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Solving forward-backward stochastic differential equations explicitly — a four step scheme
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  • Published: September 1994

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

  • Jin Ma1,
  • Philip Protter1 &
  • Jiongmin Yong2 

Probability Theory and Related Fields volume 98, pages 339–359 (1994)Cite this article

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Summary

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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, 47907-1395, West Lafayette, IN, USA

    Jin Ma & Philip Protter

  2. Department of Mathematics, Fudan University, 200433, Shanghai, People's Republic of China

    Jiongmin Yong

Authors
  1. Jin Ma
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  2. Philip Protter
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  3. Jiongmin Yong
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Additional information

Supported in part by U.S. NSF grant# DMS-9301516

Supported in part by U.S. NSF grant # DMS-9103454

Supported in part by NSF of China and Fok Ying Tung Education Foundation; part of this work was performed while visiting the IMA, University of Minnesota, Minneapolis, MN 55455

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Ma, J., Protter, P. & Yong, J. Solving forward-backward stochastic differential equations explicitly — a four step scheme. Probab. Th. Rel. Fields 98, 339–359 (1994). https://doi.org/10.1007/BF01192258

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  • Received: 07 May 1993

  • Revised: 16 September 1993

  • Issue Date: September 1994

  • DOI: https://doi.org/10.1007/BF01192258

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Mathematics Subject Classification (1991)

  • 60H10
  • 60H20
  • 60G44
  • 35K55
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