Density behaviour related to L\'evy processes

TL;DR

This paper investigates the asymptotic behaviors of densities related to Lévy processes, their supremum, and excursion laws, establishing relationships and conditions under which these densities are comparable to the Lévy measure's density.

Contribution

It provides new relationships between the asymptotic behaviors of process densities and the Lévy measure density, under mild conditions, for small times and large positions.

Findings

Asymptotic behaviors of $p_t(x)$, $f_t(x)$, and $q_t^*(x)$ are related for small $t$ and large $x.

Under mild conditions, if $p_t(x)$ is comparable to $t u(x)$, then $f_t(x)$ is also comparable to $t u(x)$.

The asymptotic behaviors of these densities are compared to the Lévy measure density for large $x$.

Abstract

Let $p_t(x)$, $f_t(x)$ and $q_t^*(x)$ be the densities at time $t$ of a real L\'evy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of $p_t(x)$, $f_t(x)$ and $q_t^*(x)$, when $t$ is small and $x$ is large. Then for large $x$, these asymptotic behaviours are compared to this of the density of the L\'evy measure. We show in particular that, under mild conditions, if $p_t(x)$ is comparable to $t\nu(x)$, as $t\rightarrow0$ and $x\rightarrow\infty$, then so is $f_t(x)$.