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Asymptotic expansions as control variates for deep solvers to fully-coupled forward-backward stochastic differential equations

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  • Makoto Naito
  • Taiga Saito
  • Akihiko Takahashi
  • Kohta Takehara

Abstract

Coupled forward-backward stochastic differential equations (FBSDEs) are closely related to financially important issues such as optimal investment. However, it is well known that obtaining solutions is challenging, even when employing numerical methods. In this paper, we propose new methods that combine an algorithm recently developed for coupled FBSDEs and an asymptotic expansion approach to those FBSDEs as control variates for learning of the neural networks. The proposed method is demonstrated to perform better than the original algorithm in numerical examples, including one with a financial implication. The results show that the proposed method exhibits not only faster convergence but also greater stability in computation.

Suggested Citation

  • Makoto Naito & Taiga Saito & Akihiko Takahashi & Kohta Takehara, 2025. "Asymptotic expansions as control variates for deep solvers to fully-coupled forward-backward stochastic differential equations," PLOS ONE, Public Library of Science, vol. 20(5), pages 1-17, May.
    • Handle: RePEc:plo:pone00:0321778
    DOI: 10.1371/journal.pone.0321778
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    References listed on IDEAS

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    1. Kenichiro Shiraya & Akihiko Takahashi & Toshihiro Yamada, 2012. "Pricing Discrete Barrier Options Under Stochastic Volatility," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 19(3), pages 205-232, September.
    2. Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2015. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 239-260, September.
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