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February 2011 Multivariate supOU processes
Ole Eiler Barndorff-Nielsen, Robert Stelzer
Ann. Appl. Probab. 21(1): 140-182 (February 2011). DOI: 10.1214/10-AAP690

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 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.

Citation

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Ole Eiler Barndorff-Nielsen. Robert Stelzer. "Multivariate supOU processes." Ann. Appl. Probab. 21 (1) 140 - 182, February 2011. https://doi.org/10.1214/10-AAP690

Information

Published: February 2011
First available in Project Euclid: 17 December 2010

zbMATH: 1235.60032
MathSciNet: MR2759198
Digital Object Identifier: 10.1214/10-AAP690

Subjects:
Primary: 60G10, 60H20
Secondary: 60E07, 60G51, 60G57

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.21 • No. 1 • February 2011
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