  In this work, we consider the problem of estimating the probability
distribution, the quantile or the conditional expectation above the quantile,
the so called conditional-value-at-risk, of output quantities of complex random
differential models by the MLMC method. We follow the approach of (reference),
which recasts the estimation of the above quantities to the computation of
suitable parametric expectations. In this work, we present novel computable
error estimators for the estimation of such quantities, which are then used to
optimally tune the MLMC hierarchy in a continuation type adaptive algorithm. We
demonstrate the efficiency and robustness of our adaptive continuation-MLMC in
an array of numerical test cases.
