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Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme

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  • Masaaki Fujii
  • Akihiko Takahashi
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    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.

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    File URL: http://arxiv.org/pdf/1106.0123
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1106.0123.

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    Date of creation: Jun 2011
    Date of revision: Jan 2012
    Handle: RePEc:arx:papers:1106.0123

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    Web page: http://arxiv.org/

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    Cited by:
    1. Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2012. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," Papers 1211.5867, arXiv.org.
    2. Masaaki Fujii & Akihiko Takahashi, 2013. "Making Mean-Variance Hedging Implementable in a Partially Observable Market -with supplementary contents for stochastic interest rates-," CIRJE F-Series CIRJE-F-891, CIRJE, Faculty of Economics, University of Tokyo.
    3. Masaaki Fujii & Akihiko Takahashi, 2013. "Making Mean-Variance Hedging Implementable in a Partially Observable Market," Papers 1306.3359, arXiv.org, revised Nov 2013.
    4. Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2012. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," CARF F-Series CARF-F-302, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Masaaki Fujii & Akihiko Takahashi, 2013. "Making Mean-Variance Hedging Implementable in a Partially Observable Market," CARF F-Series CARF-F-321, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.

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