ANTITHETIC MULTILEVEL MONTE CARLO ESTIMATION FOR MULTI-DIMENSIONAL SDES WITHOUT LEVY AREA SIMULATION

作者：Giles Michael B^*; Szpruch Lukasz

来源：Annals of Applied Probability, 2014, 24(4): 1585-1620.

DOI：10.1214/13-AAP957

• Monte Carlo
• multilevel
• Levy area
• stochastic differential equation

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 Levy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of Levy 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.

• 出版日期2014-8