Exact and asymptotic n-tuple laws at first and last passage
Exact and asymptotic n-tuple laws at first and last passage
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第一个和最后一个通道的精确且渐近的 n 元组定律
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
V. Rivero
中科院分区：
文献类型：
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作者：
A. Kyprianou;Juan Carlos Pardo;V. Rivero
了解 Levy 过程如何第一次（实际上是最后一次）跨越恒定障碍的时空特征是应用概率中许多模型的核心问题，例如排队论、金融和精算数学、最优停止问题、分支过程理论等。在多尼和基普里亚努[Ann。应用。很可能。 16 (2006) 91-106] 为首次低于固定水平的一般征税程序制定了新的五重法。在本文中，我们使用五元组法则为 Levy 过程建立一系列相关的联合法则，我们将其称为 n 元组法则，Levy 过程的条件是在第一次和最后一次超过固定水平时保持正向和正自相似马尔可夫过程。这里整数n通常在三到七的范围内。此外，我们还研究了渐近过冲和下冲分布，并将它们与从原点发出的正自相似马尔可夫过程的过冲和下冲分布联系起来。尽管由于 Lamperti 变换，Levy 过程和正自相似马尔可夫过程的 n 元组定律之间的关系很简单，但通过将（条件）稳定过程作为（条件）Levy 过程和正自相似马尔可夫过程相互作用，我们获得了一套完全明确的所谓 Lamperti 稳定 Levy 过程的第一和最后一个通道恒等式。这进一步导致引入更一般的 Levy 过程族，我们称之为超几何 Levy 过程，可以考虑类似的显式恒等式。
Understanding the space time features of how a Levy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes, to name but a few. In Doney and Kyprianou [Ann. Appl. Probab. 16 (2006) 91-106] a new quintuple law was established for a general Levy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call n-tuple laws, for Levy processes, Levy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer n typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive self-similar Markov processes issued from the origin. Although the relation between the n-tuple laws for Levy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by interplaying the role of a (conditioned) stable processes as both a (conditioned) Levy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable Levy processes. This leads further to the introduction of a more general family of Levy processes which we call hypergeometric Levy processes, for which similar explicit identities may be considered.
关于 Lamperti 稳定过程
DOI：
10.48550/arxiv.0802.0851
发表时间：
2008
期刊：
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影响因子：
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作者：
Caballero M
通讯作者：
Caballero M