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The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations


  • Shuaiqiang Liu
  • Lech A. Grzelak
  • Cornelis W. Oosterlee


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 a 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. Error analysis confirms 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. 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, 2020. "The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations," Papers 2009.03202,, revised Sep 2021.
  • Handle: RePEc:arx:papers:2009.03202

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    References listed on IDEAS

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    5. 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.
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    7. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469,, revised Sep 2018.
    8. 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.
    9. Maximilian Gaß & Kathrin Glau & Mirco Mahlstedt & Maximilian Mair, 2018. "Chebyshev interpolation for parametric option pricing," Finance and Stochastics, Springer, vol. 22(3), pages 701-731, July.
    10. Luca Capriotti & Mike Giles, 2010. "Fast Correlation Greeks by Adjoint Algorithmic Differentiation," Papers 1004.1855,
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    Cited by:

    1. T. van der Zwaard & L. A. Grzelak & C. W. Oosterlee, 2024. "On the Hull-White model with volatility smile for Valuation Adjustments," Papers 2403.14841,
    2. Leonardo Perotti & Lech A. Grzelak, 2022. "On Pricing of Discrete Asian and Lookback Options under the Heston Model," Papers 2211.03638,, revised Feb 2024.
    3. Grzelak, Lech A., 2022. "Sparse grid method for highly efficient computation of exposures for xVA," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    4. Lech A. Grzelak, 2021. "Sparse Grid Method for Highly Efficient Computation of Exposures for xVA," Papers 2104.14319,, revised May 2022.

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