The asymptotic behavior of the density of the supremum of L\'evy processes
The asymptotic behavior of the density of the supremum of L\'evy processes
复制标题
Levy 过程上界密度的渐近行为
DOI：
10.1214/15-aihp674
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
J. Małecki
中科院分区：
文献类型：
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作者：
L. Chaumont;J. Małecki
我们考虑一个真实的L\'evy过程X，它的转移概率是绝对连续的，并且有界密度。则X在任何确定时间t之前的过去上确界的定律在（0，\infty）上绝对连续。我们证明了它的密度f_t（x）在（0，\infty）上连续当且仅当向上阶梯高度过程的势密度h'在（0，\infty）上连续.然后我们证明f_t在0处和h '一样。当t趋于无穷大时，我们还描述了f_t的渐近行为.在此基础上，得到了曲折线的密度和L 'evy过程的入口律在满足正性条件下的密度的相关结果.
Let us consider a real L\'evy 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 L\'evy process conditioned to stay positive.