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Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations

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  • Teng, Long

Abstract

In this work we propose new algorithms for solving high-dimensional backward stochastic differential equations (BSDEs). Based on the general theta-discretization for the time-integrands, we show how to efficiently use eXtreme Gradient Boosting (XGBoost) regression to approximate the resulting conditional expectations in a quite high dimension. A rigorous analysis of the convergence and time complexity is provided. Numerical results illustrate the efficiency and accuracy of our proposed algorithms for solving very high-dimensional (up to 10,000 dimensions) nonlinear BSDEs. Notably, our new algorithms works also quite well on the problems with highly complex structure in high dimension, which cannot be tackled with most of the state-of-art numerical methods.

Suggested Citation

  • Teng, Long, 2022. "Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    • Handle: RePEc:eee:apmaco:v:426:y:2022:i:c:s009630032200203x
    DOI: 10.1016/j.amc.2022.127119
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    References listed on IDEAS

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    1. Huyên Pham & Xavier Warin & Maximilien Germain, 2021. "Neural networks-based backward scheme for fully nonlinear PDEs," Partial Differential Equations and Applications, Springer, vol. 2(1), pages 1-24, February.
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    3. Christian Bender & Nikolaus Schweizer & Jia Zhuo, 2017. "A Primal–Dual Algorithm For Bsdes," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 866-901, July.
    4. 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.
    5. Ma, Jin & Zhang, Jianfeng, 2005. "Representations and regularities for solutions to BSDEs with reflections," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 539-569, April.
    6. Friedman, Jerome H., 2002. "Stochastic gradient boosting," Computational Statistics & Data Analysis, Elsevier, vol. 38(4), pages 367-378, February.
    7. Gobet, E. & Turkedjiev, P., 2017. "Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1171-1203.
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    Cited by:

    1. 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.

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