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 (1984). There are few examples of nonlinear systems for which the optimal filter is finite dimensional, in particular the Kalman, Benes, and Daum filters. In the present paper, we construct new classes of scalar nonlinear systems 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 system admits a finite-dimensional filter evolving in the prescribed exponential family, provided the coefficients of the exponential family include the observation function and its square.