Probability Theory and Related Fields

, Volume 98, Issue 3, pp 339–359 | Cite as

Solving forward-backward stochastic differential equations explicitly — a four step scheme

  • Jin Ma
  • Philip Protter
  • Jiongmin Yong


In this paper 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 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 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.

Mathematics Subject Classification (1991)

60H10 60H20 60G44 35K55 


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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Jin Ma
    • 1
  • Philip Protter
    • 1
  • Jiongmin Yong
    • 2
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsFudan UniversityShanghaiPeople's Republic of China

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