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The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations

Author

Listed:
  • Shuaiqiang Liu

    (Applied Mathematics (DIAM), Delft University of Technology, 2628 CD Delft, The Netherlands)

  • Lech A. Grzelak

    (Applied Mathematics (DIAM), Delft University of Technology, 2628 CD Delft, The Netherlands
    Rabobank, 3500 HG Utrecht, The Netherlands)

  • Cornelis W. Oosterlee

    (Applied Mathematics (DIAM), Delft University of Technology, 2628 CD Delft, The Netherlands
    Centrum Wiskunde & Informatica (CWI), 1098 XG Amsterdam, The Netherlands
    Current address: Mathematical Institute, Utrecht University, 3584 CD Utrecht, The Netherlands.)

Abstract

We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.

Suggested Citation

  • Shuaiqiang Liu & Lech A. Grzelak & Cornelis W. Oosterlee, 2022. "The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations," Risks, MDPI, vol. 10(3), pages 1-27, February.
  • Handle: RePEc:gam:jrisks:v:10:y:2022:i:3:p:47-:d:756448
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    References listed on IDEAS

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    1. L. A. Grzelak & J. A. S. Witteveen & M. Suárez-Taboada & C. W. Oosterlee, 2019. "The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘expensive’ distributions," Quantitative Finance, Taylor & Francis Journals, vol. 19(2), pages 339-356, February.
    2. Leitao, Álvaro & Grzelak, Lech A. & Oosterlee, Cornelis W., 2017. "On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 461-479.
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    Cited by:

    1. Leonardo Perotti & Lech A. Grzelak, 2022. "On Pricing of Discrete Asian and Lookback Options under the Heston Model," Papers 2211.03638, arXiv.org, revised Feb 2024.
    2. Grzelak, Lech A., 2022. "Sparse grid method for highly efficient computation of exposures for xVA," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    3. Lech A. Grzelak, 2021. "Sparse Grid Method for Highly Efficient Computation of Exposures for xVA," Papers 2104.14319, arXiv.org, revised May 2022.

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