Mathematics – Statistics Theory
Scientific paper
2010-02-23
Bernoulli 2010, Vol. 16, No. 1, 80-115
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.3150/09-BEJ196 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti
Scientific paper
10.3150/09-BEJ196
Multivariate $\operatorname {COGARCH}(1,1)$ processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate L\'{e}vy process and the latent time-varying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the $\operatorname {COGARCH}(1,1)$ 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 L\'{e}vy 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 $\operatorname {COGARCH}(1,1)$ process and their "squares".
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