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October 2021 Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation
Pierre Henry-Labordère, Nizar Touzi
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Ann. Appl. Probab. 31(5): 2350-2375 (October 2021). DOI: 10.1214/20-AAP1649

Abstract

We provide probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for an approximation of the solution by the standard Monte Carlo method, whose error estimate is controlled by the standard central limit theorem, thus partly bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein–Gordon equation, a simplified scalar version of the Yang–Mills equation, a fourth-order nonlinear beam equation and the Gross–Pitaevskii PDE as an example of nonlinear Schrödinger equations.

Funding Statement

This work benefits from the financial support of the ERC Advanced Grant 321111, and the Chairs Financial Risk and Finance and Sustainable Development.

Citation

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Pierre Henry-Labordère. Nizar Touzi. "Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation." Ann. Appl. Probab. 31 (5) 2350 - 2375, October 2021. https://doi.org/10.1214/20-AAP1649

Information

Received: 1 January 2018; Revised: 1 October 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1649

Subjects:
Primary: 35A99, 35C15, 60J85, 65C05

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 5 • October 2021
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