Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

TL;DR

This paper demonstrates that the multilevel Monte Carlo Euler method diverges for certain nonlinear SDEs with superlinear growth, and proposes a modified approach combining it with a tamed Euler method to ensure convergence.

Contribution

The paper proves divergence of the multilevel Monte Carlo Euler method for specific nonlinear SDEs and introduces a combined method with a tamed Euler scheme that guarantees convergence.

Findings

Multilevel Monte Carlo Euler diverges for some nonlinear SDEs with superlinear growth.

Classical Monte Carlo Euler converges despite divergence on rare events.

Combining multilevel Monte Carlo with a tamed Euler method achieves convergence for challenging nonlinear SDEs.

Abstract

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.