Linear Quadratic Stochastic Optimal Control of Forward Backward Stochastic Control System Associated with Lévy Process

Abstract

This paper analyzes one kind of linear quadratic (LQ) stochastic control problem of forward backward stochastic control system associated with Lévy process. We obtain the explicit form of the optimal control, then prove it to be unique, and get the linear feedback regulator by introducing one kind of generalized Riccati equation. Finally, we discuss the solvability of the generalized Riccati equation, and its existence and uniqueness of the solutions are proved in a special case.

A sufficient condition for optimal control problem of fully coupled forward‐backward stochastic systems with jumps: A state‐constrained control approach

“…We also note that unlike the existing results on LQ control for FBSDEs mentioned above (see References 1,31‐40), our LQ results (see the statement in (c)) need neither the monotonicity assumption for the FBSDE nor the restriction on the objective functional. In fact, our paper provides the sufficient condition to characterize the optimal solution to the LQ problem for FBSDEs without such additional assumptions.…”

“…Remark As in Assumption 1, we assume that there is a unique solution of (), which requires the additional monotonicity assumptions for the coefficients of () in addition to Assumption 2 1,31‐33,35 (see Remarks 19 and 21). We should mention that $$$ \left(\mathbf{LQ}-{\mathbf{P}}^{\prime}\right) $$$, which is the equivalent forward problem of (LQ‐P) defined below, does not need such additional monotonicity assumptions.…”

“…Remark Unlike existing literatures on LQ optimal control for FBSDEs studied in References 1,31‐33,35,39, (LQ‐P) does not need any additional assumptions. In fact, Assumption 3 is standard in various (indefinite) LQ control problems.…”

“…In fact, Assumption 3 is standard in various (indefinite) LQ control problems. Note that in the earlier results on LQ control for FBSDEs, the additional monotonicity assumptions for the coefficients of the FBSDE are crucial to construct and verify the corresponding optimal solutions 1,31‐33,35,39 As mentioned in Section 1.3, the previous approaches to solve (LQ‐P) is the maximum principle or the completion of squares method (see References 1,31‐33,35‐39 and the references therein), where the former is only the necessary condition, while the latter requires additional restrictions on the objective functional.…”

“…Concerning stochastic control problems for FBSDEs, the LQ problems were studied in various literature (see References 1,31‐40 and the references therein). In most of LQ problems for FBSDEs, the monotonicity assumption of the coefficients for FBSDEs (e.g.…”