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

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Abstract : 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.
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Cited literature [18 references]

https://hal.archives-ouvertes.fr/hal-00870233
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Submitted on : Sunday, October 6, 2013 - 4:05:18 PM
Last modification on : Wednesday, October 20, 2021 - 3:18:52 AM
Long-term archiving on: : Tuesday, January 7, 2014 - 4:26:31 AM

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• HAL Id : hal-00870233, version 1

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

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