Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

We introduce our method $\sigma$-antithetic multilevel Monte Carlo for multi-dimensional stochastic differential equations driven by Brownian motions with drifts. Here $\sigma$ is a specific permutation of order $m$, with $m\in\mathbb{N}\backslash\{0,1\}$. Following the spirit of Giles and Szpruch in [b], we consider the Milstein scheme without Lévy area. Our aim is to prove the stable convergence for the $\sigma$-antithetic multilevel error between two consecutive levels. To do so, we chose the triangular array approach using the limit theorem of Jacod [c]. After that, a central limit theorem of Lindeberg-Feller type for the multilevel Monte Carlo method associated with the new discretization scheme is shown. As in the case of Ben Alaya and Kebaier [a], we then give new optimal parameters leading to the convergence of the central limit theorem and a complexity of the multilevel Monte Carlo algorithm is carried out.

Joint work with Mohamed Ben-Alaya and Ahmed Kebaier.

[a] BEN ALAYA, M. and KEBAIER, A. Central limit theorem for multilevel Monte Carlo Euler method, the Annals of Applied Probability, 2015, Vol.25, No.1, 211-234.
[b] GILES, M.B and SZPRUCH, L.  Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation}. The Annals of Applied Probability, 2014, Vol.24, No.4, 1585-1620.
[c] JACOD.J. On continuous conditional gaussian martingales  and stable convergence in law,  Séminaire de probabilités (Strasbourg) tome 31 (1997), p. 232-246.