20 09 A finite dimensional filter with exponential conditional density ∗

In this paper we consider the continuous–time nonlinear filtering problem, which has an infinite–dimensional solution in general, as proved by Chaleyat–Maurel and Michel. There are few examples of nonlinear systems for which the optimal filter is finite dimensional, in particular Kalman’s, Beneš’, and Daum’s filters. In the present paper, we construct new classes of scalar nonlinear filtering problems admitting finite–dimensional filters. We consider a given (nonlinear) diffusion coefficient for the state equation, a given (nonlinear) observation function, and a given finite–dimensional exponential family of probability densities. We construct a drift for the state equation such that the resulting nonlinear filtering problem admits a finite–dimensional filter evolving in the prescribed exponential family augmented by the observaton function and its square. keywords Scalar Nonlinear Diffusion Processes, Finite Dimensional Families, Exponential Families, Stochastic Differential Equations, Scalar Nonlinear Filtering Problem, Finite– Dimensional Filters.