Explicit Filters for Diffusions with Certain Nonlinear Drifts

摘要

A common problem in the analysis of stochastic systems is the estimation of the system's state given only noise-corrupted or incomplete observations. For instance, examples occur in communications theory when one wants to estimate a signal sent over a noisy channel. The problem of filtering is to build an estimate, i.e. filter, that provides the best information about the state given the observations. The most desirable solution to a filtering problem is a recursive, physically realizable algorithm that computes the best mean-square error estimate, and thus it is important to find models for which such algorithms exist. Recently, Benes defined a class of filtering problems that allow explicit computation of the conditional density of the signal given the past of the observations. This paper extends his result and then uses it to build exact, recursive algorithms for estimating any moment of the signal and for estimating polynomial-type transformations of the signal.