Persistence of integrated stable processes

TL;DR

This paper calculates the persistence exponent of the integral of a stable Lévy process using its self-similarity and positivity parameters, solving a problem posed by Shi and extending classical results for Brownian motion.

Contribution

It provides a new formula for the persistence exponent of integrated stable processes, linking it to the process's parameters and extending classical Brownian motion results.

Findings

Derived the persistence exponent in terms of process parameters

Extended classical formulas to stable processes

Analyzed the law of the process at the first hitting time

Abstract

We compute the persistence exponent of the integral of a stable L\'evy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Z. Shi (2003). Along the way, we investigate the law of the stable process L evaluated at the first time its integral X hits zero, when the bivariate process (X,L) starts from a coordinate axis. This extends classical formulae by McKean (1963) and Gor'kov (1975) for integrated Brownian motion.