A Trek Rule for the Lyapunov Equation

TL;DR

This paper introduces a new trek rule for the Lyapunov equation that relates the graphical structure of a Gaussian Markov process to its steady-state covariance matrix, providing explicit formulas and insights into variance behavior.

Contribution

It presents a novel trek rule for the Lyapunov equation, linking graph structure to covariance entries, and derives explicit formulas and variance bounds for specific cases.

Findings

The trek rule links the process's graph to its covariance matrix.

Explicit formulas for covariance entries in certain models.

A new lower bound on variances in acyclic models.

Abstract

The Lyapunov equation is a linear matrix equation characterizing the cross-sectional steady-state covariance matrix of a Gaussian Markov process. We show a new version of the trek rule for this equation, which links the graphical structure of the drift of the process to the entries of the steady-state covariance matrix. In general, the trek rule is a power series expansion of the covariance matrix in the entries of the drift and volatility matrices. For acyclic models it simplifies to a polynomial in the off-diagonal entries of the drift matrix. Using the trek rule we can give relatively explicit formulas for the entries of the covariance matrix for some special cases of the drift matrix. These results illustrate notable differences between covariance models entailed by the Lyapunov equation and those entailed by linear additive noise models. To further explore differences and similarities between these two model classes, we use the trek rule to derive a new lower bound on the marginal variances in the acyclic case. This sheds light on the phenomenon, well known for the linear additive noise model, that the variances in the acyclic case tend to increase along a topological ordering of the variables.