Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes

Abstract

In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating Lévy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.

Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function

“…Since the advent of Multilevel Monte Carlo (MLMC), introduced by Giles in [16] and subsequently developed in [2,6,7,15,17], Lévy area approximation has become less prominent in the literature. In particular, the antithetic MLMC method introduced by Giles and Szpruch in [17] achieves the optimal complexity for the weak approximation of multidimensional SDEs without the need to generate Brownian Lévy area.…”