Variance reduced multilevel path simulation: going beyond the complexity $\varepsilon^{-2}$

TL;DR

This paper introduces a modified multilevel Monte Carlo method that employs control variates to significantly reduce computational complexity, achieving an order of $ ext{O}( ext{epsilon}^{-2+ ext{delta}})$ for any $ ext{delta} ext{ in } [0,1)$, with demonstrated numerical benefits.

Contribution

The paper proposes a novel variance reduction technique for multilevel Monte Carlo, improving complexity bounds beyond the traditional $ ext{epsilon}^{-2}$ rate.

Findings

Complexity order reduced to $ ext{epsilon}^{-2+ ext{delta}}$ with proper control variates.

Numerical examples confirm theoretical complexity improvements.

Method outperforms standard MLMC in various scenarios.

Abstract

In this paper a novel modification of the multilevel Monte Carlo approach, allowing for further significant complexity reduction, is proposed. The idea of the modification is to use the method of control variates to reduce variance at level zero. We show that, under a proper choice of control variates, one can reduce the complexity order of the modified MLMC algorithm down to $\varepsilon^{-2+\delta}$ for any $\delta\in [0,1)$ with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.