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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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