Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations

Abstract

Abstract. This paper deals with the optimal control problem in which the controlled system is described by a fully coupled anticipated forward-backward stochastic differential delayed equation. The maximum principle for this problem is obtained under the assumption that the diffusion coefficient does not contain the control variables and the control domain is not necessarily convex. Both the necessary and sufficient conditions of optimality are proved. As illustrating examples, two kinds of linear quadratic control problems are discussed and both optimal controls are derived explicitly.Mathematics Subject Classification. 93E20, 60H10, 34K50.

A Global Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Systems

“…Theorem 2.6 (see [8,16])Suppose that Assumptions 2.1, 2.3, 2.4 and 2.5 hold. If there exists an admissible controlū(·) satisfying (2.3), where (h(·), m(·), n(·)) is defined in (2.4), thenū(·) is the unique optimal control for the FBLQ problem (2.1)-(2.2).…”

Non-zero sum differential games of anticipated forward-backward stochastic differential delayed equations under partial information and application

“…Moreover, the existence and uniqueness of (19) is equivalent to the existence and uniqueness of (18). According to the monotonicity condition in [19,28], it is easy to check that DFBSDDE (19) satisfies the condition and it has a unique solution. So TFBSDDE (18) admits a unique solution.…”

“…Recently, Huang and Shi [28] discussed the optimal control problem based on the AFBSDDE system. Our work distinguished itself from the above-mentioned ones in the following aspects.…”

Maximum Principle for Delayed Stochastic Linear-Quadratic Control Problem with State Constraint

“…Along this line, [13] studied the maximum principle for delayed stochastic optimal control problems in which the control domain is assumed to be convex and both the control variable and its delay part enter the diffusion coefficient. After that, [14] studied the optimal control problem in which the control system is described by a fully coupled anticipated forward-backward stochastic differential delayed equation, and then [15] generalized [13] to the case when the system involves both continuous and impulse controls and the coefficients are random. In practice, sometimes state constraints are inevitably encountered in stochastic optimal control problems; see, for example, [6,10,16,17].…”