  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.
