Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
From MaRDI portal
Publication:373839
DOI10.1214/12-AAP890zbMATH Open1283.60098arXiv1105.0226OpenAlexW2137519698MaRDI QIDQ373839
Author name not available (Why is that?)
Publication date: 25 October 2013
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/1105.0226
No records found.
No records found.
This page was built for publication: Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q373839)
