Limit theorems for weighted and regular Multilevel estimators

TL;DR

This paper analyzes the almost sure convergence and weak rates of multilevel Monte Carlo estimators, including weighted variants, providing theoretical foundations and applications to diffusion discretizations and nested Monte Carlo.

Contribution

It establishes strong law and central limit theorem results for multilevel estimators, extending their theoretical understanding and applicability.

Findings

Proves strong law of large numbers for MLMC and ML2R estimators.

Derives central limit theorems for these estimators.

Applies results to diffusion schemes and nested Monte Carlo.

Abstract

We aim at analyzing in terms of a.s. convergence and weak rate the performances of the Multilevel Monte Carlo estimator (MLMC) introduced in [Gil08] and of its weighted version, the Multilevel Richardson Romberg estimator (ML2R), introduced in [LP14]. These two estimators permit to compute a very accurate approximation of $I_0 = \mathbb{E}[Y_0]$ by a Monte Carlo type estimator when the (non-degenerate) random variable $Y_0 \in L^2(\mathbb{P})$ cannot be simulated (exactly) at a reasonable computational cost whereas a family of simulatable approximations $(Y_h)_{h \in \mathcal{H}}$ is available. We will carry out these investigations in an abstract framework before applying our results, mainly a Strong Law of Large Numbers and a Central Limit Theorem, to some typical fields of applications: discretization schemes of diffusions and nested Monte Carlo.