Perpetual Integrals for Lévy Processes

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• 作者：Leif Döring ; Andreas E. Kyprianou
• 关键词：Lévy processes ; Fluctuation theory ; Perpetual integral
• 刊名：Journal of Theoretical Probability
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：29
• 期：3
• 页码：1192-1198
• 全文大小：360 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230
• 卷排序：29

文摘

Given a Lévy process \(\xi \), we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral \(\int _0^\infty f(\xi _s)\hbox {d}s\), where \(f\) is a positive locally integrable function. If \(\mu =\mathbb {E}[\xi _1]\in (0,\infty )\) and \(\xi \) has local times we prove the 0–1 law $$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned}$$with the exact characterization $$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned}$$The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.KeywordsLévy processesFluctuation theoryPerpetual integralLeif Döring was supported by an Ambizione Grant of the Swiss Science Foundation.