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Neural networks-based algorithms for stochastic control and PDEs in finance

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  • Maximilien Germain
  • Huy^en Pham
  • Xavier Warin

Abstract

This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and derivative pricing in financial engineering. We survey recent results in the literature, present new developments, notably in the fully nonlinear case, and compare the different schemes illustrated by numerical tests on various financial applications. We conclude by highlighting some future research directions.

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  • 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.
    • Handle: RePEc:arx:papers:2101.08068
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    References listed on IDEAS

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    1. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," Papers 1710.07030, arXiv.org, revised Mar 2019.
    2. 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.
    3. Achref Bachouch & Côme Huré & Nicolas Langrené & Huyen Pham, 2019. "Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications," Working Papers hal-01949221, HAL.
    4. repec:dau:papers:123456789/5524 is not listed on IDEAS
    5. Stefan Kremsner & Alexander Steinicke & Michaela Szolgyenyi, 2020. "A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics," Papers 2010.15757, arXiv.org, revised Dec 2020.
    6. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs (Forthcoming in Asia-Pacific Financial Markets)," CARF F-Series CARF-F-456, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    7. Jiequn Han & Ruimeng Hu, 2021. "Recurrent Neural Networks for Stochastic Control Problems with Delay," Papers 2101.01385, arXiv.org, revised Jun 2021.
    8. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 391-408, September.
    9. Idris Kharroubi & Thomas Lim & Xavier Warin, 2020. "Discretization and Machine Learning Approximation of BSDEs with a Constraint on the Gains-Process," Papers 2002.02675, arXiv.org.
    10. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    11. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Carl Remlinger & Joseph Mikael & Romuald Elie, 2022. "Robust Operator Learning to Solve PDE," Working Papers hal-03599726, HAL.
    2. Lukas Gonon, 2022. "Deep neural network expressivity for optimal stopping problems," Papers 2210.10443, arXiv.org.
    3. Zhu, Yichen & Escobar-Anel, Marcos, 2022. "Polynomial affine approach to HARA utility maximization with applications to OrnsteinUhlenbeck 4/2 models," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    4. Alexandre Roch, 2023. "Optimal Liquidation Through a Limit Order Book: A Neural Network and Simulation Approach," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-29, March.
    5. William Lefebvre & Grégoire Loeper & Huyên Pham, 2023. "Differential learning methods for solving fully nonlinear PDEs," Digital Finance, Springer, vol. 5(1), pages 183-229, March.
    6. Ivan Guo & Nicolas Langren'e & Jiahao Wu, 2023. "Simultaneous upper and lower bounds of American option prices with hedging via neural networks," Papers 2302.12439, arXiv.org, revised Apr 2024.
    7. Zhipeng Huang & Balint Negyesi & Cornelis W. Oosterlee, 2024. "Convergence of the deep BSDE method for stochastic control problems formulated through the stochastic maximum principle," Papers 2401.17472, arXiv.org.
    8. Luca Di Persio & Emanuele Lavagnoli & Marco Patacca, 2022. "Calibrating FBSDEs Driven Models in Finance via NNs," Risks, MDPI, vol. 10(12), pages 1-19, November.
    9. William Lefebvre & Gr'egoire Loeper & Huy^en Pham, 2022. "Differential learning methods for solving fully nonlinear PDEs," Papers 2205.09815, arXiv.org.
    10. Mohamed Hamdouche & Pierre Henry-Labordere & Huyen Pham, 2023. "Policy gradient learning methods for stochastic control with exit time and applications to share repurchase pricing," Papers 2302.07320, arXiv.org.
    11. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2021. "Stability of backward stochastic differential equations: the general case," Papers 2107.11048, arXiv.org, revised Apr 2023.
    12. Maximilien Germain & Mathieu Laurière & Huyên Pham & Xavier Warin, 2021. "DeepSets and their derivative networks for solving symmetric PDEs ," Working Papers hal-03154116, HAL.
    13. Antoine Jacquier & Zan Zuric, 2023. "Random neural networks for rough volatility," Papers 2305.01035, arXiv.org.
    14. Maximilien Germain & Mathieu Lauri`ere & Huy^en Pham & Xavier Warin, 2021. "DeepSets and their derivative networks for solving symmetric PDEs," Papers 2103.00838, arXiv.org, revised Jan 2022.
    15. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    16. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.
    17. Tao Chen & Mike Ludkovski & Moritz Vo{ss}, 2022. "On Parametric Optimal Execution and Machine Learning Surrogates," Papers 2204.08581, arXiv.org, revised Oct 2023.
    18. Lukas Gonon, 2021. "Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality," Papers 2106.08900, arXiv.org.
    19. Ren'e Carmona & Mathieu Lauri`ere, 2021. "Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance," Papers 2107.04568, arXiv.org.

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