On partially observed jump diffusions II. The filtering density

Abstract

A partially observed jump diffusion Z " pXt, Ytq tPr0,T s 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 Xt given the observations pYsq sPr0,T s exists and belongs to Lp if the conditional density of X0 given Y0 exists and belongs to Lp.