Functionals of a Lévy Process on Canonical and Generic Probability Spaces

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• 作者：Alexander Steinicke
• 刊名：Journal of Theoretical Probability
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：29
• 期：2
• 页码：443-458
• 全文大小：454 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

We develop an approach to Malliavin calculus for Lévy processes from the perspective of expressing a random variable \(Y\) by a functional \(F\) mapping from the Skorohod space of càdlàg functions to \(\mathbb {R}\), such that \(Y=F(X)\) where \(X\) denotes the Lévy process. We also present a chain-rule-type application for random variables of the form \(f(\omega ,Y(\omega ))\). An important tool for these results is a technique which allows us to transfer identities proved on the canonical probability space (in the sense of Solé et al.) associated to a Lévy process with triplet \((\gamma ,\sigma ,\nu )\) to an arbitrary probability space \((\varOmega ,\mathcal {F},\mathbb {P})\) which carries a Lévy process with the same triplet.KeywordsLévy processesMalliavin calculus for Lévy processesCanonical Lévy process