Non-globally Lipschitz Counterexamples for the stochastic Euler scheme

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz coefficients and even with coefficients which grow at most linearly. For super-linearly growing coefficients convergence in the strong and numerically weak sense remained an open question. In this article we prove for many stochastic differential equations with super-linearly growing coefficients that Euler’s approximation does not converge neither in the strong L-sense nor in the numerically weak sense to the exact solution. Even worse, the difference of the exact solution and of the numerical approximation diverges to infinity in the strong L-sense and in the numerically weak sense.