Journal of Applied Mathematics and Stochastic Analysis 
Volume 14 (2001), Issue 2, Pages 113-138
doi:10.1155/S1048953301000090

Reflected forward-backward SDEs and obstacle problems with boundary conditions

Jin Ma1 and Jakša Cvitanić2

 1Purdue University, Department of Mathematics, West Lafayette 47907-1395, IN, USA
2Columbia University, Department of Statistics, New York 10027, NY, USA
Received 1 March 1999; Revised 1 October 1999

Abstract

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.