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Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients

Published online by Cambridge University Press:  09 March 2006

Romain Abraham  and
Olivier Riviere
Romain Abraham
Affiliation:
Laboratoire MAPMO, Université d'Orléans, B.P. 6759, 45067 Orléans Cedex 2, France; Romain.Abraham@univ-orleans.fr
Olivier Riviere
Affiliation:
Laboratoire MAP5, UFR de Mathématiques et d'Informatique, Université René Descartes, 45 rue des Saints Pères, 75270 Paris Cedex 06, France; Olivier.Riviere@math-info.univ-paris5.fr
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Abstract

We consider a system of fully coupled forward-backward stochasticdifferential equations. First we generalize the results ofPardoux-Tang [7] concerning the regularity of the solutions withrespect to initial conditions. Then, we prove that in some particularcases this system leads to aprobabilistic representation of solutions of a second-order PDE whosesecond order coefficients depend on the gradient of the solution. Wethen give some examples in dimension 1 and dimension 2 for which theassumptions are easy to check.

Keywords

Forward-backward stochastic differential equationspartial differential equations.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 184 - 205
Copyright
© EDP Sciences, SMAI, 2006

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