Stochastic Verification Theorem of Forward-Backward Controlled Systems for Viscosity Solutions

TL;DR

This paper develops a new verification theorem for forward-backward stochastic systems using viscosity solutions, broadening applicability and enabling analysis of optimal stochastic feedback controls without requiring derivatives of value functions.

Contribution

It introduces a novel verification theorem for controlled forward-backward systems based on viscosity solutions, avoiding derivative requirements and expanding classical methods.

Findings

Derived a new verification theorem applicable to wider classes of systems.

Established conditions for optimal stochastic feedback controls.

Demonstrated broader applicability compared to classical theorems.

Abstract

In this paper, we investigate the controlled system described by forward-backward stochastic differential equations with the control contained in drift, diffusion and generator of BSDE. A new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions. It is worth to pointing out that this theorem has wider applicability than the restrictive classical verification theorems. As a relevant problem, the optimal stochastic feedback controls for forward-backward system are discussed as well.