Three Robustness Results for the Nonlinear Filtering Problem

We show three different robustness resultswith respect to the modelling of the system process for the optimal filter in the classical nonlinear filtering problem. More precisely it is shown that if the system process is given by a markovian SDE, then under rather strict assumptions, the optimal filter is L_p-continuous with respectto the sup-norm on thecoefficients in the system process. Then it is shown that if the system process is given by a nonmarkovian SDEthe filter is pathwise continuous with respect to the drift term for fixed diffusion term bounded away from 0. Lastly it is proven that for general RCLL system processes, there iscontinuity with respect to the weak topology on probability measures in the sense that if a sequence of RCLL processes converges in distribution to the system process, a sequence of probability measures corresponding to theerroneous filters converges weakly towards a probability measure associated with the correct filter.