  In this article we study multivariate continuous-time autoregressive
moving-average (MCARMA) processes with values in convex cones. More
specifically, we introduce matrix-valued MCARMA processes with L\'evy noise and
present necessary and sufficient conditions for processes from this class to be
cone valued. We derive specific hands-on conditions in the following two cases:
First, for classical MCARMA on $\mathbb{R}_{d}$ with values in the positive
orthant $\mathbb{R}_{d}^{+}$. Second, for MCARMA processes on real square
matrices taking values in the cone of symmetric and positive semi-definite
matrices. Both cases are relevant for applications and we give several examples
of positivity ensuring parameter specifications. In addition to the above, we
discuss the capability of positive semi-definite MCARMA processes to model the
spot covariance process in multivariate stochastic volatility models. We
justify the relevance of MCARMA based stochastic volatility models by an
exemplary analysis of the second order structure of positive semi-definite
well-balanced Ornstein-Uhlenbeck based models.

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