Analytical Approximation for Non-Linear FBSDEs with Perturbation Scheme

35 Pages Posted: 2 Jun 2011 Last revised: 23 Jan 2012

See all articles by Masaaki Fujii

Masaaki Fujii

University of Tokyo - Faculty of Economics

Akihiko Takahashi

University of Tokyo - Faculty of Economics

Date Written: January 20, 2012

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.

Keywords: BSDE, FBSDE, Four Step Scheme, Asymptotic Expansion, Malliavin Derivative, Non-linear PDE, CVA

JEL Classification: C60, G13

Suggested Citation

Fujii, Masaaki and Takahashi, Akihiko, Analytical Approximation for Non-Linear FBSDEs with Perturbation Scheme (January 20, 2012). Available at SSRN: https://ssrn.com/abstract=1856444 or http://dx.doi.org/10.2139/ssrn.1856444

Masaaki Fujii (Contact Author)

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033
Japan

Akihiko Takahashi

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033
Japan

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