Statistical Skorohod embedding problem and its generalizations

TL;DR

This paper addresses the statistical Skorohod embedding problem for Lévy processes, proposing a Mellin-Laplace transform-based estimator for the distribution of an independent random time, with proven convergence rates and applications to mixture models.

Contribution

It introduces a novel estimator for the distribution of a random time in Lévy processes using Mellin and Laplace transforms, with proven optimal convergence rates.

Findings

Proposed a consistent estimator for the distribution of T.

Derived convergence rates depending on the Mellin transform decay.

Applied results to variance-mean mixture models and time-changed Lévy processes.

Abstract

Given a L\'evy 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 $L_{T}.$ Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose a consistent estimator for the density of $T,$ derive its 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 L\'evy processes.