The risk-sensitive maximum principle for controlled forward–backward stochastic differential equations

Abstract

In this paper, we consider the risk-sensitive optimal control problem for forward–backward stochastic differential equations (FBSDEs). We consider two different cases: (Case 1) the drift and diffusion terms of the forward part are independent of the backward part (not fully-coupled FBSDE) and the running cost does not include the backward part; (Case 2) the fully-coupled FBSDE with terminal and initial costs. In both cases, the risk-sensitive objective functional is considered, the corresponding control domain is not necessarily convex, and the diffusion term of the forward part depends on control and state variables. In both cases, by using the nonlinear transformations of the equivalent risk-neutral problems, we obtain two (different) risk-sensitive maximum principles, which are characterized in terms of the variational inequalities. We show that the risk-sensitive maximum principle of (Case 1) consists of the two-coupled adjoint equations with an additional scalar adjoint equation, which are backward stochastic differential equations (BSDEs). The risk-sensitive maximum principle of (Case 2) consists of the two-coupled adjoint equations, which are coupled FBSDEs, with an additional scalar adjoint equation and optimality conditions. In both cases, the additional scalar adjoint equations are BSDEs, which are induced due to the nonlinear transformations of the adjoint processes in the equivalent risk-neutral problems. Through the application of results, we consider the linear–quadratic risk-sensitive optimal control problem for the linear FBSDE, and by using the risk-sensitive maximum principle of (Case 1), we characterize an explicit optimal solution.