A maximum principle for controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal sate constraints

TL;DR

This paper establishes a stochastic maximum principle for optimal control of time-symmetric forward-backward doubly stochastic differential equations with initial-terminal state constraints, using perturbation methods and Ekeland's principle.

Contribution

It introduces a necessary optimality condition for complex stochastic systems with symmetric and boundary constraints, expanding control theory in stochastic differential equations.

Findings

Derived a stochastic maximum principle for the system.

Applied the principle to linear-quadratic control models.

Provided insights into boundary-constrained stochastic control problems.

Abstract

In this paper, we study the optimal control problem of a controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal sate constraints. Applying the terminal perturbation method and Ekeland's variation principle, a necessary condition of the stochastic optimal control, i.e., stochastic maximum principle is derived. Applications to backward doubly stochastic linear-quadratic control models are investigated.