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

Romain Abraham; Olivier Riviere

ESAIM: Probability and Statistics (2006)

  • Volume: 10, page 184-205
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Abraham, Romain, and Riviere, Olivier. "Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients." ESAIM: Probability and Statistics 10 (2006): 184-205. <http://eudml.org/doc/249748>.

@article{Abraham2006,
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. },
author = {Abraham, Romain, Riviere, Olivier},
journal = {ESAIM: Probability and Statistics},
keywords = {Forward-backward stochastic differential equations; partial differential equations.; forward-backward stochastic differential equations; partial differential equations},
language = {eng},
month = {3},
pages = {184-205},
publisher = {EDP Sciences},
title = {Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients},
url = {http://eudml.org/doc/249748},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Abraham, Romain
AU - Riviere, Olivier
TI - Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients
JO - ESAIM: Probability and Statistics
DA - 2006/3//
PB - EDP Sciences
VL - 10
SP - 184
EP - 205
AB - 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.
LA - eng
KW - Forward-backward stochastic differential equations; partial differential equations.; forward-backward stochastic differential equations; partial differential equations
UR - http://eudml.org/doc/249748
ER -

References

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  1. F. Antonelli, Backward forward stochastic differential equations. Ann. Appl. Probab.3 (1993) 777–793.  Zbl0780.60058
  2. F. Delarue, On the existence and uniqueness of solutions to fbsdes in a non-degenerate case. Stochastic Process. Appl.99 (2002) 209–286.  Zbl1058.60042
  3. F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab.16 (2006).  Zbl1097.65011
  4. J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitely – a four step scheme. Probab. Th. Rel. Fields98 (1994) 339–359.  Zbl0794.60056
  5. J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications. Springer, Berlin. Lect. Notes Math.1702 (1999).  Zbl0927.60004
  6. 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.  Zbl0893.60036
  7. E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic pdes. Probab. Th. Rel. Fields114 (1999) 123–150.  Zbl0943.60057
  8. P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intell.12 (1990) 629–639.  

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