)dx =: A21 + A22 + A23.
Owing to (4.3.4) and Hölder’s inequality, we have
A22 + A23 ≤ N
∫
Z1
((l1)2 + (K1)2)π1(dz)||ϕ||20.
For β > 0, we have
A21 ≤ N
∫
Z1
[
ρ1sym
]
β
2
(K1)
β
2π1(dz)∥ϕ∥20 ≤ N
∫
Z1
(
(l˜1)2 + (K1)β
)
π1(dz)∥ϕ∥20 ≤ N∥ϕ∥
2
0
and for β = 0, using Holder’s inequality, we get
A21 ≤ N∥ϕ∥20
∫
Z1
(l1)2π1(dz).
By the divergence theorem, we have
2⟨ϕ,L2ϕ⟩0 = B1 + B2 + B3,
where
B1 :=
∫
Rd1
(
ϕlσ2;iϱσ2; jϱ∂i jϕ
l + 2σ2;iϱυ2;ll¯ϱ∂iϕ
l¯
)
dx,
118 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
B2 := 2
∫
Rd1
∫
Z2
ϕl
(
ϕl(ζ˜2) − ϕl − ζ2;i∂iϕl
)
π2(dz)dx,
B3 := 2
∫
Rd1
∫
Z2
ϕlρ2;ll¯
(
ϕl¯(ζ˜2) − ϕl¯
)
π2(dz)dx.
Owing to the divergence theorem, we have
(ϕ, σ2;iϱσ2; jϱ∂i jϕ)0 = −
((
σ2;iϱσ2; jϱ∂iϕ + ϕ
(
σ2; jϱ divσ2;ϱ + σ2;iϱ∂iσ
2; jϱ
))
, ∂ jϕ
)
0
(ϕσ2;iϱ∂iσ
2; jϱ, ∂ jϕ)0 = −
1
2
(
ϕ
(
∂ jσ
2;iϱ∂iσ
2; jϱ + σ2; jϱ∂ j divσ
2;ϱ
)
, ϕ
)
0
,
(ϕσ2; jϱ∂ j divσ
2;ϱ, ϕ)0 = −(ϕ| divσ2|2, ϕ)0 + 2(ϕσ2;iϱ divσ2;ϱ, ∂ jϕ)0,
and hence,
(ϕ, σ2;iϱt σ
2; jϱ
t ∂i jϕ)0 = −
(
σ2;iϱσ2; jϱ∂iϕ
l∂ jϕ
l + 2ϕσ2; jϱ divσ2;ϱ, ∂ jϕ
)
0
+
1
2
(
ϕ
(
∂ jσ
2;iϱ∂iσ
2; jϱ − | divσ2|2
)
, ϕ
)
0
.
Thus, by Young’s inequality,
B1 ≤ −
1
2
∫
∂iϕ
lσ2;iϱσ2; jϱ∂ jϕ
ldx + N∥ϕ∥20.
Once again making use of the identity 2a(b − a) = −|b − a|2 + |b|2 − |a|2, a, b ∈ R, we get
2ϕl(ϕl(ζ˜2) − ϕl − ζ2;i∂iϕl) = −|ϕ(ζ˜2) − ϕ|2 + |ϕ(ζ˜2)|2 − |ϕ|2 − ζ
2;i
t ∂i|ϕ|
2.
Changing the variable of integration and applying the divergence theorem, we obtain
B2 = −
∫
Rd1
∫
Z2
|ϕ(ζ˜2) − ϕ|2π2(dz)dx
+
∫
Rd1
∫
Z2
|ϕ|2
(
det∇ζ˜2;−1 − 1 + div ζ2(ζ˜2;−1)
)
π2(dz)dx
+
∫
Rd1
∫
Z2
|ϕ|2
(
div ζ2 − div ζ2(ζ˜2;−1)
)
π2(dz)dx.
Changing the variable of integration in the last term of B2, we get
∫
Rd1
∫
Z2
|ϕ|2
(
div ζ2 − div ζ2(ζ˜2;−1)
)
π2(dz)dx
=
∫
Rd1
∫
Z2
(
|ϕ|2 − |ϕ(ζ˜)|2 det∇ζ˜2
)
div ζ2π2(dz)dx
=
∫
Rd1
∫
Z2
|ϕ(ζ˜2)|2
(
1 − det∇ζ˜2
)
div ζ2π2(dz)dx
4.3. The L2-Sobolev theory for degenerate SIDEs 119
+
∫
Rd1
∫
Z2
(ϕl − ϕl(ζ˜2))(ϕl + ϕl(ζ˜2)) div ζ2π2(dz)dx =: B21 + B22.
Clearly,
B21 ≤ N
∫
Z2
(K¯2)2π2(dz)∥ϕ∥20,
and applying Hölder’s inequality,
B22 ≤ N
∫
Rd1
(∫
Z2
|ϕ(ζ˜2) − ϕ|2π2(dz)
) 1
2
(∫
Z2
(
|ϕ|2 + |ϕ(ζ˜2)|2
)
(K¯2)2π2(dz)
) 1
2
dx.
Hence, by Remark 4.3.5 and Young’s inequality,
B2 ≤ −
1
2
∫
Rd1
∫
Z2
|ϕ(ζ˜2) − ϕ|2π2(dz)dx + N∥ϕ∥20.
By Hölder’s inequality,
B3 ≤ N
∫
Rd1
(∫
Z2
|ϕ(ζ˜2) − ϕ|2π2(dz)
) 1
2
(∫
Z2
(l2)2π2(dz)
) 1
2
|ϕ|dx.
Applying Young’s inequality again and combining B2 and B3, we derive
2⟨ϕ,L2ϕ⟩1 ≤ N∥ϕ∥20 −
1
4
∫
∂iϕ
lσ2;iϱσ2; jϱ∂ jϕ
ldx −
1
4
∫ ∫
Z2
|ϕ(ζ˜2) − ϕ|2π2(dz)dx. (4.3.8)
By the divergence theorem, we have
2⟨ϕ, bi∂iϕ + c·l¯ϕl¯ + f ⟩0 = 2 (ϕ, f )0 + (ϕ, ϕ div b)0 + 2(ϕ, cϕ)0 ≤ N(∥ϕ∥
2
0 + ∥ f ∥
2
0). (4.3.9)
Combining (4.3.8) and (4.3.9), we obtain (4.3.6). To obtain the estimate (4.3.7), we use
(4.3.6) and (4.3.7), and estimate the additional terms:
D :=
(
σ1;iϱ∂iϕ + υ
1;·l¯ϱϕl¯, gϱ
)
0
and
2
∫
Z1
((
ϕ(ζ˜1) − ϕ, h
)
0
+ (ρ1ϕ(ζ˜1), h)0
)
π1(dz) =: E1 + E2.
By the divergence theorem and Hölder’s inequality,|D| ≤ N
(
||ϕ||20 + ||g||
2
1
)
.Applying Hölder’s
inequality and changing the variable of integration, we get
E2 ≤
∫
Rd1
∫
Z1
(
|ρ1ϕ(ζ˜1)|2 + |h|2
)
π1(dz)dx ≤ N
(
||ϕ||20 +
∫
Z1
||h (z) ||20π
1(dz)
)
.
120 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
Then by (4.3.4), Hölder’s inequality, and Lemma 4.3.10,
E1 = 2
∫
Z1
∫
Rd1
ϕl
(
hl(ζ˜1;−1)(det∇ζ˜1;−1 − 1) + hl(ζ˜1;−1) − h
)
dxπ1(dz)
≤ N
(
∥ϕ∥20 +
∫
Rd1
∫
Z1
(
|h|2 + |h(ζ˜1;−1) − h|2
)
π1(dz)dx
)
≤ N
(
∥ϕ∥20 +
∫
Z1
||h(z)||2β
2
π1(dz)
)
.
This completes the proof. □
In the following lemma, we verify that Assumption 4.2.2(0, 1) holds for (4.3.1). Recall
thatW0,1 is the space of all H0-valued strongly càdlàg processes v : Ω × [0,T ]→ H0 that
belong to L2(Ω × [0,T ],OT , dVtdP; H1).
Lemma 4.3.8. Let Assumption 4.3.2(d2) hold. Then there is a constant N = N(d1, d2,N0, η, β)
such that for all v ∈ W0,1, P-a.s.:
(1)
2⟨vt,Ltvt⟩1dVt + ∥Ntvt∥20dVt +
∫
Z1
∥It,zvt−∥20η(dt, dz)
+2(vt,N
ϱ
t vt)0dw
ϱ
t + 2
∫
Z1
(vt−,It,zvt−)0η˜ (dt, dz)
≤
(
N ||vt||20dVt +
∫
Z1
Nκt(z)||vt−||20η(dt, dz) + 2(vt,N
ρ
t vt)0dw
ϱ
t +
∫
Z1
Gt,z(v)η˜(dt, dz)
)
,
where
|Gt,z(v)|dVt ≤ κ¯t(z)||vt−||20dVt, ∀z ∈ Z
1, |(vt,Ntvt)0|dVt ≤ N ||vt||20 dVt,
and κt and κ¯t are PT ×Z1-measurable processes such that for all t ∈ [0,T ],
∫
Z1
(
κt(z) + κ˜t(z)
2
)
π1t (dz) ≤ N;
(2)
2(Nϱt vt, g
ϱ
t )0dVt + 2
∫
Z1
(It,zvt−, ht(z))0η(dt, dz) + 2(vt, g
ϱ
t )0dw
ϱ
t + 2
∫
Z1
(vt−, ht(z))0η˜(dt, dz)
≤
(
N∥vt−∥0rtdVt +
∫
Z1
N∥vt−∥0∥ht(z)∥0κˆt(z)η(dt, dz) + 2(vt, g
ϱ
t )0dw
ϱ
t + 2
∫
Z1
G¯t,z(v)η˜(dt, dz)
)
,
4.3. The L2-Sobolev theory for degenerate SIDEs 121
where
rt := ∥gt∥1 +
∣∣∣∣∣
∣∣∣∣∣
∫
Z1
(
ht(ζ˜
1;−1
t (z), z) − ht(z)
)
π1t (dz)
∣∣∣∣∣
∣∣∣∣∣
0
, t ∈ [0,T ],
|(vt, gt)0|dVt ≤ N∥vt∥0∥gt∥0dVt,
G¯t,z(v)dVt ≤ N ||vt−||0||ht(z)||0, dVt, ∀z ∈ Z1,
and κˆt is a PT ×Z1-measurable process such that for all t ∈ [0,T ],
∫
Z1
κˆt(z)
2π1t (dz) ≤ N.
Proof. (1) Owing to the divergence theorem, we have
2(vt,N
ϱ
t vt)0 = (vt, ut divσ
1ϱ
t )0 + 2(vt, υ
1ϱ
t vt)0, ∀ϱ ∈ N,
and hence P-a.s.,
|2(vt,Ntvt)0|dVt ≤ N||vt||2dVt.
By virtue of Lemma 4.3.7(1), it suffices to estimate
Q := 2⟨vt,J1t,zvt⟩1dVt +
∫
Z1
∥It,zvt−∥20η(dt, dz) + 2
∫
Z1
(vt−,It,zvt−)0η˜ (dt, dz) .
An application of divergence theorem shows that
Q =
∫
Z1
Pt,z(u)η(dt, dz) +
∫
Z1
Gt,z(v)η˜(dt, dz),
where
Gt,z(v) := 2(vt−, ρ
1
t (z)vt−)0 − (vt−, vt− div ζ
1
t (z))0,
and Pt,z(v) := D1 + D2 + D3 with
D1 := 2(vt−(ζ˜
1
t (z)), ρ
1
t (z)vt−(ζ˜
1
t ))0 − 2(vt−, ρ
1
t (z)vt−)0,
D2 := ∥vt−(ζ˜1t (z))∥
2
0 − ∥vt−∥
2
0 + (vt−, vt− div ζ
1
t (z))0, D3 := ∥ρ
1
t (z)vt−(ζ˜
1
t (z))∥
2
0.
Given our assumptions, it is clear that P-a.s.,
Gt,z(v)dVt ≤ N
(
l1t (z) + K¯
1
t (z)
)
∥vt−∥20dVt and D3dVt ≤ l
1
t (z)
2 ||vt−||20dVt,
where in the last inequality we used the change of variable formula. Changing the variable
122 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
of integration and using (4.3.4) and (4.3.5), we find that dP -a.s.,
D1dVt ≤ N
(
vt−, vt−
∣∣∣ρ1t (ζ˜
1;−1
t (z), z) det∇ζ˜
1;−1
t (z) − ρ
1
t (z)
∣∣∣
)
0
dVt
≤ N
(
l1t (z)K¯
1
t (z) + l˜
1
t (z)K
1
t (z)
β
2
)
∥vt−∥20dVt,
and
D2dVt =
(
vt−, vt−| det∇ζ˜
1;−1
t (z) − 1 + div ζ
1
t (z)|
)
0
dVt ≤
(
K¯1t (z)
2 + K˜1t (z)K
1
t (z)
β
2
)
N∥vt−∥20dVt.
Setting
κt(z) = l
1
t (z)
2 + l1t K¯
1
t (z) + l˜
1
t (z)K
1
t (z)
β
2 + K¯1t (z)
2 + K˜1t (z)K
1
t (z)
β
2 , κ¯t(z) = l
1
t (z) + K¯
1
t (z), z ∈ Z
1,
and appealing to our assumptions, we complete the proof (1).
(2) By the divergence theorem, we have
(gϱt ,N
ϱ
t vt)0 = (g
ϱ
t , divσ
1ϱ
t vt)0 + (σ
1;iϱ∂ig
ϱ
t , vt)0, ∀ρ ∈ N,
and thus by the Cauchy-Schwarz inequality,
|(gt,Ntvt)0|dVt ≤ N∥vt∥0∥gt∥1dVt.
Changing the variable of integration, we obtain
(It,zvt−, ht(z))0 =
(
ht(z),
(
vt−(ζ˜
1
t (z)) − vt− + ρ
1
t (z)vt−(ζ˜
1
t (z))
))
0
= (ht(ζ˜
1;−1
t (z), z) − ht(z), vt−)0 + (ht(ζ˜
1;−1
t (z), z), (det∇ζ˜
1;−1
t (z) − 1)vt−)0
+ (ht(z), ρ
1
t (z)vt−(ζ˜
1
t (z)))0.
A simple calculation shows that P-a.s.,
2
∫
Z1
(It,zvt−, ht(z))0η(dt, dz) + 2
∫
Z1
(vt−, ht(z))0η˜(dt, dz)
≤ 2∥vt−∥0r1t dVt +
∫
Z1
P¯t,z(v)η(dt, dz) +
∫
Z1
G¯t,z(v)η˜(dt, dz),
where
r1t :=
∣∣∣∣∣
∣∣∣∣∣
∫
Z1
(ht(ζ˜
1;−1
t (z), z) − ht(z))π
1
t (dz)
∣∣∣∣∣
∣∣∣∣∣
0
, G¯t,z(v) = (ht(ζ˜
1;−1
t (z), z), vt),
and
Pt,z(v) := (ht(ζ˜
1;−1
t (z)), vt−(det∇ζ˜
1;−1
t (z) − 1))0 + (ht(z), ρ
1
t (z)vt−(ζ˜
1
t (z)))0.
4.3. The L2-Sobolev theory for degenerate SIDEs 123
Applying the change of variable formula and Hölder’s inequality, P-a.s. we obtain
∫
Z1
P˜t,z(u)η(dt, dz) ≤ N∥vt−∥0
∫
Z1
(K¯1t (z) + l
1
t (z))∥ht(z)∥0η(dt, dz)
and
|G˜t,z(v)|dVt ≤ N∥vt−∥0∥ht(z)∥0dVt.
This completes the proof. □
Let d ∈ N. For a function v ∈ Hm(Rd1 ,Rd), define the linear operator Dv ∈ Hm−1(Rd1;
Rd(d1+1)) by
Dv =
(
∂0v, ∂1v, . . . , ∂d1v
)
= v˜
with v˜l0 = vl and v˜l j = ∂ jvl, 1 ≤ l ≤ d, 0 ≤ j ≤ d1 (recall ∂0v = v). We define Dnv for
n ∈ N by iteratively applyingD n-times. Recall that Λ = (I − ∆)
1
2 . It is easy to check that
for each n ∈ N and all u, v ∈ Hn+1(Rd1 ,Rd),
(u, v)n,d = (Λ
nu,Λnv)0,d = (Dnu,Dnv)0,dd¯n1 , (4.3.10)
(Λu,Λ−1v)n,d = (Λ
n+1u,Λn−1v)0,d = (DnΛu,DnΛ−1v)0,dd¯n1 ,
(Dnu,Dnv)−1,dd¯n1 = (D
nΛ−1u,DnΛ−1v)0,dd¯n1 = (u, v)n−1,d,
where d¯1 = d1 + 1. Let us introduce the operators E(L), E(N), and E(Iz) acting on ϕ =
(ϕl j)1≤l≤d2,1≤ j≤d¯1 ∈ C
∞
c (R
d1 ,Rd2d¯1) that are defined as L,N , and I, respectively, but with
the d2 × d2-dimensional coefficients υkt , ρ
k, and c replaced by the d2d¯1 × d2d¯1-dimensional
coefficients given by
υk;l j,l¯ j¯ϱ = υk;ll¯ϱδ j j¯ + 1 j≥1(∂ jσ
k; j¯ϱδll¯ + ∂ jυ
k;ll¯ϱδ j¯0),
ρk;l j,l¯ j¯ = ρk;ll¯t δ j¯ j + 1 j≥1(∂ jρ
k;ll¯δ j¯0 + (δll¯ + ρ
k;ll¯)∂ jζ
k; j¯),
and
cl j,l¯ j¯ = cll¯δ j¯ j + ∂ jb
j¯δll¯ + ∂ jc
ll¯δ j¯0 +
2∑
k=1
(
υk;ll¯ϱ∂ jσ
k; j¯ϱ +
∫
Zk
ρk;ll¯∂ jζ
k; j¯πkt (dz)
)
,
for 1 ≤ l, l¯ ≤ d2 and 0 ≤ j, j¯ ≤ d1. The coefficients σk, b, and functions ζk, k ∈ {1, 2},
remain unchanged in the definition of E(L), E(N), and E(I). We define En(L), En(N),
and En(I), for n ∈ N by iteratively applying E n-times by the rules above with σk, b, and
ζk, k ∈ {1, 2}, unchanged. A simple calculation shows that for all v ∈ H2(Rd1; Rd2),
D[Lv] = E(L)Dv, D[Nϱv] = E(Nϱ)Dv, ϱ ∈ N, D[Izv] = E(Iz)Dv.
124 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
Continuing, for all v ∈ Hn+1(Rd1; Rd) we have
Dn[Lv] = En(L)Dnv, Dn[Nϱv] = En(Nϱ)Dnv, ϱ ∈ N, Dn[Izv] = En(Iz)Dnv.
(4.3.11)
If Assumption 4.3.1(m, d2) holds, it can readily be verified by induction and the definitions
(4.3.2)-(4.3.2) that Assumption 4.3.2(0, d2d¯m1 ) holds for the coefficients of the operators
Em(L), Em(N), and Em(I). Moreover, owing to our assumptions on the input data, we
have
Dmϕ ∈ H0(Rd1 ; Rd2d¯
m
1 ;F0), Dm f ∈ H0(Rd1; Rd2d¯
m
1 )
Dmg ∈ ζ1(Rd1; ℓ2(Rd2d¯
m
1 )), Dmh ∈ H
β
2 (Rd1; Rd2d¯
m
1 ; π1).
Making use of (4.3.10), (4.3.11) and applying Lemma 4.3.6 to Em(L), for all v ∈ Hm+1, we
obtain
∥Lv∥m−1 = ∥Dm[Lv]∥−1 = ∥Em(L)Dmv∥−1,d2d¯m1 ≤ N∥D
mv∥1,d2d¯m1 = N∥v∥m+1.
Likewise, for all v ∈ Hm+1, we derive
∥Nv∥m ≤ N∥v∥m+1,
∫
Z1
||Iv||2mπ
1(dz) ≤ N∥v∥2m+1.
By virtue of Lemma 4.3.7, we have that for all v ∈ Hm+1,
2(Λv,Λ−1Ltv)m + ||Ntv||2m +
∫
Z1
||It,zv||2mπ
1 (dz)
= 2(DmΛv,DmΛ−1[Ltv])0 + ||Dn[Ntv]||20 +
∫
Z1
||Dm[It,zv]||20π
1(dz)
2⟨Dmv,Em(Lt)Dmv⟩1 + ||Em(Nt)Dmv||20 +
∫
Z1
||Em(It,z)Dnv||20π
1(dz) ≤ N ||Dmv||20,d2d¯m1 = N ||v||
2
m
Using a similar argument, we find that for all v ∈ Hm+1,
2(Λv,Λ−1(Ltv + ft))m + ||Ntv + gt||2m +
∫
Z1
||It,zv + ht(z)||2mπ
1(dz) ≤ N||v||2m + N f¯t,
where
f¯t = ∥ ft∥2m + ∥gt∥
2
m+1 +
∫
Z1
∥ht(z)∥2m+ β2
π1t (dz).
Therefore, Assumption 4.2.1(m, d2) holds for the equation (4.3.1). Similarly, using Lem-
mas 4.3.8 and 4.3.10, we find that Assumption 4.2.2(m, d2) holds for equation (4.3.1) as
well. The statement of the theorem then follows directly from Theorem 4.2.4.
4.3. The L2-Sobolev theory for degenerate SIDEs 125
4.3.3 Appendix
For each κ ∈ (0, 1) and tempered distribution f on Rd1 , we define
∂κ f = F −1[| · |κF f (·)],
where F denotes the Fourier transform and F −1 denotes the inverse Fourier transform.
Lemma 4.3.9 (cf. Lemma 2.1 in [Kom84]). Let f : Rd1 → R be smooth and bounded.
Then for all κ ∈ (0, 1), there are constants N1 = N1(d1, κ), N2 = N2(d1, κ), and N3 =
N2(d1, κ) such that for all x, y, z ∈ Rd1 ,
∂κ f (x) = N1
∫
Rd
( f (x + z) − f (x))
dz
|z|d+δ
and
f (x + y) − f (x) = N2
∫
Rd1
∂κ f (x − z)k(κ)(y, z)dz,
where
k(κ)(y, z) = |y + z|κ−d − |z|κ−d and
∫
Rd1
|k(κ)(y, z)|dz = N3|z|κ.
Lemma 4.3.10. Let (Z,Z, π) be a sigma-finite measure space. Let H : Rd1 × Z → Rd1 be
B(Rd) ⊗Z-measurable and assume that for all (x, z) ∈ Rd1 × Z,
|ζ(x, z)| ≤ K(z) and |∇ζ(x, z)| ≤ K¯(z)
where K, K¯ : Z → R+ is a Z-measurable function for which there is a positive constant
N0 such that for some fixed β ∈ (0, 2],
sup
z∈Z
K(z) + sup
z∈Z
K¯(z) +
∫
Z
(
K(z)β + K¯(z)2
)
π(dz) < N0
Assume that there is a constant η < 1 such that (x, z) ∈ {(x, z) ∈ Rd1 × Z : |∇ζ(x, z)| > η},
|
(
Id1 + ∇ζt(x, z)
)−1 | ≤ N0.
Then there is a constant N = N(d1,N0, β, η) such that for all B(Rd1) ⊗ Z-measurable
h : Rd1 × Z → Rd2 with h ∈ L2(Z,Z, π; H
β
2 (Rd1; Rd2)),
∫
Rd1
∣∣∣∣∣
∫
Z
(h(x + ζ(x, z), z) − h(x, z)) π(dz)
∣∣∣∣∣
2
dx ≤ N
∫
Z
∥h(z)∥2β
2
π(dz).
126 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
Proof. It is easy to see that for any B(Rd1) ⊗Z-measurable h : Rd1 × Z → Rd2 such that
∫
Z
sup
x∈Rd1
|∇h(x, z)|2π(dz) < ∞, (4.3.12)
the integral
∫
Z
(h(x+ζ(x, z), z)−h(x, z))π(dz) is well-defined. Moreover, for anyB(Rd1)⊗Z-
measurable h : Rd1 × Z → R with h ∈ L2(Z,Z, π; H
β
2 (Rd1; Rd2)), we can always find
a sequence (hn)n∈N of B(Rd1) ⊗ Z-measurable processes such that each element of the
sequence is smooth with compact support in x and satisfies (4.3.12) and
lim
n→∞
∫
Z
∥h(z) − hn(z)∥2β
2
π(dz) = 0.
Thus, if we prove this lemma for h that is smooth with compact support in x and satisfies
(4.3.12), then we can conclude that the sequence
∫
Z
(hn(x + ζ(x, z), z) − hn(x, z)) π(dz), n ∈ N,
is Cauchy in H0(Rd1; Rd2). We then define
∫
Z
(h(x + ζ(x, z), z) − h(x, z)) π(dz)
for any B(Rd1) ⊗Z-measurable h : Rd1 × Z → R with h ∈ L2(Z,Z, π; H
β
2 (Rd1; Rd2)) to be
the unique H0(Rd1; Rd2) limit of the Cauchy sequence. Hence, it suffices to consider h that
is smooth with compact support in x and satisfies (4.3.12). First, let us consider the case
β ∈ (0, 2). By Lemma 4.3.9, we have
∫
Rd1
∣∣∣∣∣
∫
Z
(
h(ζ˜(x, z), z) − h(x, z)
)
π(dz)
∣∣∣∣∣
2
dx
= N22
∫
Rd1
∣∣∣∣∣
∫
Z
∫
Rd1
∂
β
2 h(x − y, z)k(
β
2 )(ζ(x, z), y) dyπ(dz)
∣∣∣∣∣
2
dx
=: N22
∫
Rd1
|
∫
Z
A(x, z)π(dz)|2dx.
Applying Hölder’s inequality and Lemma 4.3.9, for all x and z, we have
A(x, z) ≤
(∫
Rd1
|∂β/2h(x − y, z)|2k(
β
2 )(ζ(x, z), y)) dy
) 1
2
(∫
Rd1
k(
β
2 )(ζ(x, z), y)) dy
) 1
2
=
√
N3
(∫
Rd1
|∂β/2h(x − y, z)|2k(
β
2 )(ζ(x, z), y) dy
) 1
2
|ζt(x, z)|
β
4
4.3. The L2-Sobolev theory for degenerate SIDEs 127
≤ K(z)
β
2
√
N3
(∫
Rd1
|∂
β
2 h(x − y, z)|2k(
β
2 )(ζ(x, z), y) dyK(z)−
β
2
) 1
2
.
Using Hölder’s inequality again, for all x, we get
∣∣∣∣∣
∫
Z
A(x, z)π(dz)
∣∣∣∣∣
2
≤ N3N0
∫
Z
∫
Rd1
|∂β/2h(x − y, z)|2k(
β
2 )(ζ(x, z), y) dyK(z)−
β
2π(dz).
For each x and z, we set
B(x, z) =
∫
|y|≤2K(z)
|∂
β
2 h(x − y, z)|2k(
β
2 )(ζ(x, z), y) dy
C(x, z) =
∫
|y|>2K(z)
|∂
β
2 h(x − y, z)|2k(
β
2 )(ζ(x, z), y) dy.
Changing the variable integration, for all x and z, we find
B(x, z) ≤
∫
|y+ζ(x,z)|≤3K(z)
|∂
β
2 h(x − y, z)|2
dy
|y + ζ(x, z)|d1−
β
2
+
∫
|y|≤2K(z)
|∂
β
2 h(x − y, z)|2
dy
|y|d1−
β
2
=: B1(x, z) + B2(x, z),
and
B1(x, z) ≤
∫
|y|≤3K(z)
|∂
β
2 h((ζ˜(x, z) − y), z)|2|
dy
|y|d1−
β
2
≤ K(z)
β
2
∫
|y|≤3
|∂
β
2 h((ζ˜(x, z) − yK(z)), z)|2
dy
|y|d1−
β
2
,
B2(x, z) ≤ K(z)
β
2
∫
|y|≤2
|∂β/2h(x − yK(z), z)|2
dy
|y|d1−
β
2
.
Owing to Remark 4.3.5, for all z, the map x 7→ x + ζ(x, z) = ζ˜(x, z) is a global diffeomor-
phism and
det∇ζ˜−1(x, z) ≤ N.
for some constant N = N(N0, d1, η). Thus, by the change of variable formula, there is a
constant N = N(d1,N0, β, η) such that
∫
Rd1
∫
Z
B1(x, z)K(z)
− β2π(dz)dx
≤
∫
Z
∫
|y|≤3
∫
Rd1
|∂β/2h((ζ˜(x, z) − yK(z)), z)|2dx
dy
|y|d1−
β
2
π(dz)
≤
∫
Z
∫
|y|≤3
∫
Rd1
|∂β/2h((x − yK(z)), z)|2| det∇ζ˜−1(x, z)|dx
dy
|y|d1−
β
2
π(dz)
128 Chapter 4. The L2-Sobolev theory for parabolic SIDEs
≤ N
∫
Z
∫
Rd1
|∂
β
2 h(x, z)|2dxπ(dz)
and ∫
Rd1
∫
Z
K(z)
β
2 B2(x, z)dxπ(dz) ≤ N
∫
Z
∫
Rd1
|∂
β
2 h(x, z)|2dxπ(dz).
For all x, y, and z such that |ζ(x, z)| ≤ K(z) ≤ 12 |y|, we have
∣∣∣∣∣∣∣
1
|y + ζ(x, z)|d1−
β
2
−
1
|y|d1−β/2
∣∣∣∣∣∣∣
≤
∣∣∣∣∣d1 −
β
2
∣∣∣∣∣
∣∣∣∣∣∣∣


