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Discretization and machine learning approximation of BSDEs with a constraint on the Gains-process

Author

Listed:
  • Kharroubi Idris

    (Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM), Paris, France)

  • Lim Thomas

    (ENSIIE, Laboratoire de Mathématiques et Modélisation d’Evry, CNRS UMR 8071, Evry, France)

  • Warin Xavier

    (EDF R&D and FiME, Paris, France)

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.

Suggested Citation

  • Kharroubi Idris & Lim Thomas & Warin Xavier, 2021. "Discretization and machine learning approximation of BSDEs with a constraint on the Gains-process," Monte Carlo Methods and Applications, De Gruyter, vol. 27(1), pages 27-55, March.
    • Handle: RePEc:bpj:mcmeap:v:27:y:2021:i:1:p:27-55:n:1
    DOI: 10.1515/mcma-2020-2080
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    References listed on IDEAS

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    1. Broadie, Mark & Cvitanic, Jaksa & Soner, H Mete, 1998. "Optimal Replication of Contingent Claims under Portfolio Constraints," The Review of Financial Studies, Society for Financial Studies, vol. 11(1), pages 59-79.
    2. 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.
    3. 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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