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Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

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  • Agarwal, Ankush
  • Claisse, Julien

Abstract

We study semi-linear elliptic PDEs with polynomial non-linearity in bounded domains and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. We consider several examples and estimate their solution by using the Monte Carlo method.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:5006-5036
    DOI: 10.1016/j.spa.2020.02.009
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    References listed on IDEAS

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    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.
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    4. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    5. 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.
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    Cited by:

    1. Christian Beck & Lukas Gonon & Arnulf Jentzen, 2024. "Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations," Partial Differential Equations and Applications, Springer, vol. 5(6), pages 1-47, December.

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