Conditional densities of partially observed jump diffusions

Abstract

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).