Multivariate COGARCH(1, 1) processes
Robert Stelzer
Source: Bernoulli Volume 16, Number 1
(2010), 80-115.
Abstract
Multivariate COGARCH(1, 1) processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent time-varying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices.
After defining the COGARCH(1, 1) process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving Lévy process, explicit expressions for the first and second-order moments and (asymptotic) second-order stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and second-order structure of the increments of the multivariate COGARCH(1, 1) process and their “squares”.
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Keywords: COGARCH; Lévy process; multivariate GARCH; positive definite random matrix process; second-order moment structure; stationarity; stochastic differential equations; stochastic volatility; variance mixture model
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Permanent link to this document: http://projecteuclid.org/euclid.bj/1265984705
Digital Object Identifier: doi:10.3150/09-BEJ196
Mathematical Reviews number (MathSciNet): MR2648751
Zentralblatt MATH identifier: 05815965
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