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

TL;DR

This paper investigates the properties, stationary distributions, and second-order structures of positive semidefinite Ornstein-Uhlenbeck type processes, providing insights into their defining operators, distributional properties, and moment structures for estimation purposes.

Contribution

It characterizes the linear operators defining these processes, analyzes their stationary distribution properties, and studies the moment structure of matrix subordinators, advancing understanding of their mathematical foundations.

Findings

Linear operators of the form $X o AX + XA^T$ are unique in defining these processes.

Conditions for absolute continuity of stationary distributions are established.

Results on the first and second order moments of matrix subordinators are presented.

Abstract

Several important properties of positive semidefinite processes of Ornstein--Uhlenbeck type are analysed. It is shown that linear operators of the form $X\mapsto AX+XA^{\mathrm{T}}$ 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.