Random integral representation of operator-semi-self-similar processes with independent increments

Jeanblanc et al. (Stochastic Process. Appl. 100 (2002) 223) give a representation of self-similar processes with independent increments by stochastic integrals with respect to background driving Lévy processes. Via Lampertiʹs transformation these processes correspond to stationary Ornstein–Uhlenbeck processes. In the present paper we generalize the integral representation to multivariate processes with independent increments having the weaker scaling property of operator-semi-self-similarity. It turns out that the corresponding background driving process has periodically stationary increments and in general is no longer a Lévy process. Just as well it turns out that the Lamperti transform of an operator-semi-self-similar process with independent increments defines a periodically stationary process of Ornstein–Uhlenbeck type.