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

Abstract

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.

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Notes

  1. 1.

    It is possible to extract the linear term from the driver and treat separately. Here, we simply leave it in a driver, or work in a “discounted” base to remove linear term in \(V\).

  2. 2.

    We intend to use the result of asymptotic expansion only for higher order approximations.

  3. 3.

    It is not difficult to make it a stochastic process.

  4. 4.

    Although this can be done somewhat arbitrarily, it may be natural to set \(r(t,x)\) and \(\sigma (t,x)\) as the expected dynamics of \(X\) when all the feedback effects are switched off.

  5. 5.

    As before, this is only to use higher order expansion. For the valuation of the zero-th order itself, one can use the standard Monte Carlo simulation.

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Acknowledgments

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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Correspondence to Akihiko Takahashi.

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This research is supported by CARF (Center for Advanced Research in Finance) and the global COE program “The research and training center for new development in mathematics.” All the contents expressed in this research are solely those of the authors and do not represent any views or opinions of any institutions. The authors are not responsible or liable in any manner for any losses and/or damages caused by the use of any contents in this research.

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Fujii, M., Takahashi, A. Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method. Asia-Pac Financ Markets 22, 283–304 (2015). https://doi.org/10.1007/s10690-015-9201-7

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Keywords

  • BSDE
  • FBSDE
  • Asymptotic expansion
  • Malliavin derivative
  • Interacting particle method
  • Branching diffusion