1
|y + ζ(x, z)|1+d1−
β
2
+
1
|y|1+d1−
β
2


∣∣∣∣∣∣∣
|ζ(x, z)| ≤ 3
∣∣∣∣∣d1 −
β
2
∣∣∣∣∣
|ζ(x, z)|
|y|1+d1−
β
2
,
and hence for all x and z,
C(x, z) =
∫
|y|>2K(z)
|∂
β
2 h(x − y, z)|2
∣∣∣∣∣∣∣
1
|y + ζ(x, z)|d−
β
2
−
1
|y|d−
β
2
∣∣∣∣∣∣∣
dy
≤ N
∫
|y|>2K(z)
|∂
β
2 h(x − y, z)|2
|K(z)|
|y|1+d1−
β
2
dy
≤ NK(z)
β
2
∫
|y|>2
|∂
β
2 h((x − K(z)y), z)|2
dy
|y|1+d1−
β
2
.
Estimating as above, we find that there is a constant N = N(d1,N0, β)
∫
Z
∫
Rd1
K(z)
β
2 C(x, z)dxπ(dz) ≤ N
∫
Z
∫
Rd1
|∂
β
2 h(x, z)|2dxπ(dz).
Combining the above estimates, we obtain the desired estimate for β ∈ (0, 2). Let us now
consider the case β = 2. It follows from Remark 4.3.5 that for all θ ∈ [0, 1], on the set of
z ∈ {z : K¯(z) < 12 }, the map x 7→ x + θζ(x, z) = ζ˜θ(x, z) is a global diffeomorphism and
det∇ζ˜−1θ (x, z) ≤ N,
for some constant N = N(N0, d1). Hence, making use of Taylor’s theorem and the change
of variable formula, we find
∫
Rd1
∣∣∣∣∣
∫
Z
(h(x + ζ(x, z), z) − h(x, z)) π(dz)
∣∣∣∣∣
2
dx
≤
∫
Rd1
∣∣∣∣∣∣
∫
K¯(z)≥ 12
(h(x + ζ(x, z), z) − h(x, z)) π(dz)
∣∣∣∣∣∣
2
dx
+
∫
Rd1
∣∣∣∣∣∣
∫
K¯(z)< 12
∫ 1
0
|∇h(x + θζ(x, z), z)
∣∣∣∣∣∣
dθK(z)π(dz)|2dx
4.3. The L2-Sobolev theory for degenerate SIDEs 129
≤ π
{
K¯(z) ≥
1
2
}∫
K¯(z)≥η
∫
Rd1
|h(x, z)|2| det ζ˜−1(x, z) + 1|dxπ(dz)
+N0
∫
K¯(z)< 12
∫
Rd1
∫ 1
0
|∇h(x, z)|2| det∇ζ˜−1θ (x, z)|dθdxπ(dz) ≤ N
∫
Z
∥h(z)∥21π(dz).
This completes the proof. □

