Univariate superpositions of Ornstein-Uhlenbeck (OU) type processes, called supOU processes, provide a class of continuous time processes capable of exhibiting long memory behaviour. This paper introduces multivariate supOU processes and gives conditions for their existence and finiteness of moments. Moreover, the second order moment structure is explicitly calculated, and examples exhibit the possibility of long range dependence.

Our supOU processes are defined via homogeneous and factorisable Lévy bases. We show that the behaviour of supOU processes is particularly nice when the mean reversion parameter is restricted to normal matrices and especially to strictly negative definite ones.

For finite variation Lévy bases we are able to give conditions for supOU processes to have locally bounded càdlàg paths of finite variation and to show an analogue of the stochastic differential equation of OU type processes, which has been suggested in [2] in the univariate case. Finally, as an important special case, we introduce positive semi-definite supOU processes.