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Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme

Author

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  • Masaaki Fujii

    (Graduate School of Economics, University of Tokyo)

  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

Abstract

In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out a recursive approximation to an arbitrarily higher order, where the required calculations in each order are equivalent to those for standard European contingent claims. We have also applied the perturbative method to the PDE framework following the so-called Four Step Scheme. The method is found to render the original non-linear PDE into a series of standard parabolic linear PDEs. Due to the equivalence of the two approaches, it is also possible to derive approximate analytic solution for the non-linear PDE by applying the asymptotic expansion to the corresponding probabilistic model. Two simple examples are provided to demonstrate how the perturbation works and show its accuracy relative to known numerical techniques. The method presented in this paper may be useful for various important problems which have eluded analytical treatment so far. 1

Suggested Citation

  • Masaaki Fujii & Akihiko Takahashi, 2011. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," CIRJE F-Series CIRJE-F-802, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2011cf802
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    References listed on IDEAS

    as
    1. Akihiko Takahashi & Nakahiro Yoshida, 2004. "An Asymptotic Expansion Scheme for Optimal Investment Problems," Statistical Inference for Stochastic Processes, Springer, vol. 7(2), pages 153-188, May.
    2. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CIRJE F-Series CIRJE-F-787, CIRJE, Faculty of Economics, University of Tokyo.
    3. Masaaki Fujii & Akihiko Takahashi, 2010. "Asymmetric and Imperfect Collateralization, Derivative Pricing, and CVA," CIRJE F-Series CIRJE-F-781, CIRJE, Faculty of Economics, University of Tokyo.
    4. Kenichiro Shiraya & Akihiko Takahashi & Masashi Toda, 2009. "Pricing Barrier and Average Options under Stochastic Volatility Environment," CIRJE F-Series CIRJE-F-682, CIRJE, Faculty of Economics, University of Tokyo.
    5. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CARF F-Series CARF-F-242, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jul 2011.
    6. Kenichiro Shiraya & Akihiko Takahashi & Masashi Toda, 2009. "Pricing Average Options under Stochastic Volatility Models," CARF J-Series CARF-J-059, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    7. Duffie, Darrell & Huang, Ming, 1996. "Swap Rates and Credit Quality," Journal of Finance, American Finance Association, vol. 51(3), pages 921-949, July.
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    1. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," CARF F-Series CARF-F-270, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jun 2012.
    2. Akihiko Takahashi & Yoshifumi Tsuchida & Toshihiro Yamada, 2022. "A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver (Journal of Computational Physics, published online 19 January 2022)," CARF F-Series CARF-F-532, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Feb 2022.
    3. Akihiko Takahashi & Yoshifumi Tsuchida & Toshihiro Yamada, 2021. "A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver," Papers 2101.09890, arXiv.org, revised Jan 2021.
    4. Masaaki Fujii & Akihiko Takahashi, 2011. "Derivative Pricing under Asymmetric and Imperfect Collateralization and CVA," CARF F-Series CARF-F-265, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Masaaki Fujii & Akihiko Takahashi, 2010. "Asymmetric and Imperfect Collateralization, Derivative Pricing, and CVA," CIRJE F-Series CIRJE-F-781, CIRJE, Faculty of Economics, University of Tokyo.
    6. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," Papers 1202.0608, arXiv.org, revised Sep 2012.
    7. Akihiko Takahashi & Yoshifumi Tsuchida & Toshihiro Yamada, 2021. "A New Efficient Approximation Scheme for Solving High-Dimensional Semilinear PDEs: Control Variate Method for Deep BSDE Solver," CIRJE F-Series CIRJE-F-1159, CIRJE, Faculty of Economics, University of Tokyo.
    8. Masaaki Fujii & Akihiko Takahashi, 2012. "Derivative pricing under asymmetric and imperfect collateralization and CVA," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 749-768, October.
    9. Akihiko Takahashi & Yoshifumi Tsuchida & Toshihiro Yamada, 2021. "A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver," CARF F-Series CARF-F-504, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2022.
    10. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Papers 1204.2638, arXiv.org, revised Apr 2012.
    11. 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.
    12. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," CIRJE F-Series CIRJE-F-840, CIRJE, Faculty of Economics, University of Tokyo.

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