A Global Stochastic Maximum Principle for Forward-Backward Stochastic Control Systems with Quadratic Generators

TL;DR

This paper develops a new global stochastic maximum principle for forward-backward control systems with quadratic generators, utilizing novel estimates for BSDEs with unbounded coefficients and BMO-martingales.

Contribution

It introduces a new estimate for 1D linear BSDEs with unbounded coefficients and proves solvability for multi-dimensional BSDEs, leading to a global maximum principle.

Findings

Established a new estimate for 1D linear BSDEs with unbounded coefficients.

Proved solvability for a class of multi-dimensional BSDEs with BMO-martingales.

Derived a new global stochastic maximum principle for control systems with quadratic generators.

Abstract

We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.