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$ V$-integrability, asymptotic stability and comparison property of explicit numerical schemes for non-linear SDEs


Authors: Łukasz Szpruch and Xīlíng Zhāng
Journal: Math. Comp.
MSC (2010): Primary 65C30, 65C05
DOI: https://doi.org/10.1090/mcom/3219
Published electronically: August 3, 2017
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Abstract: Khasminski [Stochastic Stability of Differential Equations, Kluwer Academic Publishers, 1980] showed that the asymptotic stability and the integrability of solutions to stochastic differential equations (SDEs) can be obtained via Lyapunov functions. These properties are, however, not necessarily inherited by standard numerical approximations. In this article we introduce a general class of explicit numerical approximations that are amenable to Khasminski's techniques and are particularly suited for non-globally Lipschitz coefficients. We derive general conditions under which these numerical schemes are bounded in expectation with respect to certain Lyapunov functions, and/or inherit the asymptotic stability of the SDEs. Finally we show that by truncating the noise it is possible to recover the comparison theorem for numerical approximations of non-linear scalar SDEs.


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Additional Information

Łukasz Szpruch
Affiliation: School of Mathematics, The University of Edinburgh, EH9 3FD, Edinburgh, United Kingdom
Email: l.szpruch@ed.ac.uk

Xīlíng Zhāng
Affiliation: School of Mathematics, The University of Edinburgh, EH9 3FD, Edinburgh, United Kingdom
Email: xiling.zhang@ed.ac.uk

DOI: https://doi.org/10.1090/mcom/3219
Keywords: Stochastic differential equations, Lyapunov functions, Euler scheme, integrability, stability, comparison theorem
Received by editor(s): February 12, 2015
Received by editor(s) in revised form: January 4, 2016, May 23, 2016, and October 17, 2016
Published electronically: August 3, 2017
Article copyright: © Copyright 2017 American Mathematical Society