Nonlinear filtering with signal dependent observation noise

Abstract

The paper studies the filtering problem for a non-classical frame- work: we assume that the observation equation is driven by a signal dependent noise. We show that the support of the conditional distri- bution of the signal is on the corresponding level set of the derivative of the quadratic variation process. Depending on the intrinsic dimension of the noise, we distinguish two cases: In the first case, the conditional distribution has discrete support and we deduce an explicit represen- tation for the conditional distribution. In the second case, the filtering problem is equivalent to a classical one defined on a manifold and we deduce the evolution equation of the conditional distribution. The re- sults are applied to the filtering problem where the observation noise is an Ornstein-Uhlenbeck process.