Strong and weak divergence in finite time of Euler's method for stochastic differential equations with nonglobally Lipschitz continuous coefficients
Abstract
The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz continuous drift and diffusion coefficient. Recent results extend this convergence to coefficients which grow at most linearly. For superlinearly growing coefficients finitetime convergence in the strong mean square sense remained an open question according to [Higham, Mao & Stuart (2002); Strong convergence of Eulertype methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40, no. 3, 10411063]. In this article we answer this question to the negative and prove for a large class of stochastic differential equations with nonglobally Lipschitz continuous coefficients that Euler's approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense.
 Publication:

arXiv eprints
 Pub Date:
 May 2009
 arXiv:
 arXiv:0905.0273
 Bibcode:
 2009arXiv0905.0273H
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 65C30
 EPrint:
 Published at http://rspa.royalsocietypublishing.org/content/early/2010/12/08/rspa.2010.0348.full.html in the Proceedings of the Royal Society A