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Discretization and Machine Learning Approximation of BSDEs with a Constraint on the Gains-Process

Author

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  • Idris Kharroubi

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique)

  • Thomas Lim

    (LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise)

  • Xavier Warin

    (EDF - EDF)

Abstract

We study the approximation of backward stochastic differential equations (BSDEs for short) with a constraint on the gains process. We first discretize the constraint by applying a so-called facelift operator at times of a grid. We show that this discretely constrained BSDE converges to the continuously constrained one as the mesh grid converges to zero. We then focus on the approximation of the discretely constrained BSDE. For that we adopt a machine learning approach. We show that the facelift can be approximated by an optimization problem over a class of neural networks under constraints on the neural network and its derivative. We then derive an algorithm converging to the discretely constrained BSDE as the number of neurons goes to infinity. We end by numerical experiments. Mathematics Subject Classification (2010): 65C30, 65M75, 60H35, 93E20, 49L25.

Suggested Citation

  • Idris Kharroubi & Thomas Lim & Xavier Warin, 2020. "Discretization and Machine Learning Approximation of BSDEs with a Constraint on the Gains-Process," Working Papers hal-02468354, HAL.
    • Handle: RePEc:hal:wpaper:hal-02468354
    Note: View the original document on HAL open archive server: https://hal.science/hal-02468354v1
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    References listed on IDEAS

    as
    1. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    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.
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    Cited by:

    1. Maximilien Germain & Huy^en Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance," Papers 2101.08068, arXiv.org, revised Apr 2021.
    2. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance ," Post-Print hal-03115503, HAL.
    3. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance ," Working Papers hal-03115503, HAL.

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    More about this item

    Keywords

    Constrainted BSDEs; discrete-time approximation; neural networks approxi- mation; facelift transformation;
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