On an Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver

Akihiko Takahashi and Toshihiro Yamada
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Akihiko Takahashi: The University of Tokyo
Toshihiro Yamada: The University of Tokyo

No CARF-F-326, CARF F-Series from Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo

Abstract: This paper presents a mathematical validity for an asymptotic expansion scheme of the solutions to the forward-backward stochastic differential equations (FBSDEs) in terms of a perturbed driver in the BSDE and a small diffusion in the FSDE. This computational scheme was proposed by Fujii-Takahashi (2012a), which has been successfully employed to solve the derivatives and optimal portfolio problems in Fujii-Takahashi (2012b,c) and Fujii et al. (2012). In particular, we represent the coefficients up to an arbitrary order expansion of the BSDE by the solution to a system of the associated BSDEs with the FSDE, and obtain the error estimate of the expansion with respect to the driver perturbation. Accordingly, we show a concrete representation for each expansion coefficient of the volatility component, that is the martingale integrand in the BSDE. Then, we apply our proposed FSDE expansion formula with its precise error estimate to the BSDE expansion coefficients to finally obtain the total residual estimate.

Pages: 17 pages
Date: 2013-09, Revised 2013-10
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Citations: View citations in EconPapers (11)

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