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

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.