Multiple integral expansions for nonlinear filtering

Abstract : In their seminal paper, Fujisaki, Kallianpur and Kunita showed how the best least squares estimate of a signal contained in additive white noise can be represented as a stochastic integral with respect to innovation process, the integral being adapted to the observation process. The difficulty with this representation is that in general this estimate is not useful for computing the estimate since the innovations process depends on the estimate of the signal itself. In this paper we discuss representation of the estimate directly in terms of the observation process. In doing so, we derive new results on multiple integral expansions for square-integrable functionals of the observation process and show the connection of this work to the theory of contraction operators on Fock space. This letter development is due to Nelson and Segal. We also present several applications of these results to determining sub-optimal filters. (Author)