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Nesting Monte Carlo for high-dimensional non-linear PDEs

Author

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  • Warin Xavier

    (EDF R&D, OSIRIS, 7 boulevard Gaspard Monge, 91120Palaiseau, France)

Abstract

A new method based on nesting Monte Carlo is developed to solve high-dimensional semi-linear PDEs. Depending on the type of non-linearity, different schemes are proposed and theoretically studied: variance error are given and it is shown that the bias of the schemes can be controlled. The limitation of the method is that the maturity or the Lipschitz constants of the non-linearity should not be too high in order to avoid an explosion of the computational time. Many numerical results are given in high dimension for cases where analytical solutions are available or where some solutions can be computed by deep-learning methods.

Suggested Citation

  • Warin Xavier, 2018. "Nesting Monte Carlo for high-dimensional non-linear PDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 225-247, December.
    • Handle: RePEc:bpj:mcmeap:v:24:y:2018:i:4:p:225-247:n:1
    DOI: 10.1515/mcma-2018-2020
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    References listed on IDEAS

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    1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    2. Bergman, Yaacov Z, 1995. "Option Pricing with Differential Interest Rates," The Review of Financial Studies, Society for Financial Studies, vol. 8(2), pages 475-500.
    3. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CIRJE F-Series CIRJE-F-1069, CIRJE, Faculty of Economics, University of Tokyo.
    4. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CARF F-Series CARF-F-423, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
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    Cited by:

    1. Yajie Yu & Bernhard Hientzsch & Narayan Ganesan, 2020. "Backward Deep BSDE Methods and Applications to Nonlinear Problems," Papers 2006.07635, arXiv.org.
    2. Yajie Yu & Narayan Ganesan & Bernhard Hientzsch, 2023. "Backward Deep BSDE Methods and Applications to Nonlinear Problems," Risks, MDPI, vol. 11(3), pages 1-16, March.

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