Persistence of integrated stable processes

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• 作者：Christophe Profeta ; Thomas Simon
• 关键词：Integrated process ; Half ; Cauchy distribution ; Hitting place ; Lower tail probability ; Mellin transform ; Persistence ; Stable Lévy process ; 60F99 ; 60G52 ; 60J50
• 刊名：Probability Theory and Related Fields
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：162
• 期：3-4
• 页码：463-485
• 全文大小：496 KB
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• 作者单位：Christophe Profeta (1)
  Thomas Simon (2) (3)
  
  1. Laboratoire de Mathématiques et Modélisation d’Evry (LaMME), Université d’Evry-Val-d’Essonne, UMR CNRS 8071 Batiment IBGBI, 23 boulevard de France, 91037?, Evry Cedex, France
  2. Laboratoire Paul Painlevé, Université Lille 1, 59655?, Villeneuve d’Ascq Cedex, France
  3. Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, 91405?, Orsay Cedex, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Mathematical and Computational Physics
  Quantitative Finance
  Mathematical Biology
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Operation Research and Decision Theory
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-2064

文摘

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Shi?(Lower tails of some integrated processes. In: Small deviations and related topics (problem panel 2003). Along the way, we investigate the law of the stable process \(L\) evaluated at the first time its integral \(X\) hits zero, when the bivariate process \((X,L)\) starts from a coordinate axis. This extends classical formul?? by McKean (J Math Kyoto Univ 2:227-35, 1963) and Gor’kov (Soviet. Math. Dokl. 16:904-08, 1975) for integrated Brownian motion.