Chapter 5
A finite difference scheme for
non-degenerate parabolic SIDEs
5.1 Introduction
Let (Ω,F ,F,P), F = (Ft)t≥0, be a complete filtered probability space such that the filtration
is right continuous and F0 contains all P-null sets of F . Let w
ϱ
t , t ≥ 0, ϱ ∈ N, be a sequence
of independent real-valued F-adapted Wiener processes. Let π1(dz) and π2(dz) be a Borel
sigma-finite measures on Rd satisfying
∫
Rd
|z|2 ∧ 1 πr(dz) < ∞, r ∈ {1, 2}.
Let q(dt, dz) = p(dt, dz) − π2(dz)dt be a compensated F-adapted Poisson random measure
on R+ × Rd. Let T > 0 be an arbitrary fixed constant. On [0,T ] × Rd, we consider finite
difference approximations for the following SIDE
dut = ((Lt + I)ut + ft) dt +
∞∑
ϱ=1
(
Nϱt ut + g
ϱ
t
)
dwϱt +
∫
Rd
(I(z)ut− + ot(z)) q(dt, dz),(5.1.1)
with initial condition
u0(x) = φ(x), x ∈ Rd,
where the operators are given by
Ltϕ(x) :=
d∑
i, j=0
ai jt (x)∂i jϕ(x),
Iϕ(x) :=
∫
Rd

ϕ(x + z) − ϕ(x) − 1[−1,1](|z|)
d∑
j=1
z j∂ jϕ(x)

 π
1(dz), (5.1.2)
Nϱt ϕ(x) :=
d∑
i=0
σ
iϱ
t (x)∂iϕ(x), I(z)ϕ(x) = ϕ(x + z) − ϕ(x).
Here, we denote the identity operator by ∂0.
131
132 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
Equation (5.1.1) arises naturally in non-linear filtering of jump-diffusion processes.
We refer the reader to [Gri82] and [GM11] for more information about non-linear filter-
ing of jump-diffusions and the derivation of the Zakai equation. Various methods have
been developed to solve stochastic partial differential equations (SPDEs) numerically. For
SPDEs driven by continuous martingale noise see, for example, [GK96, Gyö98, Gyö99,
GM09, LR04, JK10, Yan05], and for SPDEs driven by discontinuous martingale noise, see
[HM06, Hau08, Lan12, BL12]. Among the various methods considered in the literature
is the method of finite differences. For second order linear SPDEs driven by continu-
ous martingale noise it is well-known that the Lp(Ω)-pointwise error of approximation in
space is proportional to the parameter h of the finite difference (see, e.g., [Yoo00]). In
[GM09], I. Gyöngy and A. Millet consider abstract discretization schemes for stochastic
evolution equations driven by continuous martingale noise in the variational framework
and, as a particular example, show that the L2(Ω)-pointwise rate of convergence of an
Euler-Maruyuma (explicit and implicit) finite difference scheme is of order one in space
and one-half in time. More recently, it was shown by I. Gyöngy and N.V. Krylov that under
certain regularity conditions, the rate of convergence in space of a semi-discretized finite
difference approximation of a linear second order SPDE driven by continuous martingale
noise can be accelerated to any order by Richardson’s extrapolation method. For the non-
degenerate case, we refer to [GK10] and [GK11], and for the degenerate case, we refer
to [Gyö11]. In [Hal12] and [Hal13], E. Hall proved that the same method of acceleration
can be applied to implicit time-discretized SPDEs driven by continuous martingale noise.
Also, for a pathwise convergence result of a spectral scheme for SPDEs on a bounded
domain we refer the reader to [CJM92].
In the literature, finite element, spectral, and, more generally, Galerkin schemes have
been studied for SPDEs driven by discontinuous martingale noise. One of the earliest
works in this direction is a paper [HM06] by E. Hausenblas and I. Marchis concerning
Lp(Ω)-convergence of Galerkin approximation schemes for abstract stochastic evolution
equations in Banach spaces driven by Poisson noise of impulsive-type. As an applica-
tion of their result, they study a spectral approximation of a linear SPDE in L2([0, 1])
with Neumann boundary conditions driven by Poisson noise of impulsive-type and derive
Lp(Ω)-error estimates in the L2([0, 1])-norm. In [Hau08], E. Hausenblas considers finite
element approximations of linear SPDEs in polyhedral domains D driven by Poisson noise
of impulsive-type and derives Lp(Ω) error estimates in the Lp(D)-norm. In a more recent
work [Lan12], A. Lang studied semi-discrete Galerkin approximation schemes for SPDEs
of advection diffusion type in bounded domains D driven by cádlág square integrable mar-
tingales in a Hilbert Space. A. Lang showed that the rate of convergence in the Lp(Ω) and
almost-sure sense in the L2(D)-norm is of order two for a finite-element Galerkin scheme.
In [BL12], A. Lang and A. Barth derive L2(Ω) and almost-sure estimates in the L2(D)-
5.1. Introduction 133
norm for the error of a Milstein-Galerkin approximation scheme for the same equation
considered in [Lan12] and obtain convergence of order two in space and order one in time.
In the articles [Lan12, BL12, HM06, Hau08], the authors make use of the semigroup
theory of SPDEs (mild solution) and only consider SPDEs in which the principal part of
the operator in the drift is non-random. Moreover, the authors there do not address the
approximation of equations with non-local operators in the drift or noise. The principal
part of the operator in the drift of the Zakai equation is, in general, random, and hence
numerical schemes that approximate SPDEs or SIDEs with random-adapted principal part
are of importance. More precisely, the coefficients of the Zakai equation are random if the
coefficients of the SDE governing the signal depend on the observation. In this chapter,
since we use the variational framework (L2-theory) of SPDEs, we are easily able to treat
the case of random-coefficients, and hence the diffusion coefficients ai jt (x) appearing in
(5.1.1) are random.
In dimension one, a finite difference scheme for degenerate integro-differential equa-
tions (deterministic) has been studied by R. Cont and E. Voltchkova in [CV05]. The
authors in [CV05] first approximate the integral operator near the origin with a second
derivative operator. The resulting PDE is then non-degenerate and has an integral operator
of order zero. The error of this approximation is obtained by means of the probabilistic
representation of the solution of both the original equation and the non-degenerate equa-
tion. In the second step of their approximation, R. Cont and E. Voltchkova consider an
implicit-explicit finite difference scheme and obtain pointwise error estimates of order one
in space. As a consequence of the two-step approximation scheme, there are two separate
errors for the approximation
In this chapter, we consider the non-degenerate stochastic integro-differential equation
(5.1.1) with random coefficients and apply the method of finite differences in the time
and space variables. To the best of our knowledge, this article is the first to use the finite
difference method to approximate SIDEs. The approximations of the non-local integral
operators in the drift and in the noise of (5.1.1) we choose are both natural and relatively
easy to implement. In particular, we are able to treat the singularity of the integral oper-
ators near the origin directly. We consider a fully-explicit time-discretization scheme and
an implicit-explicit time-discretization scheme, where we treat part of the approximation
of the integral operator in the drift explicitly.
To obtain error estimates for our approximations, we use the approach in [Yoo00],
where the discretized equations are first solved as time-discretized SDEs in Sobolev spaces
over Rd and an error estimate is obtained in Sobolev norms. After obtaining L2(Ω) error
estimates in Sobolev norms, the Sobolev embedding theorem is used to obtain L2(Ω)-
pointwise error estimates. So, in sum, we obtain two types of error estimates: in Sobolev
norms and on the grid. Naturally, when using the Sobolev embedding to obtain the point-
134 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
wise estimates, we do not need the equation to be differentiable to obtain pointwise error
estimates, only continuous. Using the approach of first obtaining estimates in Sobolev
spaces, we are also easily able to deduce that the more regularity on the coefficients and
data we have, the stronger the error estimates we can obtain (see Corollaries 5.5.3 and
5.5.4).
The chapter is organized as follows. In the next section, we introduce the notation that
will be used throughout the chapter and state the main results. In the third section, we
present a simulation that we did to confirm our rates of convergence. In the fourth section,
we prove auxiliary results that will be used in the proof of the main theorems. In the fifth
section, we prove the main theorems of the chapter.
5.2 Statement of main results
Let us consider the following assumption for an integer m ≥ 0.
Assumption 5.2.1 (m). For i, j ∈ {0, . . . , d}, ai jt = a
i j
t (x) are real-valued functions defined
on Ω × [0,T ] × Rd that are RT ⊗ B(Rd)-measurable and σit = (σ
iϱ
t (x))
∞
ϱ=1 are ℓ2-valued
functions that are RT ⊗ B(Rd)-measurable. Moreover,
(i) for each (ω, t) ∈ Ω× [0,T ], the functions ai jt are max(m, 1)-times continuously differ-
entiable in x for all i, j ∈ {1, . . . , d}, ai0t and a
0i
t are m-time continuously differentiable
in x for all i ∈ {0, 1, . . . , d}, and σit are m-times continuously differentiable in x as
ℓ2-valued functions for all i ∈ {0, . . . , d}. Furthermore, there is a constant K > 0
such that for all (ω, t, x) ∈ Ω × [0,T ] × Rd,
|∂γai jt | ≤ K, ∀ i, j ∈ {1, . . . , d}, ∀ |γ| ≤ max(m, 1),
|∂γai0t | + |∂
γa0it | + |∂
γσit|ℓ2 ≤ K, ∀ i ∈ {0, . . . , d}, ∀|γ| ≤ m;
(ii) there exists a positive constant κ > 0 such that for all (ω, t, x) ∈ Ω × [0,T ] × Rd and
η ∈ Rd
d∑
i, j=1

2a
i j
t −
∞∑
ϱ=1
σ
iϱ
t σ
jϱ
t

 ηiη j ≥ κ|η|
2.
In this chapter, for each integer m ≥ 0, we set Hm = Hm(Rd; R), Hm(F0), Hm =
Hm(Rd; R), Hm(ℓ2) = Hm(Rd; ℓ2), Hm(π2) = Hα(Rd; R; π2), and ∥ · ∥m = ∥ · ∥m,1, (·, ·)m =
(·, ·)m,1, ⟨·, ·⟩1 = ⟨·, ·⟩1,1. We also set C∞c = C
∞
c (R
d; R.
Assumption 5.2.2 (m). We have ϕ ∈ Hm(F0), f ∈ Hm−1, g ∈ Hm(ℓ2), and o ∈ Hm(π2). Set
κ2m = E
[
∥φ∥2m
]
+ E
∫
]0,T ]
(
∥ ft∥2m−1 + ∥gt∥
2
m,ℓ2 +
∫
Rd
∥ot(z)∥2mπ
2(dz)
)
dt.
5.2. Statement of main results 135
For a real-valued twice continuous differentiable function ϕ on Rd, it is easy to see that
for all x, z ∈ Rd,
ϕ(x + z) − ϕ(x) −
d∑
j=1
z j∂ jϕ(x) =
∫ 1
0
d∑
i, j=1
ziz j∂i jϕ(x + θz)(1 − θ)dθ. (5.2.1)
For each δ ∈ (0, 1], let
ς1(δ) =
∫
|z|≤δ
|z|2π1(dz), ς2(δ) =
∫
|z|≤δ
|z|2π2(dz), and ς(δ) = ς1(δ) + ς2(δ).
Fix δ ∈ (0, 1] such that
ς(δ) < κ, (5.2.2)
and notice that
2∑
r=1
πr({|z| > δ}) < ∞. (5.2.3)
We write I = Iδ + Iδc , where
Iδϕ(x) :=
∫
|z|≤δ
∫ 1
0
d∑
i, j=1
ziz j∂i jϕ(x + θz)(1 − θ)dθπ1(dz)
and Iδc is defined as in (5.1.2) with integration over {|z| > δ} instead of Rd.
Definition 5.2.1. An H0-valued càdlàg adapted process u is called a solution of (5.1.1) if
(i) ut ∈ H1 for dP × dt-almost-every (ω, t) ∈ Ω × [0,T ];
(ii) E
∫
]0,T ]
∥ut∥21dt < ∞;
(iii) there exists a set Ω˜ ⊂ Ω of probability one such that for all (ω, t) ∈ [0,T ] × Ω˜ and
ϕ ∈ C∞c (R
d),
(ut, ϕ)0 = (φ, ϕ)0 +
∫
]0,t]


d∑
i, j=1
(
∂ jus, ∂−i(a
i j
s ϕ)
)
0
+ [ϕ, fs]0

 ds
+
∫
]0,t]
∫
|z|≤δ
∫ 1
0
d∑
i, j=1
(
z j∂ jus(· + θz), zi∂−iϕ
)
0
(1 − θ)dθπ1(dz)ds
+
∫
]0,t]
∫
|z|>δ

