Creeping of Lévy processes through curves

IF 1.3 3区 数学 Q2 STATISTICS & PROBABILITY
L. Chaumont, Thomas Pellas
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引用次数: 2

Abstract

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 $\{(t,f(t)):t\ge0\}$ 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.
lsamvy的爬行过程通过曲线
如果在其第一次通过曲线时,过程以正概率到达曲线,则称每一个过程缓慢地通过曲线。我们首先研究了二元从属变量的这一性质。给定任意连续非递增函数$f$的图$\{(t,f(t)):t\ge0\}$,使得$f(0)> $,我们给出了从0发出的二元次元$(Y,Z)$的更新函数和分量$Y$和$Z$的漂移的概率表达式。我们将这个结果应用于任何实L\' every过程的爬行概率,通过任何连续的,不增加的函数的图,在该过程也达到其过去的上限时。这个概率涉及二元向上阶梯过程的更新函数的密度以及它的漂移系数。我们还研究了在函数图下的L\' evs过程在其最后通过时间条件下保持正爬行的情况。然后给出了一些例子,并给出了稳定的Ornstein-Uhlenbeck过程爬过固定水平的概率的应用。我们还提出了几个悬而未决的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Electronic Journal of Probability
Electronic Journal of Probability 数学-统计学与概率论
CiteScore
1.80
自引率
7.10%
发文量
119
审稿时长
4-8 weeks
期刊介绍: The Electronic Journal of Probability publishes full-size research articles in probability theory. The Electronic Communications in Probability (ECP), a sister journal of EJP, publishes short notes and research announcements in probability theory. Both ECP and EJP are official journals of the Institute of Mathematical Statistics and the Bernoulli society.
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