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Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method

Author

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  • Masaaki Fujii

    (Graduate School of Economics, The University of Tokyo)

  • Akihiko Takahashi

    (Graduate School of Economics, University of Tokyo)

Abstract

In this paper, we propose an efficient Monte Carlo implementation of a non-linear FBSDE as a system of interacting particles inspired by the idea of the branching diffusion method of McKean. It will be particularly useful to investigate large and complex systems, and hence it is a good complement of our previous work presenting an analytical perturbation procedure for generic non-linear FBSDEs. There appear multiple species of particles, where the first one follows the diffusion of the original underlying state, and the others the Malliavin derivatives with a grading structure. The number of branching points are capped by the order of perturbation, which is expected to make the scheme less numerically intensive. The proposed method can be applied to semi-linear problems, such as American Options, Credit and Funding Value Adjustments, and even fully non-linear issues, such as the optimal portfolio problems in incomplete and/or constrained markets.

Suggested Citation

  • Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CARF F-Series CARF-F-278, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2015.
  • Handle: RePEc:cfi:fseres:cf278
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    File URL: http://www.carf.e.u-tokyo.ac.jp/pdf/workingpaper/fseries/F278.pdf
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    Cited by:

    1. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," Papers 1710.07030, arXiv.org, revised Oct 2017.
    2. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations with quadratic-growth drivers by Connecting the Short-term Expansions," Papers 1606.04285, arXiv.org, revised Jan 2018.
    3. Akihiko Takahashi & Toshihiro Yamada, 2016. "An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach," CIRJE F-Series CIRJE-F-1009, CIRJE, Faculty of Economics, University of Tokyo.
    4. Akihiko Takahashi & Toshihiro Yamada, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CARF F-Series CARF-F-363, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions(Revised version of CARF-F-387)," CARF F-Series CARF-F-398, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    6. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CIRJE F-Series CIRJE-F-1069, CIRJE, Faculty of Economics, University of Tokyo.
    7. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions," CARF F-Series CARF-F-387, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    8. Stéphane Crépey & Shiqi Song, 2016. "Counterparty risk and funding: immersion and beyond," Finance and Stochastics, Springer, vol. 20(4), pages 901-930, October.
    9. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions," CIRJE F-Series CIRJE-F-1016, CIRJE, Faculty of Economics, University of Tokyo.
    10. Akihiko Takahashi & Toshihiro Yamada, 2016. "An asymptotic expansion for forward-backward SDEs: a Malliavin calculus approach (Forthcoming in Asia-Pacific Financial Markets)," CARF F-Series CARF-F-394, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    11. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs," CARF F-Series CARF-F-372, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    12. Akihiko Takahashi & Toshiaki Watanabe, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver ," CIRJE F-Series CIRJE-F-976, CIRJE, Faculty of Economics, University of Tokyo.
    13. Jean-Paul Laurent & Philippe Amzelek & Joe Bonnaud, 2014. "An overview of the valuation of collateralized derivative contracts," Review of Derivatives Research, Springer, vol. 17(3), pages 261-286, October.
    14. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," Papers 1510.03220, arXiv.org, revised Dec 2015.
    15. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CARF F-Series CARF-F-423, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    16. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," CIRJE F-Series CIRJE-F-993, CIRJE, Faculty of Economics, University of Tokyo.
    17. Akihiko Takahashi & Toshihiro Yamada, 2016. "An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 23(4), pages 337-373, December.

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