An integral representation of dilatively stable processes with independent increments

T. Bhatti and P. Kern

Stochastic Processes and their Applications, 2017, vol. 127, issue 1, 209-227

Abstract: Dilative stability generalizes the property of selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. Inspired by results of Iglói (2008), we will show how dilatively stable processes with independent increments can be represented by integrals with respect to time-changed Lévy processes. Via a Lamperti-type transformation these representations are shown to be closely connected to translatively stable processes of Ornstein–Uhlenbeck-type, where translative stability generalizes the notion of stationarity. The presented results complement corresponding representations for selfsimilar processes with independent increments known from the literature.

Keywords: Dilative stability; Translative stability; Lamperti transform; Additive process; Random integral representation; Wide sense Ornstein–Uhlenbeck process; Quasi-selfsimilar process; Time-stable process; Infinite divisibility with respect to time (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:127:y:2017:i:1:p:209-227

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DOI: 10.1016/j.spa.2016.06.006

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