An Introduction to Optimal Control of FBSDE with Incomplete Information

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.

In this chapter, we develop some filtering results for the solutions to BSDEs and FBSDEs, which play an important role in studying the optimal control with incomplete information. We first state a theorem on the stochastic filtering of a general stochastic process. The proof of that result can be found in Liptser and Shiyayev [49], so we omit it here. Then, we apply this result to the stochastic filtering for the solutions to BSDEs in Section 3.​2 and to those for FBSDEs in Section 3.​3.

In this chapter, we study an optimal control problem of fully coupled FBSDE with partial information, i.e., Problem A introduced in Section 1.​2. Using the convex variation and the duality technique, we derive a stochastic maximum principle and two verification theorems for optimality of Problem A. As an application of the optimality conditions, we solve explicitly an LQ optimal control problem and a cash management problem.

In this chapter, we study an optimal control problem with state process governed by a nonlinear FBSDE and with partially observable information, i.e., Problem B introduced in Section 1.​2. For simplicity, we take the dimensions \(n = m = k =\tilde{ k} = 1\). Using a direct method and a Malliavin derivative method, we establish two versions of the stochastic maximum principle for the characterization of the optimal control. To demonstrate the applicability, we work out an illustrative example within the framework of recursive utility and then solve it via the stochastic maximum principle and the stochastic filtering.

In this chapter, we consider the so-called LQ problem with incomplete information aiming at obtaining more explicit results comparing with those of the previous chapters. We first consider this problem when the state is given by a linear FBSDE. After that we will specialize our results to the case when the state is governed by a BSDE only. In this case, explicit solution will be presented. Finally, we will apply our results to an optimal premium problem.