Central Limit Theorem for the $\sigma$-antithetic multilevel Monte Carlo method

Understand

In this paper, we introduce the $\sigma$-antithetic multilevel Monte Carlo (MLMC) estimator for a multi-dimensional diffusion which is an extended version of the original antithetic MLMC one introduced by Giles and Szpruch \cite{a}.

• Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm.
• Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without L\'evy area and its $\sigma$-antithetic version build on the finer grid and on the other hand the coarse approximation stably converges in distribution with a rate of order 1.
• We also prove that the error between the Milstein scheme without L\'evy area and its $\sigma$-antithetic version stably converges in distribution with a rate of order $1/2$.