  This paper focuses on the study of an original combination of the Multilevel
Monte Carlo method introduced by Giles [10] and the popular importance sampling
technique. To compute the optimal choice of the parameter involved in the
importance sampling method, we rely on Robbins-Monro type stochastic
algorithms. On the one hand, we extend our previous work [2] to the Multilevel
Monte Carlo setting. On the other hand, we improve [2] by providing a new
adaptive algorithm avoiding the discretization of any additional process.
Furthermore, from a technical point of view, the use of the same stochastic
algorithms as in [2] appears to be problematic. To overcome this issue, we
employ an alternative version of stochastic algorithms with projection (see
e.g. Laruelle, Lehalle and Pag\`es [20]). In this setting, we show innovative
limit theorems for a doubly indexed stochastic algorithm which appear to be
crucial to study the asymptotic behavior of the new adaptive Multilevel Monte
Carlo estimator. Finally, we illustrate the efficiency of our method through
applications from quantitative finance.
