Large deviations for clocks of self-similar processes

TL;DR

This paper studies the long-term behavior of clocks associated with positive self-similar processes via the Lamperti correspondence, extending previous results on their asymptotic properties.

Contribution

It provides new asymptotic descriptions of the exponential functionals and clocks of self-similar processes, broadening the understanding of their large deviations.

Findings

Extended Yor and Zani's results on clock asymptotics

Derived large deviation principles for exponential functionals

Characterized the asymptotic behavior of self-similar process clocks

Abstract

The Lamperti correspondence gives a prominent role to two random time changes: the exponential functional of a L\'evy process drifting to $\infty$ and its inverse, the clock of the corresponding positive self-similar process. We describe here asymptotical properties of these clocks in large time, extending the results of Yor and Zani.