us(· + z) − us − 1[−1,1](|z|)
d∑
j=1
z j∂ jus, ϕ


0
π1(dz)ds
136 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
+
∞∑
ϱ=1
∫
]0,t]
d∑
i=0
(
σiϱs ∂ius + g
ϱ
s , ϕ
)
0
dwϱs +
∫
]0,t]
∫
Rd
(us−(· + z) − us− + ot(z), ϕ)0 q(dz, ds).
Remark 5.2.2. In the above definition, instead of δ we may choose any other positive
constant.
The following existence theorem is a consequence of Theorems 2.9, 2.10, and 4.1 in
[Gyö82] and will be verified in Section 4.
Theorem 5.2.3. If Assumptions 5.2.1(m) and 5.2.2(m) hold with m ≥ 0, then there ex-
ist a unique solution u of (5.1.1). Furthermore, u is a cádlág Hm-valued process with
probability one and there is a constant N = N(d,m, κ,K,T ) such that
E
[
sup
t≤T
∥ut∥2m
]
+ E
∫
]0,T ]
∥us∥2m+1ds ≤ Nκ
2
m. (5.2.4)
Remark 5.2.4. We have used the standard definition of solution for the variational (or L2)
theory fo stochastic evolution equations. In what follows below, we will always assume
m ≥ 2, and so we have enough regularity to formulate the solution in the weak sense in
(H1,H0,H−1) without integrating by parts.
The following proposition is needed to establish the rate of convergence in time of our
approximation scheme and is proved in Section 4.
Proposition 5.2.5. Let Assumptions 5.2.1(m) and 5.2.2(m) hold for some m ≥ 1 and u be
the solution of (5.1.1). Moreover, assume that
sup
t≤T
E
[
∥gt∥2m−1,ℓ2
]
+ sup
t≤T
E
∫
Rd
∥ot(z)∥2m−1π
2(dz) ≤ K.
Then there is a constant λ = λ(d,m,K,T, κ, κ2m) such that for all s, t ∈ [0,T ],
E
[
∥ut − us∥2m−1
]
≤ λ|t − s|.
Assumption 5.2.3 (m). For m ≥ 3, in addition to Assumption 5.2.2(m), there exists a
random variable ξ with Eξ < K such that for all ω ∈ Ω, t, s ∈ [0,T ],
∥gt∥2m−1,ℓ2 +
∫
Rd
∥ot(z)∥2m−1π
2(dz) ≤ ξ
∥ ft − fs∥2m−2 + ∥gt − gs∥
2
m−2,ℓ2 +
∫
Rd
∥ot(z) − os(z)∥2m−1π
2(dz) ≤ ξ|t − s|.
5.2. Statement of main results 137
Assumption 5.2.4 (m). For m ≥ 3, in addition to Assumption 5.2.1 (i), there is a constant
C such that for all (ω, x) ∈ Ω × Rd, s, t ∈ [0,T ], i, j ∈ {0, 1, . . . , d},
|∂γ
(
ai jt − a
i j
s
)
|2 + |∂γ
(
σit − σ
i
s
)
|2ℓ2 ≤ C|t − s|, ∀|γ| ≤ m − 2.
We turn our attention to the discretisation of equation (5.1.1). For each h ∈ R−{0} and
standard basis vector ei, i ∈ {1, . . . , d}, of Rd we define the first-order difference operator
δh,i by
δh,iϕ(x) :=
ϕ(x + hei) − ϕ(x)
h
,
for all real-valued functions ϕ on Rd. We define δh,0 to be the identity operator. Notice that
for all ψ, ϕ ∈ H0, we have
(ϕ, δ−h,iψ)0 = −(δh,iϕ, ψ)0. (5.2.5)
Set
δhi :=
1
2
(δh,i + δ−h,i)
and observe that for all ϕ ∈ H0,
(ϕ, δhi ϕ)0 = 0. (5.2.6)
For each h , 0, we introduce the grid Gh := {hzk : zk ∈ Zd, k ∈ N0, z0 = 0} with step size
|h|. Let ℓ2(Gh) be the Hilbert space of real-valued functions ϕ on Gh such that
∥ϕ∥2ℓ2(Gh) := |h|
d
∑
x∈Gh
|ϕ(x)|2 < ∞.
We approximate the operators L and Nϱ by
Lht ϕ(x) :=
d∑
i, j=0
ai jt (x)δh,iδ−h, jϕ(x) and N
ϱ;h
t ϕ(x) :=
d∑
i=0
σ
iϱ
t (x)δh,iϕ(x),
respectively. In order to approximate I, we approximate Iδ and Iδc separately. For each
k ∈ N ∪ {0} and h , 0, define the rectangles in Rd
Ahk :=
(
z1k |h| −
|h|
2
, z1k |h| +
|h|
2
]
× · · · ×
(
zdk |h| −
|h|
2
, zdk |h| +
|h|
2
]
,
where zik, i ∈ {1, ..., d}, are the coordinates of zk ∈ Z
d, and set
Bhk := A
h
k ∩ {|z| ≤ δ}, B¯
h
k := A
h
k ∩ {|z| > δ}.
138 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
We approximate Iδc by
Ihδcϕ(x) :=
∞∑
k=0

(ϕ(x + hzk) − ϕ(x)) ζ¯h,k −
d∑
i=1
ξ¯ih,kδ
h
i ϕ(x)

 ,
where
ζ¯h,k := π
1(B¯hk) and ξ¯
i
h,k :=
∫
B¯hk∩[−1,1]
ziπ1(dz).
We continue with the approximation of the operator Iδ. By (5.2.1), for all x ∈ Gh,
Iδϕ(x) =
∞∑
k=0
∫
Bhk
∫ 1
0
d∑
i, j=1
ziz j∂i jϕ(x + θz)(1 − θ)dθπ1(dz),
where there are only a finite number of non-zero terms in the infinite sum over k. The
closest point in Gh to any point z ∈ Bhk is clearly hzk. This simple observation leads us to
the following (intermediate) approximation of Iδϕ(x):
∞∑
k=0
∫ 1
0
d∑
i, j=1
∫
Bhk
ziz jπ1(dz)∂i jϕ(x + θhzk)(1 − θ)dθ.
However, in order to ensure that our approximation is well-defined for functions ϕ ∈
ℓ2(Gh), we need to approximate the integral over θ ∈ [0, 1]. Fix k ∈ N0 and h , 0.
Consider the directed line segment {θhzk : θ ∈ [0, 1]} extending from the origin to the
point hzk ∈ Rd. It is clear that this line segment intersects a unique finite sequence of rect-
angles from the set {Ah
k¯
}k¯∈N0 . Denote the number of rectangles by χ(h, k). Since the line’s
start point is the origin, the first rectangle it intersects is Ah0, and since the line’s endpoint is
hzk, the last rectangle it intersects is Ahk , the center of which is the point hzk. If χ(h, k) > 2,
then in between these two rectangles, the line segment intersects χ(h, k) − 2 additional
rectangles from the set {Ah
k¯
}k¯∈N0 − {A
h
0 ∪ A
h
k}. Denote the indices of these rectangles by r
h,k
l ,
l ∈ {2, . . . , χ(h, k)−1}, and set rh,k1 = 0 and r
h,k
χ(h,k) = k; that is, {θhzk; θ ∈ [0, 1]} ⊆ ∪
χ(h,k)
l=1 A
h
rh,kl
.
Corresponding to the set of rectangles {Ah
rh,kl
}χ(h,k)l=1 is a partition 0 = θ
h,k
0 ≤ · · · ≤ θ
h,k
χ(h,k) = 1
of the interval [0, 1] such that for each l ∈ {1, . . . , χ(h, k)} and θ ∈ (θh,kl−1, θ
h,k
l ), θhzk ∈ A
h
rh,kl
.
Since the diagonal of a d-dimensional hypercube with side length |h| has length
√
dh, for
each k ∈ N0, z ∈ Bhk , and l ∈ {1, . . . , χ(h, k)},
|θz − hzrh,kl | ≤ |θz − θhzk| + |θhzk − hzrh,kl | ≤
√
d|h|, (5.2.7)
5.2. Statement of main results 139
for all θ ∈ (θh,kl−1, θ
h,k
l ). Set
ζ
i j
h,k =
∫
Bhk
ziz jπ1(dz), θ¯h,kl =
∫ θh,kl
θ
h,k
l−1
(1 − θ)dθ
and define the operator
Ihδϕ(x) =:
∞∑
k=0
χ(h,k)∑
l=1
θ¯
h,k
l
d∑
i, j=1
ζ
i j
h,kδh,iδ−h, jϕ(x + hzrh,kl ),
where there are only a finite number of non-zero terms in the infinite sum over k. Set
Ih = Ih
δ
+ Ih
δc
and introduce the martingales
ph,k,it =
∫
]0,t]
∫
Bhk
ziq(dt, dz), p¯h,kt = q(B¯
h
k , ]0, t]).
Moreover, set
θ˜
h,k
l := θ
h,k
l+1 − θ
h,k
l .
Let T ≥ 1 be an integer and set τ = T/T and tn = nτ for i ∈ {0, 1, . . . ,T }. For
any F-martingale (pt)t≤T , we use the notation ∆pn+1 := ptn+1 − ptn . Define recursively the
ℓ2(Gh)-valued random variables (uˆ
h,τ
n )Tn=0 by
uˆh,τn (x) =uˆ
h,τ
n−1(x) +
(
(Lhtn−1 + I
h)uˆh,τn−1(x) + ftn−1(x)
)
τ +
∞∑
ϱ=1
(Nϱ;htn−1 uˆ
h,τ
n−1(x) + g
ϱ
tn−1(x))∆w
ϱ
n
+
∞∑
k=0
d∑
i=1


χ(h,k)∑
l=1
θ˜
h,k
l δh,iuˆ
h,τ
n−1(x + hzrh,kl )

 ∆p
h,k,i
n +
∫
Rd
otn−1(x, z)q(]tn−1, tn], dz)
+
∞∑
k=0
(
uˆh,τn−1(x + hzk) − uˆ
h,τ
n−1(x)
)
∆p¯h,kn , n ∈ {1, . . . ,T }, (5.2.8)
with initial condition
uˆh,τ0 (x) = φ(x), x ∈ Gh
It is clear that uˆh,τn is Ftn-measurable for every n ∈ {0, 1, . . . ,T }. Define the operators
L˜ht ϕ =
d∑
i, j=0
ai jt δh,iδ−h, jϕ − π
1({|z| > δ})ϕ −
d∑
i=1
∫
δ<|z|≤1
ziπ1(dz)δhi ϕ
and
I˜hδcϕ =
∞∑
k=0
ϕ(x + hzk)ζh,k
and note that L˜h+ I˜h
δc
+Iδ = Lh+Ih. On Gh, we also consider the following implicit-explicit
140 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
discretization scheme of (5.1.1):
vˆh,τn (x) =vˆ
h,τ
n−1(x) +
(
(L˜htn + I
h
δ )vˆ
h,τ
n (x) + I˜
h
δcv
h,τ
n−1(x) + ftn(x)
)
τ
+ 1n>1
∞∑
ϱ=1
(Nϱ;htn−1 vˆ
h,τ
n−1(x) + g
ϱ
tn−1(x))∆w
ϱ
n
+ 1n>1
∞∑
k=0
d∑
i=1


χ(h,k)∑
l=1
θ˜
h,k
l δh,ivˆ
h,τ
n−1(x + hzrh,kl )

