Kac–Lévy Processes

Abstract

Markov-modulated Lévy processes with two different regimes of restarting are studied. These regimes correspond to the completely renewed process and to the process of Markov modulation, accompanied by jumps. We give explicit expressions for the Lévy–Khintchine exponent in the case of a two-state underlying Markov chain. For the renewal case, the limit distributions (as \(t\rightarrow \infty \)) are obtained. In the case of processes with jumps, we present some results for the exponential functional.

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Correspondence to Nikita Ratanov.

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Ratanov, N. Kac–Lévy Processes. J Theor Probab 33, 239–267 (2020). https://doi.org/10.1007/s10959-018-0873-6

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Keywords

  • Markov-modulated Lévy process
  • Markov-switching model
  • Goldstein–Kac process
  • Lévy–Khintchine exponent
  • Lévy–Laplace exponent
  • Mixture of distributions
  • Exponential functional

Mathematical Subject Classification (2010)

  • 60K15
  • 60J75
  • 60J27