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

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  • Denis Belomestny
  • Tigran Nagapetyan

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.

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  • Denis Belomestny & Tigran Nagapetyan, 2014. "Variance reduced multilevel path simulation: going beyond the complexity $\varepsilon^{-2}$," Papers 1412.4045, arXiv.org, revised Mar 2017.
    • Handle: RePEc:arx:papers:1412.4045
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    1. Michael B. Giles & Lukasz Szpruch, 2012. "Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation," Papers 1202.6283, arXiv.org, revised May 2014.
    2. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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