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ESAIM: P&S, 2008, Vol. 12, p. 1-11
DOI: 10.1051/ps:2007030

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

Abdel Berkaoui1, Mireille Bossy2 and Awa Diop2

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 July 20, 2005. Revised May 30 and October 8, 2006. Published online 13 November 2007

Abstract
We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form $\vert x\vert^\alpha $, $\alpha \in [1/2,1)$. 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.


Mathematics Subject Classification. 65C30, 60H35, 65C20

Key words: Discretization scheme, strong convergence, CIR process.


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