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We introduce a class of unbiased Monte Carlo estimators for the multivariatedensity of max-stable fields generated by Gaussian processes. Our estimatorstake advantage of recent results on exact simulation of max-stable fieldscombined with identities studied in the Malliavin calculus literature and ideasdeveloped in the multilevel Monte Carlo literature. Our approach allowsestimating 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) $.

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