An integral representation of dilatively stable processes with independent increments

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 Igl\'oi, we will show how dilatively stable processes with independent increments can be represented by integrals with respect to time-changed L\'evy 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.