Robustness and convergence of approximations to nonlinear filters for jump-diffusions

The paper treats numerical approximations to the nonlinear filtering problem for jump-diffusion processes. This is a key problem in stochastic systems analysis. The processes are defined, and the optimal filters described. In the general nonlinear case, the optimal filters cannot be computed, and some numerical approximation is needed. Then the weak conditions that are required for the convergence of the approximations are given and the convergence is proved. Quite weak conditions are given under which the approximating filter is continuous in the observation function, and it is shown that our canonical methods satisfy the conditions. Such continuity is essential if the approximations are to be used with confidence on actual physical data. Finally, we prove the convergence of Monte Carlo methods for approximating the optimal filters, and also show that the optimal filter is continuous in the parameters of the signal model.