[Paper Review] Divergence of the multilevel Monte Carlo method

Martin Hutzenthaler, Arnulf Jentzen+1 more|arXiv (Cornell University)|May 2, 2011

Stochastic processes and financial applications8 citations

TL;DR

This paper demonstrates that the multilevel Monte Carlo Euler method diverges for nonlinear SDEs with superlinearly growing, globally one-sided Lipschitz drift coefficients, even on events of probability one—contrary to classical Monte Carlo methods. It proposes replacing the standard Euler scheme with a tamed Euler method in the multilevel framework, which restores convergence while preserving the method's high efficiency.

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.

Motivation & Objective

• To investigate the convergence behavior of the multilevel Monte Carlo Euler method for nonlinear SDEs with superlinearly growing drift coefficients.
• To identify why the multilevel Monte Carlo method fails in such cases, despite classical Monte Carlo methods remaining convergent.
• To propose a modified numerical scheme that ensures convergence while retaining the multilevel method's computational advantages.
• To establish theoretical convergence for the multilevel Monte Carlo method when combined with a tamed Euler scheme under one-sided Lipschitz conditions.

Proposed method

• Analyzing the divergence of the multilevel Monte Carlo Euler method on a family of SDEs with superlinearly growing, globally one-sided Lipschitz drift coefficients.
• Demonstrating that divergence occurs on an event of probability one, not on rare events, unlike classical Monte Carlo methods.
• Introducing a tamed Euler method to stabilize the numerical approximation of the SDEs with nonlinear drift.
• Combining the tamed Euler method with the multilevel Monte Carlo framework to ensure convergence.
• Proving that the resulting method preserves the higher-order convergence rate observed in the Lipschitz case.

Experimental results

Research questions

• RQ1 Does the multilevel Monte Carlo Euler method converge for SDEs with superlinearly growing, globally one-sided Lipschitz drift coefficients?
• RQ2 Why does the multilevel Monte Carlo Euler method diverge in such cases, while classical Monte Carlo methods remain convergent?
• RQ3 Can the multilevel Monte Carlo method be salvaged for these nonlinear SDEs through a modification of the underlying numerical scheme?
• RQ4 Does the combination of the tamed Euler method with multilevel Monte Carlo retain the high convergence rate observed in the Lipschitz case?

Key findings

• The multilevel Monte Carlo Euler method diverges for SDEs with superlinearly growing, globally one-sided Lipschitz drift coefficients on an event of probability one.
• This divergence occurs because the numerical instability is not confined to rare events, unlike in classical Monte Carlo methods.
• The standard multilevel Monte Carlo Euler method is therefore not reliable for such nonlinear SDEs.
• Replacing the standard Euler scheme with a tamed Euler method restores convergence in the multilevel Monte Carlo framework.
• The resulting method maintains the high convergence rate characteristic of the multilevel Monte Carlo method in the Lipschitz case.