Multilevel Monte Carlo Approximation of Distribution Functions and Densities

We construct and analyze multilevel Monte Carlo methods for the approximation of distribution functions and densities of univariate random variables. Since, by assumption, the target distribution is not known explicitly, approximations have to be used. We provide a general analysis under suitable assumptions on the weak and strong convergence. We apply the results to smooth path-independent and path-dependent functionals and to stopped exit times of stochastic differential equations (SDEs).