Asia-Pacific Financial Markets

, Volume 22, Issue 3, pp 283–304 | Cite as

Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method

  • Masaaki Fujii
  • Akihiko TakahashiEmail author


In this paper, we propose an efficient Monte Carlo implementation of a non-linear FBSDE as a system of interacting particles inspired by the idea of the branching diffusion method of McKean. It will be particularly useful to investigate large and complex systems, and hence it is a good complement of our previous work presenting an analytical perturbation procedure for generic non-linear FBSDEs. There appear multiple species of particles, where the first one follows the diffusion of the original underlying state, and the others the Malliavin derivatives with a grading structure. The number of branching points are capped by the order of perturbation, which is expected to make the scheme less numerically intensive. The proposed method can be applied to semi-linear problems, such as American options, credit and funding value adjustments, and even fully non-linear issues, such as the optimal portfolio problems in incomplete and/or constrained markets.


BSDE FBSDE Asymptotic expansion Malliavin derivative Interacting particle method Branching diffusion 



The authors thank Seisho Sato of the Institute of Statistical Mathematics (ISM) for the helpful discussions about the branching diffusion method.


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Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.Graduate School of EconomicsThe University of TokyoTokyoJapan

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