Weak error for nested Multilevel Monte Carlo

TL;DR

This paper investigates the weak error behavior of nested Multilevel Monte Carlo estimators, especially when the payoff function is non-smooth, providing new theoretical insights for their effective application.

Contribution

It introduces a novel result on the weak error order for non-smooth payoff functions, enhancing the understanding of MLMC estimator assumptions.

Findings

Weak error order exceeds 1 for non-smooth functions under certain conditions

Conditions for MLMC framework compliance depend on payoff smoothness

New theoretical bounds for nested MLMC estimators with non-smooth payoffs

Abstract

This article discusses MLMC estimators with and without weights, applied to nested expectations of the form E [f (E [F (Y, Z)|Y ])]. More precisely, we are interested on the assumptions needed to comply with the MLMC framework, depending on whether the payoff function f is smooth or not. A new result to our knowledge is given when f is not smooth in the development of the weak error at an order higher than 1, which is needed for a successful use of MLMC estimators with weights.