Annals of Applied Probability

A forward–backward stochastic algorithm for quasi-linear PDEs

François Delarue and Stéphane Menozzi

Full-text: Open access

Abstract

We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward–backward SDEs, which provides an efficient probabilistic representation of this type of equation. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE. In particular, our work provides an alternative to the method described in [Douglas, Ma and Protter (1996) Ann. Appl. Probab. 6 940–968] and weakens the regularity assumptions required in this reference.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 1 (2006), 140-184.

Dates
First available in Project Euclid: 6 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1141654284

Digital Object Identifier
doi:10.1214/105051605000000674

Mathematical Reviews number (MathSciNet)
MR2209339

Zentralblatt MATH identifier
1097.65011

Subjects
Primary: 65C30: Stochastic differential and integral equations
Secondary: 35K55: Nonlinear parabolic equations 60H10: Stochastic ordinary differential equations [See also 34F05] 60H35: Computational methods for stochastic equations [See also 65C30]

Keywords
Discretization scheme FBSDEs quantization quasi-linear PDEs

Citation

Delarue, François; Menozzi, Stéphane. A forward–backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16 (2006), no. 1, 140--184. doi:10.1214/105051605000000674. https://projecteuclid.org/euclid.aoap/1141654284


Export citation

References

  • Antonelli, F. (1993). Backward–forward stochastic differential equations. Ann. Appl. Probab. 3 777–793.
  • Bally, V. and Pagès, G. (2003). A quantization algorithm for solving discrete time multi-dimensional optimal stopping problems. Bernoulli 9 1003–1049.
  • Bally, V., Pagès, G. and Printems, J. (2005). A quantization tree method for pricing and hedging multi-dimensional American options. Math. Finance 15 119–168.
  • Bouchard, B. and Touzi, N. (2004). Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl. 111 175–206.
  • Chevance, D. (1997). Numerical methods for backward stochastic differential equations. In Numerical Methods in Finance (L. C. G. Rogers and D. Talay, eds.) 232–234. Cambridge Univ. Press.
  • Delarue, F. (2002). On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch. Process. Appl. 99 209–286.
  • Delarue, F. (2003). Estimates of the solutions of a system of quasilinear PDEs. A probabilistic scheme. Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832 290–332. Springer, Berlin.
  • Delarue, F. (2004). Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients. Ann. Probab. 32 2305–2361.
  • Delarue, F. and Menozzi, S. (2005). A probabilistic algorithm for quasilinear PDEs. Available at http://hal.ccsd.cnrs.fr/ccsd-00005448/en/.
  • Douglas, J., Ma, J. and Protter, P. (1996). Numerical methods for forward–backward stochastic differential equations. Ann. Appl. Probab. 6 940–968.
  • Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Random Vectors. Springer, Berlin.
  • Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
  • Ladyzenskaja, O., Solonnikov, V. and Ural'ceva, N. (1968). Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, RI.
  • Ma, J., Protter, P. and Yong, J. (1994). Solving forward–backward stochastic differential equations explicitly–-a four step scheme. Probab. Theory Related Fields 98 339–359.
  • Ma, J. and Yong, J. (1999). Forward–Backward Stochastic Differential Equations and Their Applications. Springer, Berlin.
  • Makarov, R. N. (2003). Numerical solution of quasilinear parabolic equations and backward stochastic differential equations. Russian J. Numer. Anal. Math. Modelling 18 397–412.
  • Milstein, G. N. and Tretyakov, M. V. (1999). Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations. Math. Comp. 69 237–267.
  • Pagès, G., Pham, H. and Printems, J. (2004). Optimal quantization methods and applications to numerical problems in finance. In Handbook on Numerical Methods in Finance (S. Rachev, ed.) 253–298. Birkhäuser, Boston.
  • Pagès, G. and Printems, J. (2004). Functional quantization for numerics with an application to option pricing. Technical Report LPMA 930, Univ. Paris VI–VII. Monte Carlo Methods Appl. To appear.
  • Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • Pardoux, E. and Tang, S. (1999). Forward–backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Related Fields 114 123–150.
  • Shiryaev, A. (1996). Probability, 2nd ed. Springer, New York.
  • Whitham, N. (1974). Linear and Nonlinear Waves. Pure and Applied Mathemathics. Wiley–Interscience, New York.
  • Woyczyński, W. A. (1998). Burgers-KPZ Turbulence. Springer, Berlin.