Reflected forward–backward stochastic differential equations and obstacle problems with boundary conditions

In this paper we study a class of forward–backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may depend on time and is possibly random. The solvability of such FBSDER is studied in a fairly general way. We also prove that if the coefficients are all deterministic and the backward equation is one-dimensional, then the adapted solution of such FBSDER will give the viscosity solution of a quasilinear variational inequality (obstacle problem) with a Neumann boundary condition. As an application, we study how the solvability of FBSDERs is related to the solvability of an American game option.