Time-changed spectrally positive L\'evy processes starting from infinity

TL;DR

This paper studies the conditions under which a time-changed spectrally positive Lévy process can start from infinity and describes its behavior as it comes down from infinity, including specific cases with negative drift and regular variation.

Contribution

It provides a necessary and sufficient condition for infinity to be an entrance boundary and characterizes the process's speed of coming down from infinity under various conditions.

Findings

Infinity can be an entrance boundary under specific conditions.

The process's speed of coming down from infinity is characterized by a deterministic function.

A renormalization law for the process's infimum at small times is established.

Abstract

Consider a spectrally positive L\'evy process $Z$ with log-Laplace exponent $\Psi$ and a positive continuous function $R$ on $(0,\infty)$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $\eta(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process $X$. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the L\'evy process has a negative drift $\delta:=-\gamma<0$, sufficient conditions over $R$ and $\Psi$ are found for the process to come down from infinity along the deterministic function $(x_t,t\geq 0)$ solution to ${\rm d} x_t=-\gamma R(x_t) {\rm d} t$, with $x_0=\infty$. When $\Psi(\lambda)\sim c\lambda^{\alpha}$, with $\lambda \rightarrow 0$, $\alpha\in (1,2]$, $c>0$ and $R$ is regularly varying at $\infty$ with index $\theta>\alpha$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.