On the Markov transformation of Gaussian processes

TL;DR

This paper develops a method to transform any Gaussian process into a Markov process with identical one-dimensional marginals, using finite-time transformations, and characterizes the uniqueness of this transformation under certain conditions.

Contribution

It introduces a construction for Gaussian Markov processes from arbitrary Gaussian processes and proves the conditions for uniqueness based on decorrelation rates.

Findings

Existence of a Gaussian Markov transform for any Gaussian process.

Uniqueness of the transform when the decorrelation rate is continuous.

Characterization of the transform via the decorrelation rate.

Abstract

Given a Gaussian process $(X_t)_{t \in \mathbb{R}}$, we construct a Gaussian \emph{Markov} process with the same one-dimensional marginals using sequences of transformations of $(X_t)_{t \in \mathbb{R}}$ "made Markov" at finitely many times. We prove that there exists at least such a Markov transform of $(X_t)_{t \in \mathbb{R}}$. In the case the instantaneous decorrelation rate of $(X_t)_{t \in \mathbb{R}}$ is continuous, we prove that the Markov transform is uniquely determined and characterized through the same instantaneous decorrelation rate.