Consider a Lévy process Y(t) over an exponentially distributed time Tβ with mean 1/β. We study the joint distribution of the running maximum Ȳ(Tβ) and the time epoch G(Tβ) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector (Ȳ(Tβ),G(Tβ)) can be written as a sum of two independent vectors, the first one being (Ȳ(Tβ+ω),G(Tβ+ω)) and the second one being the running maximum and corresponding time epoch under the restriction that the Lévy process is only observed at Poisson(ω) inspection epochs (until Tβ). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that ω only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.
Bibliographical noteFunding Information:
Partly funded by the NWO Gravitation project Networks, grant number 024.002.003.Partially supported the Vigevani Chair in Statistics.
© 2023 The Author(s)
- Fluctuation theory
- Lindley recursion
- Maximum of a Lévy process
- Maximum of a random walk