The Annals of Applied Probability

Multivariate supOU processes

Ole Eiler Barndorff-Nielsen and Robert Stelzer

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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 Lévy 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 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, and we discuss the relevance of multivariate supOU processes in applications.

Article information

Ann. Appl. Probab., Volume 21, Number 1 (2011), 140-182.

First available in Project Euclid: 17 December 2010

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Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 60H20: Stochastic integral equations
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; Lévy processes 60G57: Random measures

Lévy bases long memory normal matrices Ornstein–Uhlenbeck-type processes positive semi-definite stochastic processes second-order moment structure stochastic differential equation


Barndorff-Nielsen, Ole Eiler; Stelzer, Robert. Multivariate supOU processes. Ann. Appl. Probab. 21 (2011), no. 1, 140--182. doi:10.1214/10-AAP690.

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