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

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