Optimal switching problem and system of reflected multi-dimensional FBSDEs with random terminal time

TL;DR

This paper investigates the existence and uniqueness of solutions for multi-dimensional reflected FBSDEs with random terminal time and connects these results to an optimal switching problem, advancing stochastic control theory.

Contribution

It introduces a penalization method to prove existence and links the FBSDE solutions to optimal switching problems with unbounded stopping times.

Findings

Existence of adapted solutions under mild conditions

Uniqueness proven via a verification theorem

Connection established between FBSDEs and optimal switching problems

Abstract

In this paper, we study the solvability of a class of multi-dimensional forward backward stochastic differential equations (FBSDEs) with oblique reflection and unbounded stopping time. Under some mild assumptions on the coefficients in such FBSDE, the existence result of adapted solutions is done via a penalization method. The uniqueness is obtained by a verification theorem similarly to the one used by Hu and Tang \cite{HT10}. Finally, we establish the connection with the corresponding optimal switching problem. This latter is solved by using the previous results on FBSDEs.