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Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation

Author

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  • Rasulov, A.
  • Raimova, G.
  • Mascagni, M.

Abstract

In this paper we consider the Monte Carlo solution of the Cauchy problem for a nonlinear parabolic equation. Using the fundamental solution of the heat equation, we obtain a nonlinear integral equation with solution the same as the original partial differential equation. On the basis of this integral representation, we construct a probabilistic representation of the solution to our original Cauchy problem. This representation is based on a branching stochastic process that allows one to directly sample the solution to the full nonlinear problem. Along a trajectory of these branching stochastic processes we build an unbiased estimator for the solution of original Cauchy problem. We then provide results of numerical experiments to validate the numerical method and the underlying stochastic representation.

Suggested Citation

  • Rasulov, A. & Raimova, G. & Mascagni, M., 2010. "Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(6), pages 1118-1123.
    • Handle: RePEc:eee:matcom:v:80:y:2010:i:6:p:1118-1123
    DOI: 10.1016/j.matcom.2009.12.009
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    Cited by:

    1. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2014. "A numerical algorithm for a class of BSDEs via the branching process," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1112-1140.
    2. Bouchard Bruno & Tan Xiaolu & Warin Xavier & Zou Yiyi, 2017. "Numerical approximation of BSDEs using local polynomial drivers and branching processes," Monte Carlo Methods and Applications, De Gruyter, vol. 23(4), pages 241-263, December.
    3. Agarwal, Ankush & Claisse, Julien, 2020. "Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5006-5036.

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