 ∆p
h,k,i
n +
∫
Rd
otn−1(x, z)q(]tn−1, tn], dz)
+ 1n>1
∞∑
k=0
(
vˆh,τn−1(x + hzk) − vˆ
h,τ
n−1(x)
)
∆p¯h,kn , n ∈ {1, . . . ,T }, (5.2.9)
with initial condition
vˆh,τ0 (x) = φ(x), x ∈ Gh,
where 1n>1 = 0 if n = 1 and 1n>1 = 1 if n ≥ 2. A solution (vˆ
h,τ
n )Mn=0 of (5.2.9) is understood
as a sequence of ℓ2(Gh)-valued random variables such that vˆ
h,τ
n is Ftn-measurable for every
n ∈ {0, 1, . . . ,M} and satisfies (5.1.1).
Remark 5.2.6. Under Assumptions 5.2.2 and 5.2.3, for m > 2 + d/2, by virtue of the
embedding Hm−2 ↪→ ℓ2(Gh), the free-terms f , g, and o(z) are continuous ℓ2(Gh) valued
processes, and consequently the above schemes make sense. Moreover, for 0 < |h| < 1,
there is a constant N independent of h such that -
∥ϕ∥ℓ2(Gh) ≤ N∥ϕ∥m−2. (5.2.10)
Assumption 5.2.5. The parameters h , 0 and T are such that
d
τ
h2
<
κ − ς(δ)
(
2
(
supt,x,ω
∑d
i, j=1 |a
i j
t (x)|2
)1/2
+ ς1(δ)
)2 . (5.2.11)
Remark 5.2.7. We have assumed that the coefficients ai jt were bounded uniformly in ω, t,
and x, so that quantity in in denominator (5.2.11) is well-defined.
The following are our main theorems.
Theorem 5.2.8. Let Assumptions 5.2.1(m) through 5.2.4(m) hold for some m > 2 + d2 and
let Assumption 5.2.5 hold. Let u be the solution of (5.1.1) and let (uˆh,τn )Tn=0 be defined by
(5.2.8). Then there is a constant N = N(d,m, κ,K,T,C, λ, κ2m, δ) such that for any real
number h with 0 < |h| < 1,
E
[
max
0≤n≤T
sup
x∈Gh
|utn(x) − uˆ
h,τ
n (x)|
2
]
+ E
[
max
0≤n≤T
∥utn − uˆ
h,τ
n ∥
2
ℓ2(Gh)
]
≤ N
(
|h|2 + τ
)
.
5.3. Simulation 141
Theorem 5.2.9. Let Assumptions 5.2.1(m) through 5.2.4(m) hold for some m > 2 + d2
and let u be a solution of (5.1.1). There exists a constant R = R(d,m, κ,K, δ) such that
if T > R, then there exists a unique solution (vˆh,τn )Tn=0 of (5.2.9) and a constant N =
N(d,m, κ,K,T,C, λ, κ2m, δ) such that for any real number h with 0 < |h| < 1,
E
[
max
0≤n≤T
sup
x∈Gh
|utn(x) − vˆ
h,τ
n (x)|
2
]
+ E
[
max
0≤n≤T
∥utn − vˆ
h,τ
n ∥
2
ℓ2(Gh)
]
≤ N
(
|h|2 + τ
)
.
Remark 5.2.10. In fact, with more regularity, as consequence of Corollaries 5.5.3 and
5.5.4, we can obtain stronger error estimates with difference operators in the error norms.
This is an immediate consequence of the approach we use to obtaining these estimates.
5.3 Simulation
Let us consider finite difference approximations for the following SIDE on [0,T ] × Rd:
dut(x) =
((
σ¯21
2
+
σ¯22
2
)
∂21ut(x) +
∫
R
(ut(x + z) − ut(x) − ∂1ut(x)z)π(dz)
)
dt + σ¯2∂1ut(x)dwt
+
∫
R
(u(x + z) − u(x)) q(dt, dz),
u0(x) =
1
√
2πσ¯0
exp
(
−
x2
σ¯21σ¯
2
0
)
, (5.3.1)
where π(dz) = c− exp (−β−z) dz|z|1+α− 1(−∞,0)(z) + c+ exp (−β+z)
dz
|z|1+α+ 1(0,∞)(z). It is easily veri-
fied that for (t, x) ∈ [0,T ] × Rd,
vt(x) =
1
√
π(2σ¯20 + 4t)
exp
(
x2
σ¯21(σ¯
2
0 + 2t)
)
solves
dvt(x) =
σ¯21
2
∂21vt(x)dt, v0(x) =
1
√
2πσ¯0
exp
(
−
x2
σ¯21σ¯
2
0
)
.
Moreover, applying Itô’s formula, we find that
ut(x) = vt
(
x + σ¯2wt +
∫
Rd
zq(dt, dz)
)
(5.3.2)
solves (5.3.1). Thus, we can compare our finite difference approximations with (5.3.2).
In our numerical simulations, we used MATLAB 2013a and made the following pa-
142 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
rameter specification:
σ¯1 =
1
2
, σ¯2 =
1
4
, σ¯0 =
1
2
, c− = c+ = 1, β− = β+ = 1, α− = α+ = 1.1, T = 1.
We also made a few practical simplifications. Both the explicit and implicit-explicit ap-
proximations were assumed to take the value zero on (−∞, 8] ∪ [8,∞). We also restricted
the support of π(dz) to [−3, 3]. We would like to investigate the associated error with these
reductions in the future. Regarding the first reduction, we mention that a good heuristic is
to choose the size of domain according to the support of the free terms and the exit time
of the diffusion for which the drift of the SIDE (up to zero order terms) is the infinitesimal
generator of. In fact, it is more than a heuristic and we aim to address this in a future work.
In our simulation, we took δ = 1100 . It follows that κ = σ¯
2
1 =
1
2 and
ς(δ) = c−
∫ δ
0
exp(−β−z)z1−α−dz + c+
∫ δ
0
exp(−β+z)z1−α+dz + z
= c−β
α−−2
− γ(2 − α−, β−δ) + c+β
α+−2
+ γ(2 − α+, β+δ) ≈ 0.0082,
where γ(η, z) denotes the lower incomplete gamma function. Thus, the right-hand-side
of (5.2.11) is approximately 1.0559, and hence we can always set τ = h2. The quan-
tities ζ11h,k, ζ¯h,k, and ξ
1
h,k can all be calculated using MATLAB’s built-in upper and lower
incomplete gamma functions, or by implementing an appropriate numerical integration
procedure. The calculation of θh,kl , θ¯
h,k
l , and θ˜
h,k
l are all straightforward in one-dimension.
Some more thought would need to spent on how to calculate these quantities in higher
dimensions. Of course as an alternative, one could set δ = h2 , but then the schemes are
not guaranteed to converge as h tends to zero. This is the drawback of taking δ = h2 and
not including the additional terms in Iδ (see the paragraph at the bottom of page 1620 in
[CV05]). It does seem that the method we propose to discretise Iδ is novel in this respect.
In our error analysis, we have considered h ∈ {2−2, 2−3, 2−4, 2−5, 2−6, 2−7} and τ = h2.
The term ∫
|z|>δ
(ut(x + z) − ut(x)) π(dz)
in the drift of (5.3.1) can be cancelled with the compensator of the compensated Poisson
random measure term. We get a similar cancellation in the corresponding finite difference
equations, and thus we can replace p¯h,kt = q(B¯
h
k , ]tn, tn+1]) with pˆ
h,k
t = p(B¯
h
k , ]tn, tn+1]) in the
explicit 5.2.8 and implicit-explicit (5.2.9) scheme.
In order to simulate
∆ph,kn =
∫
]tn,tn+1]
∫
Bhk
zq(dt, dz), pˆh,kt = p(B¯
h
k , ]tn, tn+1]),
5.3. Simulation 143
for the finest time step size τ = 2−14, we used the algorithm discussed in Section 4
of [KM11]. In this algorithm, a parameter ϵ is chosen for which the process ∆ph,0n =∫
]0,t]
∫
|z|<ϵ
zq(dt, dz) is approximated by a Wiener process with infinitesimal variance
∫
|z|<ϵ
z2π(dz). We chose the parameter ϵ = 2−8, which is one-half times the smallest step size h
under consideration in our error analysis. The process
∫
]0,t]
∫
|z|>ϵ
zq(dt, dz) =
∫
]0,t]
∫
|z|>ϵ
zp(dt, dz)
(we have used symmetry of the measure π(dz))) is a compound Poisson process with jump
intensity
λ := 2
∫ 3
ϵ
π(dz) ≈ 68.9676
and jump-size density
f¯ (z) =
1
λ
(
c− exp (−β−z)
dz
|z|1+α−
1(−3,2−8)(z) + c+ exp (−β+z)
dz
|z|1+α+
1(2−8,3)(z)
)
.
The underlying Poisson process was simulated using MATLAB’s built-in Poisson random
variable generator; of course there are other simple methods that one can use as an alter-
native (e.g. exponential times or uniform times for fixed number of jumps). We sampled
random variables from the density f¯ by sampling the positive and negative parts separately
and using an acceptance-rejection algorithm with a Pareto random variable. We refer to
[KM11] for more details. Once we simulated the point process on [0,T ]× [−3,−ϵ]∪[ϵ, 3],
we then computed
∫
]0,t]
∫
|z|>ϵ
zp(dt, dz). In order to compute pˆh,kt = p(B¯
h
k , ]tn, tn+1]), we ran
a histogram with the intervals B¯hk .
The quantity ∆ph,kn =
∫
]tn,tn+1]
∫
Bhk
zq(dt, dz) is zero for k , 0 when h < δ2 (for h ∈
{2−2, 2−3, 2−4, 2−5}) since Bhk = ∅ for k , 0 when h <
δ
2 . For h ∈ {2
−6, 2−7}, ∆ph,kn is
non-zero for k ∈ {−1, 0, 1}. A similar analysis holds for the quantity ζ11h,k. As mentioned
above, we set pˆh,0t equal to the Weiner process approximating the small jumps. To compute∫
]tn,tn+1]
∫
Bhk
zq(dt, dz) for k ∈ {−1, 1} in the case h ∈ {2−6, 2−7}, we summed the jump sizes in
their respective bins and compensated. To obtain the above quantities for coarser time step
sizes, we cumulatively summed the finer increments and took the union of jump sizes.
Lastly, we made use of the Fast Fourier Transform to compute terms of the form
∞∑
k=0
ϕ(x + hzk)∆ pˆ
h,k
n ,
which would be quite computationally expensive otherwise. In our error analysis, we ran
3000 simulations of the explicit and implicit-explicit schemes on 30 CPUs and computed
the following quantities:
By our main theorems and the relation τ = h2, these errors should proportional to h
(i.e. O(h)). This is precisely what we observe in Figure 5.1. The slight bump down at the
finest two spatial step-sizes h ∈ {2−6, 2−7} is most likely due to the increase in the number
144 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
-7 -6 -5 -4 -3 -2−6
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
log2(h = τ2)
log
2(e
rro
r(h
))
Rate of convergence (3000 simulations)
√∑3000m=1 max0≤n≤T supx∈Gh |utn(x) − uˆh,τn (x)|2√∑3000m=1 max0≤n≤T ∥utn − uˆh,τn ∥2ℓ2(Gh)√∑3000m=1 max0≤n≤T supx∈Gh |utn(x) − vˆh,τn (x)|2√∑3000m=1 max0≤n≤T ∥utn − vˆh,τn ∥2ℓ2(Gh)Line of slope 1
Figure 5.1 Simulated errors with respect to the space discretization and a line as reference
slope on a log2 scale
.
of terms in the approximation of Ih
δ
(three to be precise) and the analogous small jump
term in the noise.
5.4 Auxiliary results
In this section, we present some results that will be needed for the proof of Theorems 5.2.8
and 5.2.9. Introduce the operators
Iδ;h(z)ϕ(x) :=
∞∑
k=0
1Bhk (z)
χ(h,k)∑
l=1
d∑
i=1
θ˜
h,k
l z
iδh,iϕ(x + hzrh,kl ),
Iδ
c;h(z)ϕ(x) :=
∞∑
k=0
1B¯hk (z)(ϕ(x + hzk) − ϕ(x)),
Ih(z)ϕ(x) := Iδ;h(z)ϕ(x) + Iδ
c;h(z)ϕ(x).
Consider the following explicit and implicit-explicit schemes in H0:
uh,τn =u
h,τ
n−1 +
(
(Lhtn−1 + I
h)uh,τn−1 + ftn−1
)
τ +
∞∑
ϱ=1
(Nϱ;htn−1u
h,τ
n−1 + g
ϱ
tn−1)∆w
ϱ
n
+
∫
Rd
(
Ih(z)uh,τn−1 + otn−1(z)
)
q(dz, ]tn−1, tn]), n ∈ {1, . . . ,T }, (5.4.1)
5.4. Auxiliary results 145
and
vh,τn =v
h,τ
n−1 +
(
(L˜htn + I
h
δ )v
h,τ
n + I˜
h
δcv
h,τ
n−1 + ftn
)
τ + 1n>1
∞∑
ϱ=1
(Nϱ;htn−1v
h,τ
n−1 + g
ϱ
tn−1)∆w
ϱ
n
+ 1n>1
∫
Rd
(
Ih(z)vh,τn−1 + otn−1(z)
)
q(dz, ]tn−1, tn]), n ∈ {1, . . . ,T }, (5.4.2)
with initial condition
uh,τ0 (x) = v
h,τ
0 (x) = φ(x), x ∈ R
d.
We now prove some lemmas that will help us to establish the consistency of our approxi-
mations. The following lemma is well-known and we omit the proof (see, e.g., [GK10]).
Lemma 5.4.1. For each integer m ≥ 0, there is a constant N = N(d,m) such that for all
u ∈ Hm+2 and v ∈ Hm+3,
∥δh,iu − ∂iu∥m ≤
1
2
|h|∥u∥m+2,
∥δh,iδ−h, jv − ∂i jv∥m ≤ N|h|∥v∥m+3.
Lemma 5.4.2. For each integer m ≥ 0, there is a constant
N = N(d,m, δ) such that for all u ∈ Hm+3, we have
∥Iu − Ihu∥m ≤ N|h|∥u∥m+3. (5.4.3)
Proof. It suffices to show (5.4.3) for u ∈ C∞c (R
d). We begin with m = 0. A simple
calculation shows that
Iδcu(x) − Ihδcu(x) =
∞∑
k=0
∫
B¯hk
∫ 1
0
d∑
i=1
(zi − hzik)∂iu(x + hzk + θ(z − hzk))dθπ
1(dz)
−
∞∑
k=0
∫
B¯hk∩[−1,1]
d∑
i=1
zi(∂iu(x) − δhi u(x))π
1(dz).
By Minkowski’s inequality, we get
∥Iδcu − Ihδcu∥0 ≤
∞∑
k=0
∫
B¯hk
d∑
i=1
|zi − hzik|∥∂iu∥0π
1(dz)
+
∞∑
k=0
∫
B¯hk∩[−1,1]
d∑
i=1
|zi|∥∂iu(x) − δhi u(x)∥0π
1(dz) ≤ N|h|∥u∥3 + N
d∑
i=1
∥∂iu(x) − δhi u(x)∥0,
since |z − hzk| ≤ |h|
√
d/2 and (5.2.3) holds. Thus, by Lemma 5.4.1, we have
∥Iδcu − Ihδcu∥0 ≤ N |h|∥u∥3. (5.4.4)
146 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
We also have
Iδu(x) − Ihδu(x) =
∞∑
k=0
∫
Bhk
χ(h,k)∑
l=1
∫ θh,kl
θ
h,k
l−1
d∑
i, j=1
ziz j
(
∂i ju(x + θz) − δh,iδ−h, ju(x + hzrh,kl )
)
(1 − θ)dθπ1(dz). (5.4.5)
Note that
∂i ju(x + θz) − δh,iδ−h, ju(x + hzrh,kl )
= ∂i ju(x + θz) − ∂i ju(x + hzrh,kl ) + ∂i ju(x + hzrh,kl ) − δh,iδ−h, ju(x + hzrh,kl )
=
∫ 1
0
d∑
q=1
(
θzq − hzq
rh,kl
)
∂q∂i ju
(
x + hzrh,kl + ρ(θz − hzrh,kl )
)
dρ
+∂i ju(x + hzrh,kl ) − δh,iδ−h, ju(x + hzrh,kl ).
By (5.2.7), we have |θzq − hzq
rh,kl
| ≤ N |h|. Hence, substituting the above relation in (5.4.5),
using Minkowski’s inequality, (5.2.2), and Lemma 5.4.1, we obtain
∥Iδu − Ihδu∥0 ≤ |h|N∥u∥3. (5.4.6)
Combining (5.4.4) and (5.4.6), we have (5.4.3) for m = 0. The case m > 0 follows from
the case m = 0, since for a multi-index γ, we have
∂γ(Iu − Ihu) = I∂γu − Ih∂γu.
□
Lemma 5.4.3. For each integer m ≥ 0, there is a constant N = N(d,m, δ), there is a
constant such that for all u ∈ Hm+2, we have
∫
Rd
∥Ih(z)u − I(z)u∥2mπ
2(dz) ≤ N |h|2∥u∥2m+2. (5.4.7)
Proof. It suffices to prove the lemma for u ∈ C∞c (R
d) and m = 0. We have
Iδ(z)u(x) − Iδ;h(z)u(x) =
∞∑
k=0
1Bhk (z)
χ(h,k)∑
l=1
∫ θh,kl
θ
h,k
l−1
d∑
i=1
zi(∂iu(x + θz) − δh,iu(x + hzrh,kl ))dθ.
5.4. Auxiliary results 147
Notice that
∂iu(x + θz) − δh,iu(x + hzrh,kl ) =
∫ 1
0
d∑
i, j=1
∂i ju(x + ρ(θz − hzrh,kl ))(θz
j − hz j
rh,kl
)dρ
+∂iu(x + hzrh,kl ) − δh,iu(x + hzrh,kl ).
Thus, by Remark 5.2.7 and Lemma 5.4.1, we get
∥Iδ;h(z)u − Iδ(z)u∥20 ≤ 1|z|≤δ|z|
2N|h|2∥u∥22,
and hence by (5.2.2), we obtain
∫
Rd
∥Iδ;h(z)u − Iδ(z)u∥20π
2(dz) ≤ N |h|2∥u∥22. (5.4.8)
We also have
|Iδ
c
(z)u(x) − Iδ
c;h(z)u(x)| =
∞∑
k=0
1B¯hk (z)|u(x + z) − u(x + hzk)|
≤
∞∑
k=0
1B¯hk (z)
∫ 1
0
d∑
i=1
|∂iu(x + hzk + ρ(z − hzk))∥zi − hzik|dρ.
Consequently,
∥Iδ
c;h(z)u − Iδ
c
(z)u∥20 ≤ 1|z|>δN |h|
2∥u∥21,
which implies by (5.2.3) that
∫
Rd
∥Iδ
c;h(z)u − Iδ
c
(z)u∥20π
2(dz) ≤ N |h|2∥u∥21. (5.4.9)
Combining (5.4.9) and (5.4.8), we have (5.4.7) for m = 0. The case m > 0 follows from
the case m = 0, since for a multi-index γ, we have
∂γ(Iu − Ihu) = I∂γu − Ih∂γu.
□
Lemma 5.4.4. If Assumption 5.2.1(m) holds for some m ≥ 0, then for any ϵ ∈ (0, 1) there
exists constants N1 = N1(d,m, κ,K, δ, ϵ) and N2 = N2(d,m, κ,K, δ, ϵ) such that for any
u ∈ Hm,
G(m)t (u) := 2(u,L
h
t u)m + ∥N
h
t u∥
2
m,ℓ2 + 2(u, I
hu)m +
∫
Rd
∥Ih(z)u∥2mπ
2(dz)
148 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
≤ −(κ − ς(δ) − ϵ)
d∑
i=1
∥δh,iu∥2m + N1∥u∥
2
m, (5.4.10)
and
(u, L˜ht u)m + (u, I
h
δu)m ≤ −(κ − ς1(δ) − ϵ)
d∑
i=1
∥δh,iu∥2m + N2∥u∥
m
n . (5.4.11)
Proof. By virtue of Lemma 3.1 and Theorem 3.2 in [GK10], under Assumption 5.2.1,
there is a constant N = N(d,m, κ) such that for any u ∈ Hm and ϵ > 0,
2(u,Lht u)m + ∥N
h
t u∥
2
m,ℓ2 ≤ −(κ − ϵ)
d∑
i=1
∥δh,iu∥2m + N∥u∥
2
m.
Therefore, it suffices to show that there is a constant N = N(δ) such that for all u ∈ C∞c (R
d),
2(u, Ihu)m +
∫
Rd
∥Ih(z)u∥2mπ
2(dz) ≤ ς(δ)
d∑
i=1
∥δh,iu∥2m + N∥u∥
2
m. (5.4.12)
We start with m = 0. Since
(u, Ihδu)0 =
∞∑
k=0
∫
Bhk
χ(h,k)∑
l=1
d∑
i, j=1
θ¯
k,h
l z
iz j
∫
Rd
δh,iδ−h, ju(x + hzrh,kl )u(x)dxπ
1(dz)
and ∫
Rd
δh,iδ−h, ju(x + hzrh,kl )u(x)dx = −
∫
Rd
δh,iu(x + hzrh,kl )δh, ju(x)dx,
by Hölder’s inequality, we get
2(u, Ihδu)0 ≤
∫
|z|≤δ
|z|2π1(dz)
d∑
i=1
∥δh,iu∥20 = ς1(δ)
d∑
i=1
∥δh,iu∥20.
In addition, owing to Holder’s inequality and (5.2.6), we have
2(u, Ihδcu)0 =
∞∑
k=0
∫
B¯hk
∫
Rd

u(x + hzk) − u(x) − 1[−1,1](z)
d∑
i=1
ziδhi u(x)

 u(x)dxπ
1(dz) ≤ 0.
By Minkowski’s inequality, we have
∥Iδ;h(z)u∥2 ≤
∞∑
k=0
1Bhk (z)|z|
2
d∑
i=1
∥δh,iu∥20 and ∥I
δc;h(z)u∥20 ≤ 4
∞∑
k=0
1B¯hk (z)∥u∥
2
0
and hence
∫
Rd
∥Ih(z)u∥20π
2(dz) ≤ ς2(δ)
d∑
i=1
∥δh,iu∥20 + 4π
1({|z| > δ})∥u∥20,
5.4. Auxiliary results 149
which proves (5.4.12) for m = 0. The case m > 0 follows by replacing u with ∂γu for
|γ| ≤ m. This proves (5.4.10), which implies (5.4.11). □
Remark 5.4.5. It follows that for m ≥ 0, there is a constant N5 = N5(d,m,K, δ) such that
for any u ∈ Hm,
∥Nϱ;ht u∥
2
m +
∫
Rd
∥Ih(z)u∥2mπ
2(dz) ≤ N5
d∑
i=0
∥δh,iu∥2m (5.4.13)
≤ N5
(
1 +
4d
h2
)
∥u∥2m. (5.4.14)
Lemma 5.4.6. For any m ≥ 0 and u ∈ Hm,
∥I˜hδcu∥
2
m ≤ π
1({|z| > δ})2∥u∥2m. (5.4.15)
Moreover, if Assumption 5.2.1 holds for some m ≥ 0, then for any ϵ > 0 and u ∈ Hm,
∥(Lht + I
h)u∥2m ≤ (1 + ε)
N3d
h2
d∑
i=1
∥δh,iu∥2m + N4
(
1 +
1
h2
)
∥u∥2m (5.4.16)
where
N3 :=


