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

TL;DR

This paper investigates the regularity of the filtering density in a partially observed jump diffusion model, extending previous results to establish conditions under which the density belongs to a Sobolev space.

Contribution

It provides new regularity results for the filtering density of jump diffusions, building on prior work and under specific smoothness and growth conditions.

Findings

Conditional density belongs to a Sobolev space under certain conditions

Established regularity results for jump diffusion filtering equations

Extended previous results to more general jump diffusion models

Abstract

The filtering equations associated to a partially observed jump diffusion model $(Z_t)_{t\in [0,T]}=(X_t,Y_t)_{t\in [0,T]}$, 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.