A new nonlinear filtering formula for discrete time measurements

Abstract

An exact formula for computing the conditional mean of a random variable is derived for discrete time observations. This formula is analogous to the well-known result of Fujisaki-Kallianpur-Kunita for continuous time observations, and it is similar to the discrete time formula recently derived by Takeuchi and Akashi. The derivation of the new formula is extremely elementary, and it is based on the judicious choice of a certain homotopy function.