# On optimal control of forward backward stochastic differential equations

## TL;DR

This paper addresses the optimal control problem for systems governed by coupled forward-backward stochastic differential equations, establishing existence of relaxed controls and conditions for strict control realization.

## Contribution

It introduces a framework for relaxed controls in forward-backward SDEs and proves the existence of optimal controls using weak convergence techniques.

## Key findings

- Existence of relaxed optimal controls for coupled FBSDEs.
- Under convexity, relaxed controls can be realized by strict controls.
- Utilizes tightness and weak convergence in Jakubowski S-topology.

## Abstract

We consider a control problem where the system is driven by a decoupled as well as a coupled forward-backward stochastic differential equation. We prove the existence of an optimal control in the class of relaxed controls, which are measure-valued processes, generalizing the usual strict controls. The proof is based on some tightness properties and weak convergence on the space D of c\`adl\`ag functions, endowed with the Jakubowsky S-topology. Moreover, under some convexity assumptions, we show that the relaxed optimal control is realized by a strict control.