On an Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver
Abstractã€€ã€€ This paper presents a mathematical validity for an asymptotic expansion scheme of the solutions to the forwardbackward 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 and Takahashi (2012a), which has been successfully employed to solve the derivatives and optimal portfolio problems in Fujii and 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.
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Bibliographic InfoPaper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-902.
Length: 17 pages
Date of creation: Sep 2013
Date of revision:
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This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-09-24 (All new papers)
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- Masaaki Fujii & Akihiko Takahashi, 2012. "ANALYTICAL APPROXIMATION FOR NON-LINEAR FBSDEs WITH PERTURBATION SCHEME," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(05), pages 1250034-1-1.
- Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Papers 1204.2638, arXiv.org, revised Apr 2012.
- Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2012. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," Papers 1211.5867, arXiv.org.
- Kenichiro Shiraya & Akihiko Takahashi, 2013. "Pricing Basket Options under Local Stochastic Volatility with Jumps," CARF F-Series CARF-F-336, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised May 2014.
- Kenichiro Shiraya & Akihiko Takahashi, 2014. "Pricing Basket Options under Local Stochastic Volatility with Jumps," CIRJE F-Series CIRJE-F-913, CIRJE, Faculty of Economics, University of Tokyo.
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