Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

Details Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

2009-05-04

arxiv.org/abs/0905.0273v3

Mathematics

Numerical Analysis

Published at http://rspa.royalsocietypublishing.org/content/early/2010/12/08/rspa.2010.0348.full.html in the Proceedings of

Scientific paper

10.1098/rspa.2010.0348

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 finite-time convergence in the strong mean square sense remained an open question according to [Higham, Mao & Stuart (2002); Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40, no. 3, 1041-1063]. In this article we answer this question to the negative and prove for a large class of stochastic differential equations with non-globally 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.

Affiliated with

Also associated with

No associations

Rating

Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-198854