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

Ankush Agarwal and Julien Claisse

Stochastic Processes and their Applications, 2020, vol. 130, issue 8, 5006-5036

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.

Keywords: Automatic differentiation formula; Branching diffusion processes; Elliptic; Monte Carlo method; Partial differential equation; Semi-linear (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2020.02.009

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