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

Loïc Chaumont; Jacek Malecki

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arXiv · 1310.1587

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

Loïc Chaumont·Jacek Malecki

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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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BibTeX RIS

Loïc Chaumont, Jacek Malecki. 2013-10-06. The asymptotic behavior of the density of the supremum of Lévy processes. https://arxiv.org/abs/1310.1587

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