On the definition, stationary distribution and second order structure of positive semidefinite Ornstein--Uhlenbeck type processes

Several important properties of positive semidefinite processes of Ornstein–Uhlenbeck type are analysed. It is shown that linear operators of the form $X\mapsto AX+X{A}^{\mathrm{T}}$ with $A\in M_d(\mathbb{R})$ are the only ones that can be used in the definition provided one demands a natural non-degeneracy condition. Furthermore, we analyse the absolute continuity properties of the stationary distribution (especially when the driving matrix subordinator is the quadratic variation of a $d$-dimensional L'{e}vy process) and study the question of how to choose the driving matrix subordinator in order to obtain a given stationary distribution. Finally, we present results on the first and second order moment structure of matrix subordinators, which is closely related to the moment structure of positive semidefinite Ornstein–Uhlenbeck type processes. The latter results are important for method of moments based estimation.