Prove general variance-reduction rates for multiple-ensemble MLMC

Prove that the variance-reduction rates required by the multiple-ensemble multilevel Monte Carlo methods cited for McKean–Vlasov equations hold in general.

Background

The paper surveys multiple-ensemble multilevel Monte Carlo algorithms for McKean–Vlasov equations, which simulate separate ensembles at different levels and apply the telescoping correction to a quantity of interest over the law of the solution. Several such methods have obtained favorable complexity bounds, but those bounds depend on variance-reduction rates for coupled estimators.

The authors explicitly state that the rates needed for some of these complexity results have not yet been established in general. Establishing them would provide a rigorous foundation for the claimed performance of these multiple-ensemble methods beyond the particular settings in which the rates are currently known.

References

For instance, the results in require variance reduction rates that, to our knowledge, have not yet been proven to hold~in~\mbox{general}.

— Convergence of a single-ensemble multilevel scheme for McKean-Vlasov SDEs  (2609.10000 - Bouillon et al., 9 Sep 2026) in Section 1, subsection “Multilevel Monte Carlo”