On partially observed jump diffusions II. The filtering density

TL;DR

This paper proves the existence and regularity of the filtering density for partially observed jump diffusions, showing that the conditional density of the unobserved process exists and belongs to certain function spaces under general conditions.

Contribution

It establishes the existence and $L_p$ regularity of the filtering density for jump diffusions with Lipschitz coefficients, extending previous results to more general stochastic models.

Findings

Conditional density of unobserved component exists

Density belongs to $L_p$ space under certain conditions

Results hold for general Lipschitz and growth conditions

Abstract

A partially observed jump diffusion $Z=(X_t,Y_t)_{t\in[0,T]}$ given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component $X_t$ given the observations $(Y_s)_{s\in[0,T]}$ exists and belongs to $L_p$ if the conditional density of $X_0$ given $Y_0$ exists and belongs to $L_p$.