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Weak Convergence of the Euler Scheme for Stochastic Differential Delay Equations

Published online by Cambridge University Press:  01 February 2010

Evelyn Buckwar ,
Rachel Kuske ,
Salah-Eldin Mohammed  and
Tony Shardlow
Evelyn Buckwar
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom, E.Buckwar@hw.ac.uk
Rachel Kuske
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada, rachel@math.ubc.ca
Salah-Eldin Mohammed
Affiliation:
Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL 62901, USA, salah@sfde.math.siu.edu
Tony Shardlow
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom, shardlow@maths.man.ac.uk

Abstract

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We study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 60 - 99
Copyright
Copyright © London Mathematical Society 2008

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