  This paper focuses on studying the multilevel Monte Carlo method recently
introduced by Giles [Oper. Res. 56 (2008) 607-617] which is significantly more
efficient than the classical Monte Carlo one. Our aim is to prove a central
limit theorem of Lindeberg-Feller type for the multilevel Monte Carlo method
associated with the Euler discretization scheme. To do so, we prove first a
stable law convergence theorem, in the spirit of Jacod and Protter [Ann.
Probab. 26 (1998) 267-307], for the Euler scheme error on two consecutive
levels of the algorithm. This leads to an accurate description of the optimal
choice of parameters and to an explicit characterization of the limiting
variance in the central limit theorem of the algorithm. A complexity of the
multilevel Monte Carlo algorithm is carried out.
