Solving Forward-backward Stochastic Diierential Equations Explicitly | a Four Step Scheme

In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic diierential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an \ordinary" sense over an arbitrarily prescribed time duration, via a direct \Four Step Scheme". Using this scheme, we further prove that the backward components of the adapted solution are determined explicitly by the forward component via the solution of a certain quasi-linear parabolic PDE system. Moreover the uniqueness of the adapted solutions (over an arbitrary time duration), as well as the continuous dependence of the solutions on the parameters, can all be proved within this uniied framework. Some special cases are studied separately. In particular, we derive a new form of the integral representation of the Clark-Haussmann-Ocone type for functionals (or functions) of diiusions, in which the conditional expectation is no longer needed.