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Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes

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Abstract

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg–Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (SIAM J Numer Anal 40(3):1041–1063, 2002) and a newly proposed explicit variant of the Euler–Maruyama scheme, the so called projected Euler–Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.

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Acknowledgments

The authors wish to thank R. D. Grigorieff for calling our attention to the reference [22] and thereby pointing us to the concept of C-stability. Further, the first two authors gratefully acknowledge financial support by the DFG-funded CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’. The same holds true for the third named author, who has been supported by the research center Matheon. He also likes to thank the CRC 701 for making possible a very fruitful research stay at Bielefeld University, during which essential parts of this work were written. Finally, the authors wish to thank an anonymous referee and the associated editor for very helpful suggestions and comments.

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Authors and Affiliations

  1. Universität Bielefeld, Fakultät für Mathematik, Postfach 100 131, 33501, Bielefeld, Germany

    Wolf-Jürgen Beyn & Elena Isaak

  2. Technische Universität Berlin, Institut für Mathematik, Secr. MA 5-3, Straße des 17. Juni 136, 10623, Berlin, Germany

    Raphael Kruse

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  1. Wolf-Jürgen Beyn
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  2. Elena Isaak
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  3. Raphael Kruse
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Correspondence to Raphael Kruse.

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Beyn, WJ., Isaak, E. & Kruse, R. Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes. J Sci Comput 67, 955–987 (2016). https://doi.org/10.1007/s10915-015-0114-4

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