Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



An interpolated stochastic algorithm for quasi-linear PDEs

Authors: François Delarue and Stéphane Menozzi
Journal: Math. Comp. 77 (2008), 125-158
MSC (2000): Primary 65C30; Secondary 60H10, 60H35
Published electronically: July 26, 2007
MathSciNet review: 2353946
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we improve the forward-backward algorithm for quasi-linear PDEs introduced in Delarue and Menozzi (2006). The new discretization scheme takes advantage of the standing regularity properties of the true solution through an interpolation procedure. For the convergence analysis, we also exploit the optimality of the square Gaussian quantization used to approximate the conditional expectations involved.

The resulting bound for the error is closely related to the Hölder exponent of the second order spatial derivatives of the true solution and turns out to be more satisfactory than the one previously established.

References [Enhancements On Off] (What's this?)

  • 1. F. Antonelli, Backward-Forward Stochastic Differential Equations, Ann. Appl. Prob. 3-3 (1993), 777-793. MR 1233625 (95a:60079)
  • 2. V. Bally, G. Pagès, and J. Printems, A quantization tree method for pricing and hedging multi-dimensional American options, Math. Finance 15 (2005), 119-168. MR 2116799 (2005k:91142)
  • 3. C. Bender and J. Zhang, Time discretization and Markovian iteration for coupled FBSDEs, Technical report, Weierstrass Institute, Berlin, and University of Southern California, Los Angeles (2006).
  • 4. B. Bouchard and N. Touzi, Discrete time approximation and Monte-Carlo simulation of Backward Stochastic Differential Equations, Stoch. Proc. Appl. 111 (2004), 175-206. MR 2056536 (2005b:65007)
  • 5. S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Second edition. Texts in Applied Mathematics, Springer-Verlag, New York, 2002. MR 1894376 (2003a:65103)
  • 6. C. de Boor, A practical guide to splines. revised edition, Springer-Verlag, New York, 2001. MR 1900298 (2003f:41001)
  • 7. F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stoch. Proc. Appl. 99 (2002), 209-286. MR 1901154 (2003c:60108)
  • 8. F. Delarue and G. Guatteri, Weak existence and uniqueness for FBSDEs, Stoch. Proc. Appl. 116 (2006), 1712-1742. MR 2307056
  • 9. F. Delarue and S. Menozzi, A Forward Backward Stochatic Algorithm for Quasilinear PDEs, Ann. Appl. Prob. 16-1 (2006), 140-184. MR 2209339 (2006m:60096)
  • 10. J. Douglas, J. Ma, and P. Protter, Numerical methods for Forward-Backward Stochastic Differential Equations, Ann. Appl. Prob. 6 (1996), 940-968. MR 1410123 (97k:60160)
  • 11. S. Graf, H. Luschgy, and G. Pagès, Distortion mismatch in the quantization of probability measures, Technical report, no 1051, Laboratoire PMA, Universités Paris 6 et 7 (2006),
  • 12. S. Graf and H. Lushgy, Foundations of quantization for random vectors, LNM-1730, Springer-Verlag, 2000.
  • 13. Y. Hu, P. Imkeller, and M. Müller, Utility maximization in incomplete markets, Ann. Appl. Prob. 15 (2005). MR 2152241 (2006b:91071)
  • 14. J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes, second edition, Springer-Verlag, 2004. MR 1943877 (2003j:60001)
  • 15. M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892.
  • 16. M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Prob. 28 (2000). MR 1782267 (2001h:60110)
  • 17. O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, 1967. MR 0241822 (39:3159b)
  • 18. J.P. Lemor, E. Gobet, and X. Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations, Bernoulli 12 (2006) no. 5, 889-916. MR 2265667
  • 19. -, A regression-based Monte-Carlo method to solve backward stochastic differential equations, Ann. Appl. Prob. 15 (2005), 2172-2002. MR 2152657 (2006c:60078)
  • 20. J. Ma, P. Protter, and J. Yong, Solving Forward-Backward Stochastic Differential Equations explicitly - a four step scheme, Prob. Th. Rel. Fields 98 (1994), 339-359. MR 1262970 (94m:60118)
  • 21. J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, LNM-1702, Springer-Verlag, 1999. MR 1704232 (2000k:60118)
  • 22. G. N. Milstein and M. V. Tretyakov, Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations, Math. Comp. 69-229 (1999), 237-267. MR 1653966 (2000i:65160)
  • 23. -, Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations, IMA J Numer Anal 10.1093/imanum/drl019 (2006),
  • 24. -, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comp. 28 (2006), 561-582. MR 2231721
  • 25. G.N. Milstein and M.V. Tretyakov, Stochastic numerics for mathematical physics, Springer-Verlag, Berlin, 2004. MR 2069903 (2005f:60004)
  • 26. E. Pardoux and S.G. Peng, Adapted solution of a Backward Stochastic Differential Equation, Systems Control Lett. 14-1 (1990), 55-61. MR 1037747 (91e:60171)
  • 27. O. Rivière, Equations différentielles stochastiques progressives rétrogrades couplées: équations aux dérivées partielles et discrétisation, Ph.D. Thesis, Université Paris 5 René Descartes (2005).
  • 28. A.N. Shiryaev, Probability, Second edition, Graduate Texts in Mathematics, 95, Springer-Verlag, New York, 1996. MR 1368405 (97c:60003)
  • 29. W.A. Woyczynski, Burgers-KPZ turbulence, LNM-1700, Springer-Verlag, 1998. MR 1732301 (2000j:60077)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65C30, 60H10, 60H35

Retrieve articles in all journals with MSC (2000): 65C30, 60H10, 60H35

Additional Information

François Delarue
Affiliation: Université Paris 7, UFR de Mathématiques, Case 7012, 2, Place Jussieu, 75251 Paris Cedex 05, France

Stéphane Menozzi
Affiliation: Université Paris 7, UFR de Mathématiques, Case 7012, 2, Place Jussieu, 75251 Paris Cedex 05, France

Received by editor(s): March 30, 2006
Received by editor(s) in revised form: October 31, 2006
Published electronically: July 26, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society