| Abstract: | We deal with a filtering problem of a general jump-diffusion process, X, when the observations process, Y, is a correlated jump-diffusion having common jump times with X. In this frame, at any at time t the sigma-algebra generated by Y provides all the available information about X and the central goal is to characterize the filter, that is the conditional distribution of X given observations. To this aim, we prove that the filter solves Kushner-Stratonovich equation and by applying the Filtered Martingale Problem approach, that it is the unique weak solution to this equation. Under an additional hypothesis we provide also a pathwise uniqueness result. As an application, we consider a financial market where Y describes the logreturn process of a risky asset S whose dynamics depends on an unobservable stochastic factor X. Investors acting on the market can access only to the information flow generated by stock prices. Thus, we are in presence not only of an incomplete market situation but also of partial information. Assuming the price S of the risky asset modeled directly under a martingale measure we study a risk-minimizing hedging problem, under restricted information, whose solution can be computed via the filter by using a projection result.
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