On partially observed jump diffusions III: regularity of the filtering density

Abstract

The filtering equations associated to a partially observed jump diffusion model $$(Z_t)_{t\in [0,T]}=(X_t,Y_t)_{t\in [0,T]}$$ ( Z t ) t ∈ [ 0 , T ] = ( X t , Y t ) t ∈ [ 0 , T ] , driven by Wiener processes and Poisson martingale measures are considered. Building on results from two preceding articles on the filtering equations, the regularity of the conditional density of the signal $$X_t$$ X t , given observations $$(Y_s)_{s\in [0,t]}$$ ( Y s ) s ∈ [ 0 , t ] , is investigated, when the conditional density of $$X_0$$ X 0 given $$Y_0$$ Y 0 exists and belongs to a Sobolev space, and the coefficients satisfy appropriate smoothness and growth conditions.