On fluctuation-theoretic decompositions via Lindley-type recursions

Onno Boxma, Offer Kella and Michel Mandjes

Stochastic Processes and their Applications, 2023, vol. 165, issue C, 316-336

Abstract: 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.

Keywords: Lindley recursion; Fluctuation theory; Maximum of a random walk; Maximum of a Lévy process; Decomposition (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.09.004

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