On partially observed jump diffusions III. Regularity of the filtering density

The filtering equations associated to a partially observed jump diffusion model $(Z_t)_{t\in [0,T]}=(X_t,Y_t)_{t\in [0,T]}$, 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 $X_t$, given observations $(Y_s)_{s\in [0,t]}$, is investigated, when the conditional density of $X_0$ given $Y_0$ exists and belongs to a Sobolev space, and the coefficients satisfy appropriate smoothness and growth conditions.