Path decomposition of a reflected Lévy process on first passage over high levels

Research output: Contribution to journalArticlepeer-review


Let X be a real valued Lévy process and set Rt=Xt−infs≤tXs. This paper addresses the asymptotic behavior of the sample paths of the reflected process R on first passage over an arbitrarily high level u. We show that under the convolution equivalent condition of Klüppelberg et al. (2004), the sample paths of R on the first excursion which crosses over a high level u can be decomposed into two processes. The first describes the paths in a neighborhood of the origin. The process then takes a large jump into a neighborhood of u. The second process describes the subsequent paths. This sample path behavior is similar to that of X conditioned to cross level u. Using this connection many results concerning, for example, undershoots and overshoots can be easily obtained.

Original languageEnglish (US)
Pages (from-to)29-47
Number of pages19
JournalStochastic Processes and their Applications
StatePublished - Mar 2022


  • Convolution equivalence
  • First passage time
  • Lévy process
  • Overshoot
  • Reflected process

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


Dive into the research topics of 'Path decomposition of a reflected Lévy process on first passage over high levels'. Together they form a unique fingerprint.

Cite this