The asymptotic behavior of the density of the supremum of Lévy processes

Loïc Chaumont (1) , Jacek Malecki (2)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 IMCS - Institute of Mathematics and Computer Science [Wroclaw]

Let us consider a real Lévy process $X$ whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of $X$ before any deterministic time $t$ is absolutely continuous on $(0,\infty)$. We show that its density $f_t(x)$ is continuous on $(0,\infty)$ if and only if the potential density $h'$ of the upward ladder height process is continuous on $(0,\infty)$. Then we prove that $f_t$ behaves at 0 as $h'$. We also describe the asymptotic behaviour of $f_t$, when $t$ tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.

Mots clés: Density; past supremum; asymptotic behaviour; renewal function; conditioning to stay positive; meander.; meander

Domaines: Probabilités [math.PR]

Loïc Chaumont, Jacek Malecki. The asymptotic behavior of the density of the supremum of Lévy processes. 2013. ⟨hal-00870233⟩