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We introduce a continuous time autoregressive moving average process with random Lévy coefficients, termed RC-CARMA(p,q) process, of order p and q < p via a subclass of multivariate generalized Ornstein-Uhlenbeck processes. Sufficient conditions for the existence of a strictly stationary solution and the existence of moments are obtained. We further investigate second order stationarity properties, calculate the autocovariance function and spectral density, and give sufficient conditions for their existence.

Continuous time autoregressive moving average processes with random Lévy coefficients