Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence

¹ Dept of Statistics, University of Warwick, Gibbet Hill road, Coventry CV4 7AL, UK; berkaoui@stats.warwick.ac.uk
² OMEGA project, INRIA Sophia Antipolis, 2004 route des Lucioles, B.P. 93, 06902 Sophia-Antipolis Cedex, France; mireille.bossy@sophia.inria.fr; ADiop@bbspinc.com

Received: 20 July 2005
Revised: 30 May 2006
Revised: 8 October 2006

Abstract

We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form |x|^α, α ∈ [1/2,1). In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.