Limit theorems for Multilevel estimators with and without weights. Comparisons and applications.

Summary In this work, we are interested in Multilevel Monte Carlo estimators. These estimators will appear in their standard form, with weights and in a randomized form. We will recall their definitions and the existing results concerning these estimators in terms of simulation cost minimization. We will then show a strong law of large numbers and a central limit theorem. After that we will study two application frameworks. The first one is that of diffusions with antithetic discretization schemes, where we will extend the Multilevel estimators to Multilevel estimators with weights. The second is the nested framework, where we will focus on strong and weak error assumptions. We will conclude with the implementation of the randomized form of Multilevel estimators, comparing it to Multilevel estimators with and without weights.