Covariance Kernels of Gaussian Markov Processes

TL;DR

This paper characterizes the covariance kernels of Gaussian Markov processes, providing simpler semi-parametric forms, necessary and sufficient conditions for stationary increments, and explicit estimators for parameter inference and forecasting.

Contribution

It introduces a simplified semi-parametric form for covariance kernels of Gaussian Markov processes and derives explicit estimators and posterior moments for practical inference.

Findings

Covariance kernels have a simpler semi-parametric form for certain Gaussian Markov processes.

Necessary and sufficient conditions for stationary increments are established.

Closed-form maximum likelihood estimators and posterior moments are derived for discretely sampled processes.

Abstract

The solution to a multivariate linear Stochastic Differential Equation (SDE) with constant initial state is well known to be a Gaussian Markov process, but its covariance kernel involves the solution to an integral equation in the general case. We show that the covariance kernel has a simpler semi-parametric form for families of such solutions representing increments of a common process. We also show that a covariance kernel of a particular parametric form is necessary and sufficient for a solution to possess stationary increments and for a Gaussian process, in considerable generality, to have stationary increments and the Markov property. For a discretely sampled Gaussian process with such a parametric kernel, we derive closed-form expressions for unique maximum likelihood estimators of the parameter matrices that are unbiased, jointly sufficient, and easily computed regardless of the dimension. Using those estimators, we also derive closed-form expressions for posterior moments useful for forecasting.