Statistical Skorohod embedding problem and its generalizations

Given a Lévy process L, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time T based on i.i.d. sample from LT . Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose consistent estimators for the density of T, derive their convergence rates and prove their optimality. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of T. We also consider the application of our results to the problem of statistical inference for variance-mean mixture models and for time-changed Lévy processes.