The critical variational setting was recently introduced and shown to be applicable to many important SPDEs not covered by the classical variational setting. In this paper, we extend the critical variational setting in several ways. We introduce a flexibility in the range space for the nonlinear drift term, due to which certain borderline cases can now also be included. An example of this is the Allen–Cahn equation in dimension two in the weak setting. In addition to this, we allow the drift to be singular in time, which is something that naturally arises in the study of the skeleton equation for large deviation principles for SPDEs. Last but not least, we present the theory in the case of Lévy noise for which the critical setting was not available yet.

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.