  The Euler-Maruyama scheme is known to diverge strongly and numerically weakly
when applied to nonlinear stochastic differential equations (SDEs) with
superlinearly growing and globally one-sided Lipschitz continuous drift
coefficients. Classical Monte Carlo simulations do, however, not suffer from
this divergence behavior of Euler's method because this divergence behavior
happens on rare events. Indeed, for such nonlinear SDEs the classical Monte
Carlo Euler method has been shown to converge by exploiting that the Euler
approximations diverge only on events whose probabilities decay to zero very
rapidly. Significantly more efficient than the classical Monte Carlo Euler
method is the recently introduced multilevel Monte Carlo Euler method. The main
observation of this article is that this multilevel Monte Carlo Euler method
does - in contrast to classical Monte Carlo methods - not converge in general
in the case of such nonlinear SDEs. More precisely, we establish divergence of
the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly
growing and globally one-sided Lipschitz continuous drift coefficients. In
particular, the multilevel Monte Carlo Euler method diverges for these
nonlinear SDEs on an event that is not at all rare but has probability one. As
a consequence for applications, we recommend not to use the multilevel Monte
Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead
we propose to combine the multilevel Monte Carlo method with a slightly
modified Euler method. More precisely, we show that the multilevel Monte Carlo
method combined with a tamed Euler method converges for nonlinear SDEs with
globally one-sided Lipschitz continuous drift coefficients and preserves its
strikingly higher order convergence rate from the Lipschitz case.
