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

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Abstract

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in Beyn et al. (J Sci Comput 67(3):955–987, 2016. doi:10.1007/s10915-015-0114-4). As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler–Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.

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Acknowledgments

The authors would like to thank Martin Steinborn for bringing several typos to our attention. In addition, the first two authors are grateful for financial support by the DFG-funded CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’. Further, this research was carried out by the third named author in the framework of Matheon, Project A25, supported by Einstein Foundation Berlin.

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Authors and Affiliations

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

    Wolf-Jürgen Beyn & Elena Isaak

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

    Raphael Kruse

Authors
  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 Milstein-Type Schemes. J Sci Comput 70, 1042–1077 (2017). https://doi.org/10.1007/s10915-016-0290-x

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