Nonlinear filtering for jump diffusion observations

We deal with the filtering problem of a general jump diffusion process, X, when the observation process, Y, is a correlated jump diffusion process having common jump times with X. In this setting, at any time t the σ-algebra F^Y_t provides all the available information about X_t , and the central goal is to characterize the filter, π_t , which is the conditional distribution of X_t given observations F^Y_t. To this end, we prove that π t solves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see Kurtz and Ocone (1988)), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.