Multivariate supOU processes

Abstract

Univariate superpositions of Ornstein--Uhlenbeck-type processes (OU), called supOU processes, provide a class of continuous time processes capable of exhibiting long memory behavior. 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 factorizable L\'{e}vy bases. We show that the behavior 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 L\'{e}vy bases we are able to give conditions for supOU processes to have locally bounded c\`{a}dl\`{a}g paths of finite variation and to show an analogue of the stochastic differential equation of OU-type processes, which has been suggested in \cite barndorffnielsen01 in the univariate case. Finally, as an important special case, we introduce positive semi-definite supOU processes, and we discuss the relevance of multivariate supOU processes in applications.