## Properties and advances of probabilistic and statistical algorithms with applications in finance

##### Abstract

This thesis is concerned with the construction and enhancement of algorithms involving probability
and statistics. The main motivation for these are problems that appear in finance and
more generally in applied science. We consider three distinct areas, namely, credit risk modelling,
numerics for McKean Vlasov stochastic differential equations and stochastic representations of
Partial Differential Equations (PDEs), therefore the thesis is split into three parts.
Firstly, we consider the problem of estimating a continuous time Markov chain (CTMC)
generator from discrete time observations, which is essentially a missing data problem in
statistics. These generators give rise to transition probabilities (in particular probabilities of
default) over any time horizon, hence the estimation of such generators is a key problem in
the world of banking, where the regulator requires banks to calculate risk over different time
horizons. For this particular problem several algorithms have been proposed, however, through
a combination of theoretical and numerical results we show the Expectation Maximisation (EM)
algorithm to be the superior choice. Furthermore we derive closed form expressions for the
associated Wald confidence intervals (error) estimated by the EM algorithm. Previous attempts
to calculate such intervals relied on numerical schemes which were slower and less stable. We
further provide a closed form expression (via the Delta method) to transfer these errors to the
level of the transition probabilities, which are more intuitive. Although one can establish more
precise mathematical results with the Markov assumption, there is empirical evidence suggesting
this assumption is not valid. We finish this part by carrying out empirical research on non-Markov
phenomena and propose a model to capture the so-called rating momentum. This model has
many appealing features and is a natural extension to the Markov set up.
The second part is based on McKean Vlasov Stochastic Differential Equations (MV-SDEs),
these Stochastic Differential Equations (SDEs) arise from looking at the limit, as the number of
weakly interacting particles (e.g. gas particles) tends to infinity. The resulting SDE has coefficients
which can depend on its own law, making them theoretically more involved. Although MV-SDEs
arise from statistical physics, there has been an explosion in interest recently to use MV-SDEs
in models for economics. We firstly derive an explicit approximation scheme for MV-SDEs with
one-sided Lipschitz growth in the drift. Such a condition was observed to be an issue for standard
SDEs and required more sophisticated schemes. There are implicit and explicit schemes one
can use and we develop both types in the setting of MV-SDEs. Another main issue for MVSDEs
is, due to the dependency on their own law they are extremely expensive to simulate
compared to standard SDEs, hence techniques to improve computational cost are in demand.
The final result in this part is to develop an importance sampling algorithm for MV-SDEs, where
our measure change is obtained through the theory of large deviation principles. Although
importance sampling results for standard SDEs are reasonably well understood, there are several
difficulties one must overcome to apply a good importance sampling change of measure in this
setting. The importance sampling is used here as a variance reduction technique although our
results hint that one may be able to use it to reduce propagation of chaos error as well.
Finally we consider stochastic algorithms to solve PDEs. It is known one can achieve numerical
advantages by using probabilistic methods to solve PDEs, through the so-called probabilistic
domain decomposition method. The main result of this part is to present an unbiased stochastic
representation for a first order PDE, based on the theory of branching diffusions and regime
switching. This is a very interesting result since previously (Itô based) stochastic representations
only applied to second order PDEs. There are multiple issues one must overcome in order to
obtain an algorithm that is numerically stable and solves such a PDE. We conclude by showing
the algorithm’s potential on a more general first order PDE.