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

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.