Appl Math Optim 45:1–22 (2002)
2002 Springer-Verlag New York Inc.
Approximate Solvability of Forward–Backward Stochastic
and J. Yong
Department of Mathematics, Purdue University,
West Lafayette, IN 47907-1395, USA
Laboratory of Mathematics for Nonlinear Sciences, Department of Mathematics,
and Institute of Mathematical Finance, Fudan University,
Shanghai 200433, People’s Republic of China
Abstract. The solvability of forward–backward stochastic differential equations
(FBSDEs for short) has been studied extensively in recent years. To guarantee the
existence and uniqueness of adapted solutions, many different conditions, some
quite restrictive, have been imposed. In this paper we propose a new notion: the
approximate solvability of FBSDEs, based on the method of optimal control intro-
duced in our primary work . The approximate solvability of a class of FBSDEs
is shown under mild conditions; and a general scheme for constructing approximate
adapted solutions is proposed.
Key Words. Forward–backward stochastic differential equation, Approximate
solvability, Approximate adapted solution, Nodal set.
AMS Classiﬁcation. 60H15, 35R60, 34F05, 93E20.
The ﬁrst author was supported in part by the Ofﬁce of Naval Research Grant #N00014-96-1-0262 and
the U.S. National Science Foundation Grant #9971720. The second author’s work was supported in part by the
NSFC, under Grant #79790130, the National Distinguished Youth Science Foundation of China under Grant
#19725106, the Chinese Education Ministry Science Foundation under Grant #2000024605, and the Cheung
Kong Scholars Programme. Part of this work was completed when this author was visiting the Department of
Mathematics, Purdue University.