Maximum Principle for Forward-Backward Doubly Stochastic Control Systems and Applications

TL;DR

This paper develops a maximum principle for fully coupled forward-backward doubly stochastic differential equations, enabling the analysis of complex stochastic control problems without requiring convex control domains.

Contribution

It introduces a stochastic maximum principle for FBDSDEs with non-convex control domains and applies it to SPDEs, linear quadratic problems, and differential games.

Findings

Derived a maximum principle for FBDSDEs without control diffusion dependence.

Solved linear quadratic stochastic control problems explicitly.

Established optimal controls and Nash equilibria using FBDSDE solutions.

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 differential games problem.