| dc.description.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). | en |
