Malliavin-based Multilevel Monte Carlo Estimators for Densities of Max-stable Processes

We introduce a class of unbiased Monte Carlo estimators for the multivariate density of max-stable fields generated by Gaussian processes. Our estimators take advantage of recent results on exact simulation of max-stable fields combined with identities studied in the Malliavin calculus literature and ideas developed in the multilevel Monte Carlo literature. Our approach allows estimating multivariate densities of max-stable fields with precision $\varepsilon$ at a computational cost of order $O\left( \varepsilon ^{-2}\log \log \log \left( 1/\varepsilon \right) \right)$.