Journal of Applied Mathematics and Stochastic Analysis
Volume 14 (2001), Issue 2, Pages 113-138

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

Copyright © 2001 Jin Ma and Jakša Cvitanić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.