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August 2014 Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation
Michael B. Giles, Lukasz Szpruch
Ann. Appl. Probab. 24(4): 1585-1620 (August 2014). DOI: 10.1214/13-AAP957

Abstract

In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(\Delta t)$ with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of $\epsilon$ from $O(\epsilon^{-3})$ to $O(\epsilon^{-2})$. However, in general, to obtain a rate of strong convergence higher than $O(\Delta t^{1/2})$ requires simulation, or approximation, of Lévy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of Lévy areas and still achieve an $O(\Delta t^{2})$ multilevel correction variance for smooth payoffs, and almost an $O(\Delta t^{3/2})$ variance for piecewise smooth payoffs, even though there is only $O(\Delta t^{1/2})$ strong convergence. This results in an $O(\epsilon^{-2})$ complexity for estimating the value of European and Asian put and call options.

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Michael B. Giles. Lukasz Szpruch. "Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation." Ann. Appl. Probab. 24 (4) 1585 - 1620, August 2014. https://doi.org/10.1214/13-AAP957

Information

Published: August 2014
First available in Project Euclid: 14 May 2014

zbMATH: 1373.65007
MathSciNet: MR3211005
Digital Object Identifier: 10.1214/13-AAP957

Subjects:
Primary: 65C05, 65C30

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 4 • August 2014
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