Stochastic filtering of a pure jump process with predictable jumps and path-dependent local characteristics

The objective of this paper is to study the filtering problem for a system of partially observable processes $(X, Y)$, where $X$ is a non-Markovian pure jump process representing the signal and $Y$ is a general jump diffusion which provides observations. Our model covers the case where both processes are not necessarily quasi-left-continuous, allowing them to jump at predictable stopping times. The aim is to characterize the optional projection of the signal with respect to the observation filtration generated by $Y$, i.e., the filtering process. To do so, we introduce the Markovian version of the signal, which is linked to the process $X$ by a suitable bijective transformation. This allows for the computation of the SDE satisfied by the filter via the innovation approach.