Unbiased simulation method with the poisson kernel method for stochastic differential equations with reflection

  • Tomooki YuasaEmail author
Original Paper


We consider unbiased simulation methods for one-dimensional stochastic differential equations with reflection at zero. In particular, we propose improvements of the forward unbiased simulation method provided by Alfonsi et al. (Parametrix methods for one-dimensional reflected SDEs. Modern problems of stochastic analysis and statistics: selected contributions in honor of Valentin Konakov. Springer, pp 43–66, 2017). In this paper, we will apply the Poisson kernel method to improve the negativity and high variance problems of the associated simulation method. We also discuss some choices for the behavior of the approximation process near the boundary. This improvement is demonstrated through some numerical experiments.


Unbiased simulation method Stochastic differential equations Parametrix method 

Mathematics Subject Classification

Primary 60H35 



The author would like to thank Arturo Kohatsu-Higa for his helpful comments, and was supported by JSPS Grant 17J05514.


  1. 1.
    Alfonsi, A., Hayashi, M., Kohatsu-Higa, A.: Parametrix Methods for One-Dimensional Reflected SDEs. Modern Problems of Stochastic Analysis and Statistics: Selected Contributions In Honor of Valentin Konakov, pp. 43–66. Springer, Berlin (2017)CrossRefGoogle Scholar
  2. 2.
    Andersson, P., Kohatsu-Higa, A.: Unbiased simulation of stochastic differential equations using parametrix expansions. Bernoulli 23(3), 2028–2057 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andersson, P., Kohatsu-Higa, A., Yuasa, T.: Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations (preprint)Google Scholar
  4. 4.
    Bally, V., Kohatsu-Higa, A.: A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25(6), 3095–3138 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beskos, A., Papaspiliopoulos, O., Roberts, G.O., Fearnhead, P.: Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 68(3), 333–382 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, N., Huang, Z.: Brownian measures, importance sampling and unbiased simulation of diffusion extremes. Oper. Res. Lett. 40(6), 554–563 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Upper Saddle River (1964)zbMATHGoogle Scholar
  8. 8.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1998)CrossRefGoogle Scholar
  9. 9.
    Kang, W., Lee, J.M. : Unbiased estimators of the Greeks for general diffusion processes (preprint)Google Scholar
  10. 10.
    Kohatsu-Higa, A., Taguchi, D., Zhong, J.: The parametrix method for skew diffusions. Potential Anal. 45, 299–329 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lépingle, D.: Euler scheme for reflected stochastic differential equations. Math. Comput. Simul. 38(1–3), 119–126 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pilipenko, A.: An Introduction to Stochastic Differential Equations with Reflection. Potsdam University Press, Potsdam (2014)zbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRitsumeikan UniversityKusatsuJapan

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