Multivariate continuous-time autoregressive moving-average processes on cones

TL;DR

This paper introduces multivariate continuous-time autoregressive moving-average (MCARMA) processes on convex cones, providing conditions for their positivity and applications in modeling covariance in multivariate stochastic volatility.

Contribution

It develops necessary and sufficient conditions for cone-valued MCARMA processes with Lévy noise, including specific cases for positive orthants and positive semi-definite matrices, with applications in finance.

Findings

Conditions for cone-valued MCARMA processes are derived.

Examples of parameter specifications ensuring positivity are provided.

The relevance of these processes for modeling covariance in stochastic volatility is demonstrated.

Abstract

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.