Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T11:22:36.200Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

Abstract

We consider a system of fully coupled forward-backward stochastic differential equations. First we generalize the results of Pardoux-Tang [7] concerning the regularity of the solutions with respect to initial conditions. Then, we prove that in some particular cases this system leads to a probabilistic representation of solutions of a second-order PDE whose second order coefficients depend on the gradient of the solution. We then give some examples in dimension 1 and dimension 2 for which the assumptions 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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antonelli, F., Backward forward stochastic differential equations. Ann. Appl. Probab. 3 (1993) 777–793. CrossRef
Delarue, F., On the existence and uniqueness of solutions to fbsdes in a non-degenerate case. Stochastic Process. Appl. 99 (2002) 209–286. CrossRef
F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16 (2006).
Ma, J., Protter, P. and Yong, J., Solving forward-backward stochastic differential equations explicitely – a four step scheme. Probab. Th. Rel. Fields 98 (1994) 339–359. CrossRef
J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications. Springer, Berlin. Lect. Notes Math. 1702 (1999).
E. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic pdes of second order, in Stochastic Analysis and Relates Topics: The Geilo Workshop (1996) 79–127.
Pardoux, E. and Tang, S., Forward-backward stochastic differential equations and quasilinear parabolic pdes. Probab. Th. Rel. Fields 114 (1999) 123–150. CrossRef
Perona, P. and Malik, J., Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intell. 12 (1990) 629–639. CrossRef