Solving forward-backward stochastic differential equations explicitly – a four step scheme. (English) Zbl 0794.60056

Ma, Jin; Protter, Philip; Yong, Jiongmin

Probab. Theory Relat. Fields 98, No. 3, 339-359 (1994).

We investigate the nature of the adapted solutions to a class of forward- backward stochastic differential equations (SDEs for short) in which the forward equation is nondegenerate. 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 components via the solution of a certain quasilinear 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 unified 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 diffusions, in which the conditional expectation is no longer needed.

Reviewer: J.Ma

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20	Stochastic integral equations
60G44	Martingales with continuous parameter
35K55	Nonlinear parabolic equations

Keywords:

forward-backward stochastic differential equations; uniqueness; integral representation

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