Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

Authors:
Xuerong Mao;Lukasz Szpruch
Affiliations:
Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK;Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2013

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Abstract

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.