The Annals of Applied Probability

Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation

Michael B. Giles and Lukasz Szpruch

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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.

Article information

Ann. Appl. Probab., Volume 24, Number 4 (2014), 1585-1620.

First available in Project Euclid: 14 May 2014

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Zentralblatt MATH identifier

Primary: 65C30: Stochastic differential and integral equations 65C05: Monte Carlo methods

Monte Carlo multilevel Lévy area stochastic differential equation


Giles, Michael B.; Szpruch, Lukasz. Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Probab. 24 (2014), no. 4, 1585--1620. doi:10.1214/13-AAP957.

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