1. Introduction

Stochastic optimal control with incomplete information is composed of filtering and control. The filtering part is related to two stochastic processes: signal and observation. The signal process is what we want to estimate based on the observation which provides the information we can use. Kalman–Bucy filtering is the most successful result in linear filtering theory, which was obtained by Kalman and Bucy [38]. Nonlinear filtering is much more difficult to study. There have been two essentially different approaches so far. One is based on the innovation process, an observable Brownian motion, with the martingale representation theorem. This theory achieved its culmination with the celebrated paper of Fujisaki et al. [25]. See also Liptser and Shiryayev [49] and Kallianpur [36] for a systematic account of this approach. Another approach was introduced by Duncan [18], Mortensen [56], and Zakai [112] independently, who derived a linear stochastic partial differential equation (SPDE) satisfied by the unnormalized conditional density function of the signal. This SPDE is called the Duncan–Mortensen–Zakai equation, or, simply, Zakai’s equation. Unlike the Kalman–Bucy filtering, nonlinear filtering results in infinite-dimensional stochastic processes, whose analytical solutions are rarely available in general. Much effort has been devoted to finding finite-dimensional filters and numerical schemes. See, e.g., Benes̆ [5], Wonham [98], Xiong [104], and Bain and Crisan [2] for the development of this aspect.