Asymptotic behaviour of first passage time distributions for subordinators

TL;DR

This paper investigates the asymptotic behavior of first passage times for subordinators within the Feller class, providing refined local estimates and extending results to stable distribution domains of attraction.

Contribution

It establishes new local estimates for first passage times of subordinators, especially in stable distribution domains, improving upon previous results and adapting methods for monotone processes.

Findings

Refined local estimates for first passage times.

Extension of results to stable distribution domains.

Exponential tail behavior of first passage times in subordinators.

Abstract

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt in 1987. In the particular case of a subordinator in the domain of attraction of a stable distribution the results are the analogue of the results obtained by the authors for non-monotone L\'evy processes. For subordinators an approach different to that used for non-monotone L\'evy processes is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.