Creeping of Lévy processes through curves
Creeping of Lévy processes through curves
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Lévy 过程通过曲线的蠕变
DOI：
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发表时间：
2022
影响因子：
1.4
通讯作者：
Thomas Pellas
中科院分区：
文献类型：
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作者：
L. Chaumont;Thomas Pellas
一个L 'evy过程被称为爬行通过一条曲线，如果在它第一次通过这条曲线的时候，这个过程以正概率到达它。我们首先研究这个属性的二元从属。给定任何连续、非增函数$f$的图${（t，f（t））：tge 0}$，使得$f（0）>0$，我们给出了从0发出的二元从属子$（Y，Z）$根据其更新函数和分量$Y$和$Z$的漂移爬过该图的概率的表达式。我们将这一结果应用于爬行概率的任何真实的L 'evy过程通过图的任何连续的，非增函数的时间，该过程也达到其过去的上确界。这个概率包含了二元向上阶梯过程更新函数的密度和漂移系数。我们还调查的情况下，L 'evy过程的条件保持积极的爬行在其最后一个通道的时间低于图的功能。然后，我们提供了一些例子，我们给一个应用程序的概率蠕变通过固定的水平稳定的Ornstein-Uhlenbeck过程。我们还提出了几个开放的问题沿着文本。
A L'evy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph ${(t,f(t)):tge0}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real L'evy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of L'evy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.