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

TL;DR

This paper demonstrates that Euler's method fails to converge in finite time for stochastic differential equations with non-globally Lipschitz coefficients, with the approximation diverging in mean square and weak senses.

Contribution

It provides a negative answer to the open question about finite-time convergence of Euler's method for superlinearly growing coefficients, showing divergence instead.

Findings

Euler's method does not converge in finite time for certain SDEs.

The difference between exact and numerical solutions diverges to infinity.

Convergence fails in both strong mean square and weak senses.

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 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.