Description:
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 $F_t^Y$ 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 $F_t^Y$ . 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.