2

sup
t,x,ω
d∑
i, j=1
|ai j(x)|2


1/2
+ ς1(δ)


2
and N4 is a constant depending only on d,m,K, δ, and ϵ.
Proof. It suffices to prove the lemma for u ∈ C∞c (R
d). It follows that
(Lht + I
h
δ )u(x) =
∞∑
k=0
χ(h,k)∑
l=1
θ¯
h,k
l
d∑
i, j=1
ζˆ
i j
t,h,k(x)δh,iδ−h, ju(x + hzrh,kl ) +
d∑
i, j=0
i or j=0
ai jt δh,iδ−h, ju(x)
where ζˆ i jt,h,k(x) := ζ
i j
h,k for k , 0 and ζˆ
i j
t,h,0(x) := ζ
i j
h,0 + 2a
i j
t (x) (recall that θ¯
h,0
1 =
1
2 and
χ(h, 0) = 1). Moreover, for each multi-index γ with 1 ≤ |γ| ≤ m,
∂γ(Lht + I
h
δ )u(x) =
∞∑
k=0
χ(h,k)∑
l=1
θ¯
h,k
l
d∑
i, j=1
ζˆ
i j
h,k(x)δh,iδ−h, j∂
γu(x + hzrh,kl )
+
∑
{β : β<γ}
N(β, γ)
d∑
i, j=1
(
∂γ−βai jt (x)
)
δh,iδ−h, j∂
βu(x)
+
∑
{β : β≤γ}
N(β, γ)
d∑
i, j=0
i or j=0
((
∂γ−βai jt (x)
)
δh,iδ−h, j∂
βu(x)
)
150 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
=: (A1(γ) + A2(γ) + A3(γ))u(x),
where N(β, γ) are constants depending only on β and γ. By Young’s inequality and
Jensen’s inequality, for any ϵ ∈ (0, 1), we have
∥(Lht + I
h)u∥2m ≤ (1 + ϵ)
∑
|γ|≤m
∥A1(γ)u∥20
+3
(
1 +
1
ϵ
) 
∑
|γ|≤m
(
∥A2(γ)u∥20 + ∥A3(γ)u∥
2
0
)
+ ∥Ihδcu∥
2
m

 .
Applying Minkowski’s inequality and the Cauchy-Bunyakovsky-Schwarz inequality and
noting that
∑χ(h,k)
l=1 θ¯
h,k
l =
1
2 and
||δh,i∂βu||0 ≤
2
h
||∂βu||0; ∀i ∈ {0, 1 . . . , d}, ∀|β| = m,
we obtain
∥A1(γ)u∥0 ≤
∞∑
k=0
χ(h,k)∑
l=1
θ¯
h,k
l

sup
t,x,ω
d∑
i, j=1
∣∣∣ζˆ
i j
h,k(x)
∣∣∣
2


1/2 

d∑
i, j=1
∣∣∣∣
∣∣∣∣δh,iδ−h, ju(· + hzrh,kl )
∣∣∣∣
∣∣∣∣
2
m


1/2
≤
√
d
h
∞∑
k=0

sup
t,x,ω
d∑
i, j=1
|ζˆ i jh,k(x)|
2


1/2 

d∑
i=1
∥δh,i∂γu∥20


1/2
and
∞∑
k=0

sup
t,x,ω
d∑
i, j=1
|ζˆ i jh,k(x)|
2


1/2
=

sup
t,x,ω
d∑
i, j=1
∣∣∣∣∣∣
∫
Bh0
ziz jπ1(dz) + 2ai jt (x)
∣∣∣∣∣∣
2


1/2
+
∞∑
k=1


d∑
i, j=1
∣∣∣∣∣∣
∫
Bhk
ziz jπ1(dz)
∣∣∣∣∣∣
2


1/2
≤ 2

sup
t,x,ω
d∑
i, j=1
|ai jt (x)|
2


1/2
+ ς(δ).
Thus,
∑
|γ|≤m
||A1(γ)u||20 ≤
N3d
h2
d∑
i=1
∥∂h,iu∥2m.
Another application of the Cauchy-Bunyakovsky-Schwarz inequality and
Minkowski’s inequality, combined with the inequalities
||δh,i∂βu||0 ≤ ||∂i∂βu||0 ∀i ∈ {0, 1 . . . , d}, ∀|β| ≤ m − 1,
||δh,iδ−h, j∂βu||0 ≤ ||∂i j∂βu||0, ∀; i, j ∈ {1, . . . , d}, ∀|β| ≤ m − 2,
5.4. Auxiliary results 151
and
||δh,iδ−h, j∂βu||0 ≤
2
h
∥δh,iu∥m, ∀; i, j ∈ {1, . . . , d}, ∀|β| = m − 1,
yields
∑
|γ|≤m
(
∥A2(γ)u∥20 + ∥A3(γ)∥
2
0
)
≤ N
(
1 +
1
h2
)
||u||2m .
By Minkowski’s integral inequality, we have
∥Ihδcu∥m ≤
∫
Rd
∞∑
k=0
1
B
h
k
∥u(· + hzk) − u − 1[−1,1](z)
d∑
i=1
ziδh,iu∥mπ1(dz)
≤ 3

π
1({|z| > δ}) +
2d
∫
δ<|z|≤1
|z|π1(dz)
h

 ∥u∥m.
It is also easy to see that (5.4.15) holds. Combining above inequalities, we obtain (5.4.16).
□
The following theorem establishes the stability of the explicit approximate scheme
(5.4.1).
Theorem 5.4.7. Let Assumption 5.2.1 hold with m ≥ 0 and Assumption 5.2.5 hold. Let
F i ∈ Hm for i ∈ {0, ..., d}, G ∈ Hm(ℓ2), and R ∈ Hm(π2). Consider the following scheme in
Hm:
uh,τn = u
h,τ
n−1 +
∫
]tn−1,tn]

(L
h
tn−1 + I
h)uh,τn−1 +
d∑
i=0
δh,iF
i
t

 dt +
∫
]tn−1,tn]
(
Nϱ;htn−1u
h,τ
n−1 + G
ϱ
t
)
dwϱt
+
∫
]tn−1,tn]
∫
Rd
(
Ih(z)uh,τn−1 + Rt(z)
)
q(dt, dz), n ∈ {1, . . . ,T },
for any Hm−valued F0−measurable initial condition φ. If ϕ ∈ Hm(F0), then there is a
constant N = N(d,m, κ,K,T, δ) such that
E
[
max
0≤n≤T
∥uh,τn ∥
2
m
]
+ E
T∑
n=0
τ
d∑
i=0
∥δh,iuh,τn ∥
2
m ≤ NE
[
∥φ∥2m
]
+NE
∫ T
0
( d∑
i=0
∥F it∥
2
m + ∥Gt∥
2
m +
∫
Rd
∥Rt(z)∥2mπ
2(dz)
)
dt. (5.4.17)
Proof. If E
[
∥φ∥2m
]
< ∞, then proceeding by induction on n and using Young’s and Jensen’s
inequality, Itô’s isometry, (5.4.16), and (5.4.14), we get that for all n ∈ {0, 1, . . . ,T },
E
[
∥uh,τn ∥2m
]
< ∞. Applying the identity ∥y∥2m − ∥x∥
2
m = 2(x, y − x)m + ∥y − x∥
2
m, x, y ∈ H
m,
152 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
for each n ∈ {1, . . . ,T }, we obtain
∥uh,τn ∥
2
m = ∥u
h,τ
n−1∥
2
m +
6∑
i=1
Ii(tn), (5.4.18)
where
I1(tn) := 2τ(u
h,τ
n−1,
(
Lhtn−1 + I
h
)
uh,τn−1)m + ∥η(tn)∥
2
m,
I2(tn) := 2
∫
]tn−1,tn]
d∑
i=0
(uh,τn−1, δh,iF
i
t)mdt,
I3(tn) :=
∣∣∣∣∣∣∣
∣∣∣∣∣∣∣
τ
(
Lhtn−1 + I
h
)
uh,τn−1 +
∫
]tn−1,tn]
d∑
i=0
δh,iF
i
tdt
∣∣∣∣∣∣∣
∣∣∣∣∣∣∣
2
m
,
I4(tn) := 2
∫
]tn−1,tn]
(
uh,τn−1,N
ϱ;h
tn−1u
h,τ
n−1 + G
ϱ
t
)
m
dwϱt ,
I5(tn) := 2
∫
]tn−1,tn]
∫
Rd
(
uh,τn−1,I
h(z)uh,τn−1 + Rt(z)
)
m
q(dt, dz),
I6(tn) := 2
(
τ(Lhtn−1 + I
h)uh,τn−1, η(tn)
)
m
+ 2


∫
]tn−1,tn]
d∑
i=0
δh,iF
i
tdt, η(tn)


m
,
and where
η(tn) :=
∫
]tn−1,tn]
(
Nϱ;htn−1u
h,τ
n−1 + G
ϱ
t
)
dwϱt +
∫
]tn−1,tn]
∫
Rd
(
Ih(z)uh,τn−1 + Rt(z)
)
q(dt, dz).
By virtue of Assumption 5.2.5, we fix q˜ > 0 and ϵ > 0 small enough such that
q := κ − ς(δ) − ϵ − (1 + ϵ)(1 + q˜)N3d
τ
h2
− q˜ > 0,
where N3 is the constant in (5.4.6). Since the two stochastic integrals that define η are
orthogonal square-integrable martingales, by Young’s inequality and (5.4.13), for all q > 0,
E
[
∥η(tn)∥2m
]
≤ E
[
τ∥Nhtn−1u
h,τ
n−1∥
2
m,ℓ2
]
+ Eτ
∫
Rd
∥Ih(z)uh,τn−1∥
2
mπ
2(dz) + qEτ
d∑
i=0
∥δh,iu
h,τ
n−1∥
2
m
+
(
1 +
N5
q
)
E
∫
]tn−1,tn]
(
∥Gt∥2m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)
)
dt. (5.4.19)
Thus, taking q = q˜3 in (5.4.19), we have
EI1(tn) ≤ EτG
(m)
tl−1(u
h,τ
l−1) +
q˜
3
Eτ
d∑
i=0
∥δh,iu
h,τ
n−1∥
2
m
5.4. Auxiliary results 153
+
(
1 +
3N5
q˜
)
E
∫
]tn−1,tn]
(
∥Gt∥2m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)
)
dt.
Using (5.2.5) and Young’s inequality, we obtain
EI2(tn) ≤
q˜
3
Eτ
d∑
i=0
∥δh,iu
h,τ
n−1∥
2
m +
3
q˜
E
∫
]tn−1,tn]
d∑
i=0
∥F it∥
2
mdt.
An application of Young’s inequality and (5.4.16) yields
EI3(tn) ≤ (1 + ϵ)(1 + q˜)N3d
τ
h2
Eτ
d∑
i=1
∥δh,iu
h,,τ
n−1∥
2
m + (1 + q˜)N4
(
τ +
τ
h2
)
E
[
τ∥uh,τn−1∥
2
m
]
+(d + 1)
(
1 +
1
q˜
)
E
∫
]tn−1,tn]

τ∥F
0
t ∥
2
m +
4dτ
h2
d∑
i=1
∥F it∥
2
m

 dt.
Making use of the estimate (5.4.14) and noting that E∥uh,τn ∥2m < ∞, G ∈ H
m(ℓ2), and
R ∈ Hm(π2), we obtain EI4(tn) = EI5(tn) = 0. Moreover, as (Lhtn−1 + I
h)uh,τn−1 is Ftn−1-
measurable and E(η(tn)|Ftn−1) = 0, the expectation of first term in I6(tn) is zero, and hence
by Young’s inequality, for any q1 > 0,
EI6(tn) ≤ q1E
[
∥η(tn)∥2m
]
+
1
q1
E
∣∣∣∣∣∣∣
∣∣∣∣∣∣∣
∫
]tn−1,tn]
d∑
i=0
δh,iF
i
tdt
∣∣∣∣∣∣∣
∣∣∣∣∣∣∣
2
m
.
Moreover, by Jensen’s inequality, (5.4.19), and (5.4.13), for any q1 > 0 and q > 0,
EI6(tn) ≤ (q1q + q1N5)Eτ
d∑
i=0
∥δh,iu
h,τ
n−1∥
2
m
+E
∫
]tn−1,tn]


(d + 1)τ
q1
∥F0t ∥
2
m +
4d(d + 1)τ
q1h2
d∑
i=1
∥F it∥
2
m

 dt
+q1
(
1 +
N5
q
)
E
∫
]tn−1,tn]
(
∥Gt∥2m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)
)
dt.
We choose q and q1 such that q1q + q1N5 ≤ q˜/3. Thus, owing to (5.4.10), we have
EG(m)tn−1(u
h,τ
n−1) +
(
q˜ + (1 + ϵ)(1 + q˜)N3d
τ
h2
)
Eτ
d∑
i=1
∥δh,iu
h,τ
n−1∥
2
m
≤ −qEτ
d∑
i=1
∥δh,iu
h,τ
n−1∥
2
m + N1E
[
τ||uh,τn−1||
2
m
]
.
Taking the expectation of both sides of (5.4.18), summing-up, and combining the above
154 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
inequalities and identities, we find that there is a constant N = N(d,m, κ,K, δ) such that
for all n ∈ {0, 1, . . . ,T },
E
[
∥uh,τn ∥
2
m
]
≤ E
[
∥φ∥2m
]
− qE
n∑
l=1
τ
d∑
i=1
∥δh,iu
h,τ
l−1∥
2
m
+
(
N1 + q˜ + (1 + q˜)N4
(
τ +
τ
h2
))
E
n∑
l=1
τ∥uh,τl−1∥
2
m
+N
(
τ +
τ
h2
)
E
∫
]0,tn]
d∑
i=0
∥F it∥
2
mdt + NE
∫
]0,tn]
(
∥Gt∥2m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)
)
dt.
Therefore, by discrete Gronwall’s inequality, there is a constant N = N(d,m, κ,K,T, δ)
such that
E
[
∥uh,τn ∥
2
m
]
+ E
n∑
l=0
τ
d∑
i=0
∥δh,iu
h,τ
l ∥
2
m ≤ NE
[
∥φ∥2m
]
+NE
∫
]0,T ]


d∑
i=0
∥F it∥
2
m + ∥Gt∥
2
m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)

 dt. (5.4.20)
Now that we have proved (5.4.20), we will show (5.4.17). Estimating as we did above, we
get that there is a constant N such that
E max
0≤n≤T
n∑
l=1
(I1(tl) + I2(tl) + I3(tl) + I6(tl)) ≤ NE
T−1∑
l=0
τ
d∑
i=0
∥δh,iu
h,τ
l ∥
2
m
+NE
∫
]0,T ]


d∑
i=0
∥F it∥
2
m + ∥Gt∥
2
m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)

 dt.
Applying the Burkholder-Davis-Gundy inequality and Young’s inequality, we obtain
E max
0≤n≤T
n∑
l=1
I5(tl) ≤ 6E
∣∣∣∣∣∣∣
n∑
l=1
∫
]tn−1,tn]
∫
Rd
(
uh,τn−1,I
h(z)uh,τn−1 + Rt(z)
)2
m
π2(dz)dt
∣∣∣∣∣∣∣
1/2
≤
1
4
E
[
max
0≤n≤T
∥uh,τn ∥
2
m
]
+ N

