Discretization of the Lamperti representation of a positive self-similar Markov process

TL;DR

This paper develops discretization methods for the Lévy process in the Lamperti representation of positive self-similar Markov processes, establishing limit theorems and applications to simulation.

Contribution

It introduces new discretization techniques for the Lévy process in the Lamperti representation and analyzes their convergence and applications.

Findings

Limit theorems for discretized Lévy process approximations

Scaling limit of self-similar Markov processes at small times

Application to simulating positive self-similar Lévy processes

Abstract

This paper considers discretization of the L\'evy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given L\'evy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar L\'evy processes conditioned to stay positive.