This paper is the first part of a series of papers on filtering for partially observed jump diffusions satisfying a stochastic differential equation driven by Wiener processes and Poisson martingale measures. The coefficients of the equation only satisfy appropriate growth conditions. Some results in filtering theory of diffusion processes are extended to jump diffusions and equations for the time evolution of the conditional distribution and the unnormalised conditional distribution of the unobserved process at time t, given the observations until t, are presented.

The filtering equations associated to a partially observed jump diffusion model (Zt)_t∈[0,T]=(Xt,Yt)_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, given observations (Ys)_s∈[0,t], is investigated, when the conditional density of X₀ given Y₀ exists and belongs to a Sobolev space, and the coefficients satisfy appropriate smoothness and growth conditions.