Maximum principle for forward-backward doubly stochastic control systems and applications

Abstract

Abstract.The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum stochastic differential games problem.Mathematics Subject Classification. 93E20, 60H10.

Partially observed optimal controls of forward-backward doubly stochastic systems

“…Almost simultaneously Bahlali and Gherbal [2] independently proved necessary and sufficient optimality conditions for backward doubly stochastic control systems. Complete information maximum principles of forward-backward doubly stochastic control systems of (1.2) have been studied in [28,29]. They established maximum principles in the case the control domain is convex or the diffusion coefficients can not contain a control variable.…”

Necessary condition for optimal control of doubly stochastic systems

“…Recently, Hu [16] introduces a direct method for treating this similar control problem for FBSDEs, namely considering the second-order terms in the Taylor expansion of the variation for the BSDE, furthermore derives the first and second order variational equations. Inspired a series of work mentioned above, in this paper, we will improve the previous results in the following aspects and owning new feature itself: First, we shall establish two kinds of new adjoint equations, which is crucial to deal with variational inequality; Second, we shall provide a pointwise second order necessary condition for optimal controls, which actually extends the work by Zhang and Shi [42]. A range of interesting phenomena in the framework of doubly stochastic systems will be raised as well.…”

“…As for the classical control problems, Han, Peng and Wu [14] first established a Pontryagin type maximum principle for the optimal control problems for the state process driven by BDSDEs under convex control domain. Later, Zhang and Shi [42] considered the maximum principle for fully coupled forward-backward doubly stochastic control system, under the assumptions that the diffusion coefficient does not contain the control variable with non-convex control constraints. A similar work can be seen in Ji, Wei and Zhang [18] and Bahlali, Gherbal [8] for convex control domains (see references for more details).…”

“…Remark 1. The Hamilton function (32) is completely different from the one in Han, Peng and Wu [14], Zhang and Shi [42]. Specifically, in (32), f depends on z + p (σ (t, x, u) − σ (t,x,ū)) instead of z only.…”

Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations

“…[11]) and stochastic Hamiltonian systems arising in stochastic optimal control problems (cf. [12][13][14]).…”