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A new architecture of high-order deep neural networks that learn martingales

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  • Syoiti Ninomiya
  • Yuming Ma

Abstract

A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning models. The behaviour of deep neural networks based on this architecture, when applied to the problem of pricing financial derivatives, is also examined. The core of this new architecture lies in the high-order weak approximation algorithms of the explicit Runge--Kutta type, wherein the approximation is realised solely through iterative compositions and linear combinations of vector fields of the target SDEs.

Suggested Citation

  • Syoiti Ninomiya & Yuming Ma, 2025. "A new architecture of high-order deep neural networks that learn martingales," Papers 2505.03789, arXiv.org, revised Jun 2025.
    • Handle: RePEc:arx:papers:2505.03789
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    References listed on IDEAS

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    1. Mariko Ninomiya & Syoiti Ninomiya, 2009. "A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method," Finance and Stochastics, Springer, vol. 13(3), pages 415-443, September.
    2. Syoiti Ninomiya & Nicolas Victoir, 2008. "Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(2), pages 107-121.
    3. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
    4. S. Ninomiya & S. Tezuka, 1996. "Toward real-time pricing of complex financial derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 1-20.
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    7. Syoiti Ninomiya & Yuji Shinozaki, 2019. "Higher-order Discretization Methods of Forward-backward SDEs Using KLNV-scheme and Their Applications to XVA Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 26(3), pages 257-292, May.
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