E
T−1∑
l=0
τE∥δh,iu
h,τ
l ∥
2
m + E
T−1∑
l=0
τE∥uh,τl ∥
2
m


+NE
∫
]0,T ]
∫
Rd
∥Rt(z)∥2mπ
2(dz)dt.
We can estimate E max0≤n≤T
∑n
l=1 I4(tl) in similar way. Combining the above E max0≤n≤T -
estimates and (5.4.20), we obtain (5.4.17). □
The following theorem establishes the existence and uniqueness of a solution to (5.4.2)
and the stability of the implicit-explicit approximation scheme.
5.4. Auxiliary results 155
Theorem 5.4.8. Let Assumption 5.2.1 hold with m ≥ 0. Let F i ∈ Hm for i ∈ {0, ..., d},
G ∈ Hm(ℓ2) and R ∈ Hm(π2). Then there exists a constant R = R(d,m, κ,K, δ) such that if
T > R, then for any h , 0, there exists a unique Hm-valued solution (vh,τn )Tn=0 of
vh,τn = v
h,τ
n−1 +
∫
]tn−1,tn]

(L˜
h
tn + I
h
δ )v
h,τ
n + I˜
h
δcv
h,τ
n−1 +
d∑
i=0
δh,iF
i
t

 dt
+
∫
]tn−1,tn]
(
1n>1N
ϱ;h
tn−1v
h,τ
n−1 + G
ϱ
t
)
dwϱt
+
∫
]tn−1,tn]
∫
Rd
(
1n>1Ih(z)v
h,τ
n−1 + Rt(z)
)
q(dt, dz), (5.4.21)
for n ∈ {1, . . . ,T }, for any Hm−valued F0−measurable initial condition φ. Moreover, if
ϕ ∈ Hm(F0), then there is a constant N = N(d,m, κ,K,T, δ) such that
E
[
max
0≤n≤T
∥vh,τn ∥
2
m
]
+ E
T∑
n=0
τ
d∑
i=0
∥δh,ivh,τn ∥
2
m ≤ NE
[
∥φ∥2m
]
+NE
∫ T
0


d∑
i=0
∥F it∥
2
m + ∥Gt∥
2
m +
∫
Rd
∥Rt(z)∥2mπ
2(dz)

 dt. (5.4.22)
Proof. For each n ∈ {1, . . . ,T }, we write (5.4.21) as
Dnv
h,τ
n = yn−1,
where Dn is the operator defined by
Dnϕ := ϕ − τ
(
L˜htn + I
h
δ
)
ϕ
and
yn−1 := v
h,τ
n−1 +
∫
]tn−1,tn]

I˜
h
δcv
h,τ
n−1 +
d∑
i=0
δh,iF
i
t

 dt +
∫
]tn−1,tn]
(
1n>1N
ϱ;h
tn−1v
h,τ
n−1 + G
ϱ
t
)
dwϱt
+
∫
]tn−1,tn]
∫
Rd
(
(1n>1Ih(z)v
h,τ
n−1 + Rt(z)
)
q(dt, dz).
Fix ϵ1 and ϵ2 in (0, 1) such that
q1 := κ − ς1(δ) − ϵ1 > 0.
and
q2 := κ − ς(δ) − ϵ2 > 0.
156 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
Owing to Lemma 5.4.6, there is a constant N = N(d,m,K, δ) such that for all ϕ ∈ Hm,
∥Dnϕ∥2m ≤ N
(
1 + τ2
(
1
h2
+
1
h4
))
∥ϕ∥2m. (5.4.23)
Assume T > T N2. By (5.4.11), for all ϕ ∈ Hm, we have
(ϕ,Dnϕ)m ≥ (1 − τN2)∥ϕ∥2m + q1τ
d∑
i=1
∥δh,iϕ∥2m ≥ (1 − τN2)∥ϕ∥
2
m. (5.4.24)
Using Jensen’s inequality and (5.4.15), we get
∥y0∥2m ≤ 5
(
1 + π1({|z| > δ})2τ2
)
∥ϕ∥2m +
20τ
h2
∫
]0,t1]
d∑
i=0
∥F it∥
2
mdt + 5
∣∣∣∣∣∣
∣∣∣∣∣∣
∫
]0,t1]
Gϱt dw
ϱ
t
∣∣∣∣∣∣
∣∣∣∣∣∣
2
m
+ 5
∣∣∣∣∣∣
∣∣∣∣∣∣
∫
]0,t1]
∫
Rd
Rt(z)q(dt, dz)
∣∣∣∣∣∣
∣∣∣∣∣∣
2
m
. (5.4.25)
Since φ ∈ Hm, F i ∈ Hm, i ∈ {0, 1, . . . , d}, G ∈ Hm(ℓ2), and R ∈ Hm(π2), it follows that
y0 ∈ Hm. By (5.4.23), and (5.4.24), owing to Proposition 3.4 in [GM05] (p = 2), there
exists a unique vh,τ1 in H
m such that D1v
h,τ
1 = y0, and moreover
∥vh,τ1 ∥
2
m ≤ 1 +
∥y0∥2m
(1 − τN2)2
< ∞. (5.4.26)
Proceeding by induction on n ∈ {1, . . . ,T }, one can show that there exists a unique vh,τn in
Hm such that Dnv
h,τ
n = yn−1, and moreover
∥vh,τn ∥
2
m ≤ 1 +
∥yn−1∥2m
(1 − τN2)2
< ∞. (5.4.27)
Assume that E∥φ∥2m < ∞. By (5.4.25) and (5.4.26) and the fact that f
i ∈ Hm, i ∈
{0, 1, . . . , d}, g ∈ Hm(ℓ2), and R ∈ Hm(π2), it follows that E∥v
h,τ
1 ∥
2
m < ∞. By Jensen’s
inequality, (5.4.15), and (5.4.14), we have
E
[
∥yn−1∥2m
]
≤ 7N
(
1 + π1({|z| > δ})2τ2 + 1n>1τ
(
1 +
1
h2
))
E
[
∥vh,τn−1∥
2
m
]
+
28τ
h2
E
∫
]0,t1]
d∑
i=0
∥F it∥
2
mdt + 7E
∫
]0,t1]
∥Gt∥2m,ℓ2dt + 7E
∫
]0,t1]
∫
Rd
∥Rt(z)∥2mπ
2(dz)dt.(5 4.28)
Proceeding by induction on n and combining (5.4.27) and (5.4.28), we obtain
E
[
∥vh,τn ∥
2
m
]
< ∞, ∀n ∈ {0, 1, . . . ,T }. (5.4.29)
5.4. Auxiliary results 157
Applying the identity ∥y∥2m−∥x∥
2
m = 2(x, y−x)m+∥y−x∥
2
m, x, y ∈ H
m, for any n ∈ {1, . . . ,T },
we have
∥vh,τn ∥
2
m = ∥v
h,τ
n−1∥
2
m +
6∑
i=1
Ii(tn),
where
I1(tn) := 2τ(v
h,τ
n ,
(
L˜htn + I
h
δ
)
vh,τn )m + 2τ(v
h,τ
n−1, I˜
h
δcv
h,τ
n−1)m + ∥η(tn)∥
2
m,
I2(tn) := 2
∫
]tn−1,tn]
d∑
i=0
(uh,τn , δh,iF
i
t)mdt,
I3(tn) := −
∣∣∣∣∣∣∣
∣∣∣∣∣∣∣
τ
(
L˜htn + I
h
δ
)
vh,τn +
d∑
i=0
∫
[tn−1,tn]
δh,iF
i
tdt
∣∣∣∣∣∣∣
∣∣∣∣∣∣∣
2
m
+
∣∣∣
∣∣∣I˜hδcv
h,τ
n−1
∣∣∣
∣∣∣
2
m
τ2,
I4(tn) := 2
∫
]tn−1,tn]
(
vh,τn−1, 1n>1N
ϱ;h
tn−1v
h,τ
n−1 + G
ϱ
t
)
m
dwϱt ,
I5(tn) := 2
∫
]tn−1,tn]
∫
Rd
(
vh,τn−1, 1n>1I
h(z)vh,τn−1 + Rt(z)
)
m
q(dt, dz),
I6(tn) :=
(
τI˜hδcv
h,τ
n−1, η(tn)
)
m
,
and where
η(tn) :=
∫
]tn−1,tn]
(
1n>1N
ϱ;h
tn−1v
h,τ
n−1 + G
ϱ
t
)
dwϱt +
∫
]tn−1,tn]
∫
Rd
(
1n>1Ih(z)v
h,τ
n−1 + Rt(z)
)
q(dt, dz).
As in the proof Theorem 5.4.7, by Young’s inequality, (5.4.10), and (5.4.15), we have
E
[
∥vh,τn ∥
2
m
]
≤
(
1 + 2π1({|z| > δ})
)
E
[
∥φ∥2m
]
− q2E
n∑
l=1
τ
d∑
i=1
∥δh,iv
h,τ
l ∥
2
m
+ E
n∑
l=1
τ
(
N2 + 2π
1({|z| > δ}) + τπ1({|z| > δ})2
)
∥vh,τl ∥
2
m
+ NE
∫
]0,tn]


d∑
i=0
∥F it∥
2
mdt + ∥Gt∥
2
m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)

 dt.
Set
Z := N2 + 2π
1({|z| > δ}),
R := max


2π1({|z| > δ})2
√
Z2 + 4π1({|z| > δ}2 − Z
,N2

 T.
Assume T > R. Making use of (5.4.29) and applying discrete Gronwall’s lemma, we get
158 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
that there exist a constant N(d,m,K, κ,T, δ) such that
E
[
∥vh,τn ∥
2
m
]
+ E
n∑
l=1
τ
d∑
i=0
∥δh,iv
h,τ
l ∥
2
m ≤ NE
[
∥φ∥2m
]
+NE
∫
]0,T ]


d∑
i=0
∥F it∥
2
m + ∥Gt∥
2
m,ℓ2 +
∫
Rd
∥Rt(z)∥2mπ
2(dz)

 dt. (5.4.30)
Using (5.4.15) instead of (5.4.16), we obtain (5.4.22) from (5.4.30) in the same manner as
Theorem 5.4.7. Note that no bound on τ/h2 is needed in this case. □
5.5 Proof of the main theorems
Proof of Theorem 5.2.3. By virtue of Theorems 2.9, 2.10, and 4.1 in [Gyö82], in order to
obtain the existence, uniqueness, regularity, and the estimate (5.2.4), we only need to show
that (5.1.1) may be realized as an abstract stochastic evolution equation in a Gelfand triple
and that the growth condition and coercivity condition are satisfied. Indeed, since (5.1.1)
is a linear equation, the hemicontinuity condition is immediate and monotonicity follows
directly from the coercivity condition. By Holder’s inequality and Assumption 5.2.1(i),
for u, v ∈ H1, we have
d∑
i, j=0
(
∂ ju, (v∂−ia
i j
t + a
i j∂−iv)
)
0
+
∫
|z|>δ

u(· + z) − u − 1[−1,1](z)
d∑
j=1
z j∂ ju, v


0
π1(dz)
+
∫
|z|≤δ
∫ 1
0
d∑
i, j=1
(
z j∂ ju(· + θz), zi∂−iv
)
0
(1 − θ)dθπ1(dz) ≤ N∥u∥1∥v∥1.
Therefore, since the pairing ⟨·, ·⟩1 brings (H1)∗ and H−1 into isomorphism, for each (ω, t) ∈
[0,T ] ×Ω, there exists a linear operator A˜t : H1 → H−1 such that ⟨v, A˜tu⟩1 agrees with the
left-hand-side of the above inequality and for u, v ∈ H1, ∥A˜tu∥−1 ≤ N∥u∥1. By Assumption
5.2.2, the operator A defined by A(u) = A˜u+ f , maps H1 to H−1 and for u ∈ H1, ∥At(u)∥−1 ≤
N(∥u∥1 + ∥ f ∥−1).
For an integer m ≥ 1, with abuse of notation, we write
(·, ·)m = ((1 − ∆)m/2·, (1 − ∆)m/2·)0.
and ∥ · ∥m for the corresponding norm in Hm. It is well known that the above inner product
and norm are equivalent to the ones introduced in Section 1. For each m ≥ 1 and for
all u ∈ Hm+1 and v ∈ Hm, we have (u, v)m ≤ ∥u∥m+1∥v∥m−1. Since Hm+1 is dense in Hm−1,
we may define the pairing [·, ·]m : Hm+1 × Hm−1 → R by [v, v′]m = limn→∞(v, vn)m for all
5.5. Proof of the main theorems 159
v ∈ Hm+1 and v′ ∈ Hm−1, where (vn)∞n=1 ⊂ H
m+1 is such that ∥vn − v′∥m−1 → 0 as n→ ∞. It
can be shown that the mapping from Hm−1 to (Hm+1)∗ given by v′ 7→ [·, v′]m is an isometric
isomorphism. For more details, see [Roz90]. Therefore, for all m ≥ 0, (Hm+1,Hm,Hm−1)
forms a Gelfand triple with the pairing [·, ·]m, where we make the convention that ⟨·, ·⟩1 =
[·, ·]0.
For m ≥ 1 and all u ∈ Hm+1 and v ∈ Hm, using integration by parts, we get ⟨v, At(u)⟩1 =
((Lt + It)u + f , v)0 = ⟨v, (Lt + It)u + f ⟩1. Since this is true for all v ∈ H
m, which is dense in
H1, the restriction of A to Hm+1 coincides with L + I + f . Moreover, it can easily be shown
under Assumptions 5.2.1(i) and 5.2.2 that for all m ≥ 1 and u, v ∈ Hm+1, ∥At(u)∥m−1 ≤
N∥u∥m+1 + ∥ f ∥m−1, where N is a constant depending only on m, d,K, and ν, which shows
that A satisfies the growth condition. For u ∈ Hm, m ≥ 1, define Bϱt (u) = b
iϱ
t ∂iu + g
ϱ
t ,
Bt = (B
ϱ
t )
∞
ϱ=1, and Cz(u) = u(· + z) − u + ot(z), z ∈ R
d. Owing to Assumption 5.2.1
(i), Bt is an operator from Hm+1 to Hm(ℓ2). Furthermore, C is an operator from Hm+1
to L2(Rd, π2(dz); Hm) (see (5.5.2)). It is also clear that A, B, and C are appropriately
measurable. Thus, (5.1.1) may be realized as the following stochastic evolution equation
in the Gelfand triple (Hm+1,Hm,Hm−1):
ut = u0 +
∫
]0,t]
As(us)ds +
∫
]0,t]
Bϱs(us)dw
ϱ
s +
∫
]0,t]
Cz(us−)q(dz, ds), (5.5.1)
for t ∈ [0,T ]. Let u ∈ C∞c . A simple calculation shows that there is a constant N = N(δ)
such that
∫
Rd
∥u(· + z) − u∥2mπ
2(dz) ≤ ς2(δ)∥u∥2m+1 + N∥u∥
2
m. (5.5.2)
Applying Holder’s inequality and the identity (u, ∂ ju) = 0, we obtain
∫
|z|>δ′

u(· + z) − u − 1[−1,1](z)
d∑
j=1
z j∂ ju, u


m
π1(dz) ≤ 0.
By the Holder’s inequality and the Cauchy-Bunyakovsky-Schwarz inequality, we have
2
∫
|z|≤δ′
∫ 1
0
d∑
i, j=1
(
z j∂ ju(· + θz), zi∂−iu
)
m
(1 − θ)dθπ1(dz) ≤ ς1(δ)∥u∥2m+1.
There exists a constant ϵ = ϵ(κ, δ) such that
q := κ − ς(δ) − ϵ > 0.
160 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
As in Theorem 4.1.2 in [Roz90] and Lemma 5.4.4, using Holder’s and Young’s inequal-
ities, the above estimates, and Assumption 5.2.1, we find that for each ϵ > 0, there is a
constant N = N(d,m, κ,K,T, δ) such that for all (ω, t) ∈ Ω × [0,T ],
2[u, At(u)]m + ∥Bt(u)∥2m,l2 +
∫
Rd
∥Cz(u)∥2mπ
2(dz) + q∥u∥2m+1
≤ N
(
∥u∥2m + ∥ ft∥m−1 + ∥gt∥m,ℓ2 +
∫
Rd
∥ot(z)∥2mπ
2(dz)
)
.
Using the self-adjointness of (1 − ∆)1/2, the properties of the CBF [·, ·]m, and Assumption
5.2.2, for all v ∈ C∞c and u ∈ H
m+1, m ≥ 1, we have
[v, A(u)]m = ((L + I)u, (1 − ∆)mv)0 + ( f , (1 − ∆)mv)0. (5.5.3)
Owing to (5.5.3) and the denseness of (1 − ∆)−mC∞c in H
1, from Theorems 2.9, 2.10, and
4.1 in [Gyö82], we obtain the existence and uniqueness of a solution u of (5.1.1), such that
u is a càdlàg Hm-valued process satisfying (5.2.4). □
Proof of Proposition 5.2.5. Let A, B, and C be as in (5.5.1). Owing to Assumption 5.2.1,
the boundedness of the m−1-norm of g in expectation, and estimate (5.2.4), using Jensen’s
inequality and Itô’s isometry, for s, t ∈ [0,T ], we get
E


