Establish theoretical analysis for MLQMC combined with preintegration

Establish a rigorous theoretical analysis for multilevel quasi-Monte Carlo (MLQMC) combined with preintegration in nested risk estimation, including convergence rates, error bounds for level differences, and overall computational complexity for estimating probabilities of the form θ = E[ I{ E[X|ω] > c } ].

Background

The authors extend their MLMC-with-preintegration method by employing quasi-Monte Carlo (QMC) within the inner simulations and report improved empirical strong convergence—often exceeding that of standard MLMC—especially in high dimensions.

Despite these empirical findings, they explicitly note that a formal theoretical framework for MLQMC combined with preintegration has not yet been developed, identifying this as an important direction for further research.