On the Acceleration of the Multi-Level Monte Carlo Method

TL;DR

This paper introduces a modified multi-level Monte Carlo estimator that significantly reduces computational costs by leveraging different weak approximation orders, enhancing efficiency in stochastic process expectation calculations.

Contribution

It proposes a new modified estimator for the multi-level Monte Carlo method that reduces computational costs asymptotically based on approximation orders.

Findings

Reduces computational costs by a factor of $(p/\alpha)^2$

Applicable when costs grow with variance decay

Improves efficiency of stochastic expectation estimation

Abstract

The multi-level Monte Carlo method proposed by M. Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper, a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor $(p/\alpha)^2$ if weak approximation methods of orders $\alpha$ and $p$ are applied in case of computational costs growing with same order as variances decay.