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A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise

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Abstract

In this paper a family of fully implicit Milstein methods are introduced for solving stiff stochastic differential equations (SDEs). It is proved that the methods are convergent with strong order 1.0 for a class of SDEs. For a linear scalar test equation with multiplicative noise terms, mean-square and almost sure asymptotic stability of the methods are also investigated. We combine analytical and numerical techniques to get insights into the stability properties. The fully implicit methods are shown to be superior to those of the corresponding semi-implicit methods in term of stability property. Finally, numerical results are reported to illustrate the convergence and stability results.

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Acknowledgements

This work was supported by NSF of China (No. 11171352), Hunan Provincial Innovation Foundation For Postgraduate (No. CX2010B118), Singapore AcRF RG 59/08 (M52110092) and Singapore NRF 2007 IDM-IDM002-010. The first author would like to express his deep gratitude to Prof. P.E. Kloeden and Dr. A. Jentzen for their kind help and useful discussions during his stay in Goethe University of Frankfurt am Main.

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Authors and Affiliations

  1. School of Mathematical Sciences and Computing Technology, Central South University, Changsha, 410075, Hunan, China

    Xiaojie Wang & Siqing Gan

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Desheng Wang

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  1. Xiaojie Wang
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  2. Siqing Gan
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  3. Desheng Wang
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Correspondence to Siqing Gan.

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Communicated by Anders Szepessy.

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Wang, X., Gan, S. & Wang, D. A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise. Bit Numer Math 52, 741–772 (2012). https://doi.org/10.1007/s10543-012-0370-8

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  • DOI: https://doi.org/10.1007/s10543-012-0370-8

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