Gyongy, Istvan

Siska, David

Germ, Fabian

2023-01-19T12:16:02Z

2023-01-19T12:16:02Z

2023-01-19

In this thesis, we study the fi ltering problem for a partially observed jump diffusion (Zₜ)ₜɛ[ₒ,T] = (Xₜ, Yₜ)tɛ[ₒ,T] driven by Wiener processes and Poisson martingale measures, such that the signal and observation noises are correlated. We derive the fi ltering equations, describing the time evolution of the normalised conditional distribution (Pₜ(dx))tɛ[ₒ,T] and the unnormalised conditional distribution of the unobservable signal Xₜ given the observations (Yₛ)ₛɛ[ₒ,T]. We prove that if the coefficients satisfy linear growth and Lipschitz conditions in space, as well as some additional assumptions on the jump coefficients, then, if E|πₒ|ᵖLρ < ∞ for some p ≥ 2, the conditional density π = (πₜ)tɛ[ₒ,T], where πₜ = dPₜ/dx, exists and is a weakly cadlag Lp-valued process. Moreover, for an integer m ≥ 0 and p ≥ 2, we show that if we additionally impose m + 1 continuous and bounded spatial derivatives on the coefficients and if the initial conditional density E|πₒ|ᵖWρᵐ < ∞, then π is weakly cadlag as a Wρᵐ-valued process and strongly cadlag as a Wρˢ - valued process for s ɛ [0;m).

https://hdl.handle.net/1842/39732

http://dx.doi.org/10.7488/era/2980

en

The University of Edinburgh

Stochastic Partial Differential Equations

Stochastic analysis

Jump processes

Filtering

Conditional densities of partially observed jump diffusions

On conditional densities of partially observed jump diffusions

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy