  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".
