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 partially observable processes (X, Y), where X non-Markovian pure jump process representing signal and Y general diffusion which provides observations. Our model covers case both are not necessarily quasi left-continuous, allowing them at predictable stopping times. By introducing Markovian version signal, we able compute an explicit equation filter via innovations approach.