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

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

2009-09-04

arxiv.org/abs/0909.0851v1

Bernoulli 2009, Vol. 15, No. 3, 754-773

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/08-BEJ175 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ175

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.

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