Multilevel Monte Carlo for stochastic differential equations with additive fractional noise

Abstract

We adopt the multilevel Monte Carlo method introduced by M. Giles (Multilevel Monte Carlo path simulation, Oper. Res. 56(3):607-617, 2008) to SDEs with additive fractional noise of Hurst parameter H > 1/2. For the approximation of a Lipschitz functional of the terminal state of the SDE we construct a multilevel estimator based on the Euler scheme. This estimator achieves a prescribed root mean square error of order ε with a computational effort of order ε −2 .

Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

“…It was first introduced by Heinrich [26] for the computation of high-dimensional, parameter-dependent integrals, and has since been applied in many areas of mathematics related to differential equations. In particular, a lot of research has been done in stochastic differential equations [10,15,16,27,29] and several types of partial differential equations (PDEs) with random coefficients [2,6,8,17,19,22].…”

On the Acceleration of the Multi-Level Monte Carlo Method

“…Here, we want to point out that there also exist higher order weak approximation schemes, e. g. p = 3 in case of SDEs with additive noise [2], that may further improve the benefit of the modified multi-level Monte Carlo estimator. Future research will consider the application of this approach to, e.g., more general SDEs like SDEs driven by Lévy processes [3] or fractional Brownian motion [11] and to the numerical solution of SPDEs [13]. Further, the focus will be on numerical schemes that feature not only high orders of convergence but also minimized constants for the variance estimates.…”

“…The multi-level Monte Carlo method proposed in [7] approximates the expectation of some functional applied to some stochastic processes like e. g. solutions of stochastic differential equations (SDEs) at a lower computational complexity than classical Monte Carlo simulation, see also [5,8,9]. Multi-level Monte Carlo approximation is applied in many fields like mathematical finance [1,6], for SDEs driven by a Lévy process [3], by fractional Brownian motion [11] or for stochastic PDEs [13]. The main idea of this article is to reduce the computational costs additionally by applying the multi-level Monte Carlo method as a variance reduction technique for some higher order weak approximation method.…”

Improving distribution system reliability calculation efficiency using multilevel Monte Carlo method

“…The idea was later extended by Giles to reduce the computational burden for estimating an expected value arising from stochastic differential equations (SDE) in mathematical finance. Since then, it has been applied in numerous areas of solving SDE and stochastic partial differential equations with random variables. From these studies, the computational cost effectiveness of MLMC over MC method has been proved.…”