Existence of an Optimal Control for a coupled FBSDE with a non degenerate diffusion coefficient

TL;DR

This paper proves the existence of an optimal control for a coupled forward-backward stochastic differential equation with a non-degenerate diffusion matrix, using approximation and convexity methods.

Contribution

It introduces a new approach to establish the existence of relaxed and strict optimal controls for coupled FBSDEs with non-degenerate diffusion.

Findings

Existence of a sequence of feedback optimal controls for approximating systems

Establishment of a relaxed optimal control via limit passage

Existence of a strict control under Filippov's convexity condition

Abstract

We a controlled system driven by a coupled forward-backward stochastic differential equation (FBSDE) with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation (BSDE), at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition. Our results improve in some sense those of Buckdahn et al..