We give necessary and sufficient conditions for existence of proper integrals from 0 to infinity or from minus infinity to 0 of one exponentiated Lévy process with respect to another Lévy process. The results are related to the existence of stationary generalized Ornstein-Uhlenbeck processes. Finally, in the square integrable case the Wold-Karhunen representation is given.

Keywords: stochastic integration; Lévy processes; generalized Ornstein-Uhlenbeck processes.

AMS Subject Classification: 60G44; 60G57; 60H05.