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A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

Author

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  • Stefan Kremsner

    (Department of Mathematics, University of Graz, Heinrichstraße 36, 8010 Graz, Austria)

  • Alexander Steinicke

    (Department of Mathematics and Information Technology, Montanuniversitaet Leoben, Peter Tunner-Straße 25/I, 8700 Leoben, Austria)

  • Michaela Szölgyenyi

    (Department of Statistics, University of Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria)

Abstract

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

Suggested Citation

  • Stefan Kremsner & Alexander Steinicke & Michaela Szölgyenyi, 2020. "A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics," Risks, MDPI, vol. 8(4), pages 1-18, December.
    • Handle: RePEc:gam:jrisks:v:8:y:2020:i:4:p:136-:d:459366
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    References listed on IDEAS

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    1. Jamshaid Ul Rahman & Sana Danish & Dianchen Lu, 2023. "Deep Neural Network-Based Simulation of Sel’kov Model in Glycolysis: A Comprehensive Analysis," Mathematics, MDPI, vol. 11(14), pages 1-9, July.
    2. 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.
    3. Lorenc Kapllani & Long Teng, 2024. "A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations," Papers 2404.08456, arXiv.org.
    4. Julia Eisenberg & Stefan Kremsner & Alexander Steinicke, 2021. "Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate," Papers 2108.00234, arXiv.org.
    5. Julia Eisenberg & Stefan Kremsner & Alexander Steinicke, 2021. "Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate," Mathematics, MDPI, vol. 9(18), pages 1-20, September.
    6. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance ," Post-Print hal-03115503, HAL.
    7. E. Lorenzo & G. Piscopo & M. Sibillo, 2024. "Addressing the economic and demographic complexity via a neural network approach: risk measures for reverse mortgages," Computational Management Science, Springer, vol. 21(1), pages 1-22, June.
    8. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance ," Working Papers hal-03115503, HAL.
    9. Rudiger Frey & Verena Kock, 2021. "Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematics," Papers 2109.11403, arXiv.org, revised Sep 2021.

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