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


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3


  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.


  1. Bender, C., & Denk, R. (2007). A forward scheme for backward SDEs. Stochastic Processes and their Applications, 117(12), 1793–1823.

  2. Bismut, J. M. (1973). Conjugate convex functions in optimal stochastic control. Journal of Political Economy, 3, 637–654.

    Google Scholar 

  3. Bielecki, T., & Rutkowski, M. (2002). Credit risk: Modeling, valuation and hedging. Berlin: Springer Finance.

    Google Scholar 

  4. Bielecki, T., Jeanblanc, M., & Rutkowski, M. (2009). Credit risk modeling. Suita: Osaka University Press.

    Google Scholar 

  5. Bouchard, B., & Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Processes and their Applications, 111(2), 175–206.

    Article  Google Scholar 

  6. Carmona, (Ed.). (2009). Indifference pricing. Princeton: Princeton University Press.

    Google Scholar 

  7. Crépey, S., & Bielecki, T with and introductory dialogue by Brigo (2014).Counterparty risk and funding. UK: Chapman & Hall/CRC.

  8. Crépey, S., & Song, S. (2014). Counterparty risk modeling: Beyond immersion, working paper, Evry University.

  9. Del Moral, P. (2004). Feynman-Kac formula: Genealogical and interacting particle systems with applications. Berlin: Springer.

    Google Scholar 

  10. Doucet, A., de Freitas, N., & Gordon, N. (2001). Sequential Monte Carlo methods in practice. New York: Springer.

    Google Scholar 

  11. El Karoui, N., Peng, S. G., & Quenez, M. C. (1997a). Backward stochastic differential equations in finance. Mathematical Finance, 7, 1–71.

    Article  Google Scholar 

  12. Fujii, M., & Takahashi, A. (2012). Analytical approximation for non-linear FBSDEs with perturbation scheme. International Journal of Theoretical and Applied Finance, 15, 125003. (24).

    Article  Google Scholar 

  13. Fujii, M., & Takahashi, A. (2012). Perturbative expansion of FBSDE in an incomplete market with stochastic volatility. Quarterly Journal of Finance, 2(3), 1250015.

    Article  Google Scholar 

  14. Fujii, M., Sato, S., & Takahashi A. (2014). An FBSDE approach to American option pricing with an interacting particle method. Asia-Pacific Financial Markets. doi:10.1007/s10690-014-9195-6.

  15. Fujita, H. (1966). On the blowing up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). Journal of the Faculty of Science, University of Tokyo, 13, 109–124.

    Google Scholar 

  16. Gobet, E., Lemor, J.-P., & Warin, X. (2005). A regression-based Monte Carlo method to solve backward stochastic differential equations. The Annals of Applied Probability, 15(3), 2172–2202.

    Article  Google Scholar 

  17. Gobet, E., & Labart, C. (2010). Solving BSDE with adaptive control variate. SIAM Journal on Numerical Analysis, 48, 257–277.

    Article  Google Scholar 

  18. Haramiishi, M. (2013a). Bank of Japan discussion paper, No. 2013-J-11, in Japanese.

  19. Haramiishi, M. (2013b). Bank of Japan discussion paper, No. 2013-J-12, in Japanese.

  20. Henry-Labordère, P. (2012). Cutting CVA’s complexity, Risk Magazine July Issue.

  21. Ikeda, N., Nagasawa, M., & Watanabe, S. (1965). Branching Markov processes. Abstracts Proceedings of the Japan Academy, 41, 816–821.

    Article  Google Scholar 

  22. Ikeda, N., Nagasawa, M., & Watanabe, S. (1966). Branching Markov processes. Abstracts Proceedings of the Japan Academy, 42, 252–257. 370–375, 380–384, 719–724, 1016–1021, 1022–1026.

    Article  Google Scholar 

  23. Ikeda, N., Nagasawa, M., & Watanabe, S. (1968). Branching Markov processes I(II). Journal of Mathematics of Kyoto University, 8, 233–278. 365–410.

    Google Scholar 

  24. Ikeda, N. et al. (1966, 1967). Seminar on probability, vol. 23 I-II and vol. 25 I-II (in Japanese).

  25. Kunitomo, N., & Takahashi, A. (2003). On validity of the asymptotic expansion approach in contingent claim analysis. Annals of Applied Probability, 13(3), 914–952.

    Article  Google Scholar 

  26. Ma, J., & Yong, J. (2000). Forward-backward stochastic differential equations and their applications. Berlin: Springer.

    Google Scholar 

  27. Ma, J., Protter, P., & Yong, J. (1994). Solving forward-backward stochastic differential equations explicitly. Probability & Related Fields, 98, 339–359.

    Article  Google Scholar 

  28. Nagasawa, M., & Sirao, T. (1969). Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation. Transactions of the American Mathematical Society, 139, 301–310.

    Article  Google Scholar 

  29. McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Communications on Pure and Applied Mathematics, 5, 323–331.

    Article  Google Scholar 

  30. Pardoux, E., & Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 14, 55–61.

    Article  Google Scholar 

  31. Shiraya, K., & Takahashi, A. (2014). Price impacts of imperfect collateralization, CARF working paper, CARF-F-355.

  32. Takahashi, A. (1999). An asymptotic expansion approach to pricing contingent claims. Asia-Pacific Financial Markets, 6, 115–151.

    Article  Google Scholar 

  33. Takahashi, A., Takehara, K., & Toda, M. (2012). A general computation scheme for a high-order asymptotic expansion method. International Journal of Theoretical and Applied Finance, 15–6, 1250044. (25).

    Article  Google Scholar 

  34. Takahashi, A., & Yamada, T. (2013). On an asymptotic expansion of forward-backward SDEs with a perturbation driver, CARF working paper, CARF-F-326.

  35. Takahashi, A., & Yoshida, N. (2004). An asymptotic expansion scheme for optimal investment problems. Statistical Inference for Stochastic Processes, 7(2), 153–188.

    Article  Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Akihiko Takahashi.

Additional information

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fujii, M., Takahashi, A. Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method. Asia-Pac Financ Markets 22, 283–304 (2015).

Download citation


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