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Strong Convergence For Euler–Maruyama And Milstein Schemes With Asymptotic Method

Author

Listed:
  • HIDEYUKI TANAKA

    (Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan)

  • TOSHIHIRO YAMADA

    (Graduate School of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan;
    Mitsubishi UFJ Trust Investment Technology, Institute Co., Ltd., 2-6, Akasaka 4-Chome, Minato, Tokyo 107-0052, Japan)

Abstract

Motivated by weak convergence results in the paper of Takahashi & Yoshida (2005), we show strong convergence for an accelerated Euler–Maruyama scheme applied to perturbed stochastic differential equations. The Milstein scheme with the same acceleration is also discussed as an extended result. The theoretical results can be applied to analyze the multi-level Monte Carlo method originally developed by M.B. Giles. Several numerical experiments for the stochastic alpha-beta-rho (SABR) model of stochastic volatility are presented in order to confirm the efficiency of the schemes.

Suggested Citation

  • Hideyuki Tanaka & Toshihiro Yamada, 2014. "Strong Convergence For Euler–Maruyama And Milstein Schemes With Asymptotic Method," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(02), pages 1-22.
    • Handle: RePEc:wsi:ijtafx:v:17:y:2014:i:02:n:s0219024914500149
    DOI: 10.1142/S0219024914500149
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    Cited by:

    1. Akihiko Takahashi & Toshihiro Yamada, 2015. "A weak approximation with asymptotic expansion and multidimensional Malliavin weights (Revised version of CARF-F-335; Forthcoming in Annals of Applied Probability")"," CARF F-Series CARF-F-358, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Apr 2016.

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