Solving forward-backward stochastic differential equations explicitly -- a four step scheme
type: Item
instance of: scholarly article
title: Solving forward-backward stochastic differential equations explicitly -- a four step scheme
zbMATH Open document ID: 0794.60056
DOI: 10.1007/BF01192258
author: Jin Ma
author: Philip Protter
author: Jiongmin Yong
publication date: 1994-08-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 569033
zbMATH Keywords: forward-backward stochastic differential equations
zbMATH Keywords: uniqueness
zbMATH Keywords: integral representation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01192258
OpenAlex ID: W2046553535
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569033