∣∣∣∣∣∣
∣∣∣∣∣∣
∫
]s,t]
Ar(ur)ds
∣∣∣∣∣∣
∣∣∣∣∣∣
2
m−1

 ≤ |t − s|
(
NE
∫
]0,T ]
∥ut∥2m+1dt + E
∫
]0,T ]
∥ fr∥2m−1dr
)
≤ N|t − s|,
E


∣∣∣∣∣∣
∣∣∣∣∣∣
∫
]s,t]
Bϱr (ur)dw
ρ
r
∣∣∣∣∣∣
∣∣∣∣∣∣
2
m−1

 = E
∫
]s,t]
∥Br(ur)∥2m−1,ℓ2dr
≤ N|t − s|
(
sup
t≤T
E∥ut∥2m + sup
t≤T
E∥gt∥m−1,ℓ2
)
≤ N|t − s|,
and
E


∣∣∣∣∣∣
∣∣∣∣∣∣
∫
]s,t]
∫
Rd
Cz(ur−)q(dr, dz)
∣∣∣∣∣∣
∣∣∣∣∣∣
2
m−1

 = E
∫
]s,t]
∫
Rd
∥Cz(ur)∥2m−1π
2(dz)ds
≤ N|t − s|
(
sup
t≤T
E∥ut∥2m + sup
t≤T
E
∫
Rd
∥ot(z)∥2m−1π
2(dz)
)
≤ N|t − s|,
which completes the proof of the proposition. □
Theorem 5.5.1. Let Assumptions 5.2.1 through 5.2.5 hold for some m ≥ 2. Let u be
the solution of (5.1.1) and (uh,τn )Tn=0 be defined by (5.4.1). Then there is a constant N =
N(d,m, κ,K,T,C, λ, κ2m, δ) such that
E
[
max
0≤n≤T
∥utn − u
h,τ
n ∥
2
m−2
]
+ E
T∑
l=0
τ
d∑
i=0
∥δh,iutl − δh,iu
h,τ
l ∥
2
m−2ds ≤ N(τ + |h|
2). (5.5.4)
5.5. Proof of the main theorems 161
Proof. For t ∈ [0,T ], let κ1(t) := tn−1 for t ∈]tn−1, tn], and set e
h,τ
n := u
h,τ
n − utn . One can
easily verify that eh,τn satisfies in Hm−2,
eh,τn = e
h,τ
n−1 +
∫
]tn−1,tn]

(L
h
tn−1 + I
h)eh,τn−1 +
d∑
i=0
δh,iF
i
t

 dt +
∫
]tn−1,tn]
(
Nϱ;htn−1e
h,τ
n−1 + G
ϱ
t
)
dwϱt
+
∫
]tn−1,tn]
∫
Rd
(
Ih(z)eh,τn−1 + Rt(z)
)
q(dt, dz),
where
F0t := (L
h
κ1(t) − Lκ1(t))ut + (Lκ1(t) − Lt)ut + (I
h − I)ut + ( fκ1(t) − ft) + I
h
δc(uκ1(t) − ut)
+
d∑
j=1
a0 j
κ1(t)
δ−h, j(uκ1(t) − ut) +
d∑
i=0
ai0κ1(t)δh,i(uκ1(t) − ut) −
d∑
i, j=1
δ−h, j(uκ1(t) − ut)(· + h)δh,ia
i j
κ1(t)
,
F it :=
d∑
j=1
ai j
κ1(t)
δ−h, j(uκ1(t) − ut) +
∞∑
k=0
χ(h,k)∑
l=1
θ¯
k,h
l ζ
i j
k,hδ−h, j(uκ1(t) − ut)(· + hzrh,kl )
Gϱt : = (N
ϱ
κ1(t)
− Nϱt )ut + (N
ϱ;h
κ1(t)
− Nϱ
κ1(t)
)ut +N
ϱ
κ1(t)
(uκ1(t) − ut) + (g
ϱ
κ1(t)
− gϱt )
Rht (z) : =
(
Ih(z) − I(z)
)
ut− + Ih(z)(uκ1(t) − ut−) +
(
oκ1(t)(z) − ot(z)
)
.
By Theorem 5.4.7, we have
E
[
max
0≤n≤T
∥eh,τn ∥
2
m−2
]
+ E
M∑
n=0
τ
d∑
i=0
∥δh,ieh,τn ∥
2
m−2
≤ NE
∫
]0,T ]
( d∑
i=0
∥F it∥
2
m−2 + ∥Gt∥
2
m−2,ℓ2 +
∫
Rd
∥Rt(z)∥2m−2π
2(dz)
)
dt.
Using Lemmas 5.4.1, 5.4.2, and 5.4.3 and Assumptions 5.2.1(i) and 5.2.4, the right-hand-
side of the above relation can be estimated by
NE
∫
]0,T ]
(
|h|2∥ut∥2m+1 + |κ1(t) − t|∥ut∥
2
m + ∥uκ1(t) − ut∥
2
m−1
)
dt
+NE
∫
]0,T ]
(
∥ fκ1(t) − ft∥
2
m−2 + ∥gκ1(t) − gt∥m−2,ℓ2 +
∫
Rd
∥oκ1(t)(z) − ot(z)∥m−2π
2(dz)
)
dt
where N depends only on d,m, κ,K,C, λ,T, δ and ν. By virtue of (5.2.4), Proposition
162 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
5.2.5, and Assumption 5.2.3, we obtain (5.5.4), which completes the proof. □
Theorem 5.5.2. Let Assumptions 5.2.1 through 5.2.4 hold with m ≥ 2 and let u be the
solution of (5.1.1). There exists a constant R = R(d,m, κ,K, δ) such that if T > R, then
there exists a unique solution (vh,τ)Tn=0 of (5.4.2) in H
m−2. Moreover, there is a constant
N = N(d,m, κ,K,T,C, λ, κ2m, δ) such that
E
[
max
0≤n≤T
∥utn − v
h,τ
n ∥
2
m−2
]
+ E
T∑
l=0
τ
d∑
i=0
∥δh,iutl − δh,iv
h,τ
l ∥
2
m−2ds ≤ N(τ + |h|
2). (5.5.5)
Proof. The existence and uniqueness follows directly from Theorem 5.4.8. Let κ1(t) be
as in the previous proof and set κ2(t) = tn for t ∈]tn−1, tn]. Let G and R be defined as in
Theorem 5.5.1 and define F¯ i to be F i with κ1(t) replaced with κ2(t). Set e
h,τ
n = v
h,τ
n − utn . As
in the proof of Theorem 5.5.1, we have
eh,τn = e
h,τ
n−1 +
∫
]tn−1,tn]

(L˜
h
tn + I
h
δ )e
h,τ
n + I˜
h
δce
h,τ
n−1 +
d∑
i=0
δh,iF˜
i
t

 dt
+
∫
]tn−1,tn]
(
1n>1N
ϱ;h
tn−1e
h,τ
n−1 + G˜
ϱ
t
)
dwϱt +
∫
]tn−1,tn]
∫
Rd
(
1n>1Ih(z)e
h,τ
n−1 + R˜t(z)
)
q(dt, dz),
where
F˜ i = F¯ i, for i , 0, F˜0 = F¯0 + I˜hδc(uκ1(t) − uκ2(t)),
G˜ϱt = 1t≤t1(N
ϱ
t ut + g
ϱ
t ) + 1t>t1G
ϱ
t , R˜t(z) = 1t≤t1I(z)ut− + 1t>t1Rt(z).
By Theorem 5.4.8, we have
E
[
max
0≤n≤T
∥eh,τn ∥
2
m−2
]
+ E
M∑
n=0
τ
d∑
i=0
∥δh,ieh,τn ∥
2
m−2 ≤ N(A1 + A2 + A3),
where
A1 := E
∫
]0,T ]
d∑
i=0
∥F¯ it∥
2
m−2dt +
∫
]t1,T ]
(
∥Gt∥2m−2,ℓ2 +
∫
Rd
∥Rt(z)∥2m−2π
2(dz)
)
dt,
A2 := E
∫
]0,T ]
∥I˜hδc(uκ1(t) − uκ2(t))∥
2
m−2dt
A3 := E
∫
]0,t1]
(
∥Mtut + gt∥2m−2,ℓ2 +
∫
Rd
∥I(z)ut + ot(z)∥2m−2π
2(dz)
)
dt.
5.5. Proof of the main theorems 163
As in the proof of Theorem 5.5.1, we have A1 ≤ N(τ + |h|2). By Proposition 5.2.5, we get
A2 ≤ NE
∫ T
0
∥uκ1(t) − uκ2(t)∥
2
m−1dt ≤ Nτ.
Owing to (5.2.3), we have
A3 ≤ NE
∫ t1
0
(
∥ut∥2m−1 + ∥gt∥
2
m−2,ℓ2 +
∫
Rd
∥ot(z)∥2m−2π
2(dz)
)
dt
≤ NτE
∫ t1
0
(
sup
t≤T
∥ut∥2m−1 + ξ
)
dt ≤ Nτ.
Combining the above estimates yields (5.5.5). □
By virtue of Sobolev’s embedding theorem and (5.2.10), as in [GK10], we obtain the
following corollaries of Theorem 5.5.1 and Theorem 5.5.2.
Corollary 5.5.3. Suppose the assumptions of Theorem 5.5.1 hold with m > n + 2 + d/2,
where n is an integer with n ≥ 0. Then for all λ = (λ1, . . . , λn) ∈ {1 . . . , d}n and δh,λ =
δh,λ1 · · · δh,λn , there is a constant N = N(d,m, κ,K,T,C, λ, κ2m, δ) such that
E
[
max
0≤n≤T
sup
x∈Rd
|δh,λutn(x) − δh,λu
h,τ
n (x)|
2
]
+ E
[
max
0≤n≤T
∥δh,λutn − δh,λu
h,τ
n ∥
2
ℓ2(Gh)
]
≤ N(τ + |h|2).
Corollary 5.5.4. Suppose the assumptions of Theorem 5.5.2 hold with m > n + 2 + d/2,
where n is an integer with n ≥ 0. Then for all λ = (λ1, . . . , λn) ∈ {1 . . . , d}n and δh,λ =
δh,λ1 · · · δh,λn , there is a constant N = N(d,m, κ,K,T,C, λ, κ2m, δ) such that
E
[
max
0≤n≤T
sup
x∈Rd
|δh,λutn(x) − δh,λv
h,τ
n (x)|
2
]
+ E
[
max
0≤n≤T
∥δh,λutn − δh,λv
h,τ
n ∥
2
ℓ2(Gh)
]
≤ N(τ + |h|2).
Proof of Theorems 5.2.8 and 5.2.9. Let (uˆh,τn )Mn=0 be defined by (5.2.8). Denote by (·, ·)ℓ2(Gh)
the inner product of ℓ2(Gh). There exists a constant ϵ = ϵ(κ, δ) such that
q := κ − ς1(δ) − ϵ > 0.
As in (5.4.11), there is a constant N6 = N6(d, κ,K, δ) such that for all ϕ ∈ ℓ2(Gh),
(ϕ, L˜ht ϕ)ℓ2(Gh) + (ϕ, I
h
δϕ)ℓ2(Gh) ≤ −q
d∑
i=1
∥δh,iϕ∥2ℓ2(Gh) + N6∥ϕ∥
2
ℓ2(Gh).
Following the arguments in the beginning of the proof of Theorem 5.4.8, we conclude that
if T > N6T , then there exists a unique solution (vˆ
h,τ
n )Mn=0 in ℓ2(Gh) of (5.2.9). It is easy
to see that N6 < N2 (for the same choice of ϵ) for all m > 0, where N2 is the constant
164 Chapter 5. A finite difference scheme for non-degenerate parabolic SIDEs
appearing on the right-hand-side of (5.4.11), and hence N6 < R, where R is as in Theorem
5.4.8. Let (uh,τn )Mn=1 be defined by (5.4.1). By Theorem 5.5.2, there exists a unique solution
(vh,τn )Mn=1 of (5.4.2). It suffices to show that almost surely,
uh,τn (x) = uˆ
h,τ
n (x) (5.5.6)
and
vh,τn (x) = vˆ
h,τ
n (x), (5.5.7)
for all n ∈ {0, ...,M} and x ∈ Gh. Let S : Hm−2 → ℓ2(Gh) denote the embedding from
Remark 5.2.6. ApplyingS to both sides of (5.4.1), one can see thatS uh,τ and uˆh,τ satisfy
the same recursive relation in ℓ2(Gh) with common initial condition φ, and hence (5.5.6)
follows. Similarly, S vh,τ and vˆh,τ satisfy the same equation in ℓ2(Gh) and (5.5.7) follows
from the uniqueness of the ℓ2(Gh) solution of (5.2.9). □
Remark 5.5.5. It follows from Corollaries 5.5.3, 5.5.4, and relations (5.5.6) and (5.5.7)
that if more regularity is assumed of the coefficients and the data of the equation (5.1.1),
then better estimates can be obtained than the ones presented in Theorems 5.2.8 and 5.2.9.
Chapter 6
Conclusions and future work
In this thesis, we investigated existence, uniqueness, and regularity of degenerate parabolic
linear SIDEs in the whole space with adapted coefficients from a couple of standpoints.
First, we used the method of stochastic characteristics to derive classical solutions directly.
Second, we derived the integer scale L2-Sobolev theory for the equations using the varia-
tional framework of stochastic evolution equations and the method of vanishing viscosity.
We then established the rate of convergence of some finite difference schemes for a simple
class of SIDEs under the assumption of non-degenerate stochastic parabolicity.
There are myriad of future directions to consider. We will only discuss a few. As
we mentioned in Chapter 4, the integer scale Lp-Sobolev theory for degenerate SIDEs is
currently underway and will be available soon. Still though, it would be ideal to obtain
an existence and regularity theory in the full scale of Bessel potential spaces with Hölder
assumptions on the coefficients. It is also interesting to study non-degenerate linear SIDEs
under weaker regularity conditions than those required for the fully degenerate theory.
Currently, an existence theory is known only in a few special cases; we refer the reader to
[MP09, MP11, MP12, MP13, KK12b, KK12a, KKK13] for some relatively recent results
in this direction. With regards to approximations of SIDEs, we considered only equations
with non-degenerate stochastic parabolicity and where the integral operators do not depend
on the space variable. It would be interesting to relax these conditions in the future. Also,
from a practical standpoint, the error from truncating the domain should be studied in con-
junction with the numerical approximations of the equations. It is worth mentioning that
a regularity theory for SIDEs in bounded domains with degenerate stochastic parabolicity
is non-existent as far as the author knows. There is also no reason to confine oneself to
finite difference schemes, since there are other numerical methods such as finite elements
and wavelets. Although no simulations were done in this thesis, they will be considered in
a future work along with some applications to the non-linear filtering problem.
165

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