Gauss-Markov processes as space-time scaled stationary Ornstein-Uhlenbeck processes

TL;DR

This paper introduces a class of Gauss-Markov processes represented as space-time scaled stationary Ornstein-Uhlenbeck processes, providing explicit examples, density formulas for supremum locations, and conditions for zero crossings.

Contribution

It offers a novel representation of Gauss-Markov processes using Ornstein-Uhlenbeck processes, along with explicit examples and new theoretical results.

Findings

Explicit representations for Gauss bridge processes

Formula for the density of supremum locations

Conditions for zero crossings of mean-centered processes

Abstract

We present a class of Gauss-Markov processes which can be represented as space-time scaled stationary Ornstein-Uhlenbeck processes defined on the real line. We give several explicit examples of the representation for certain Gauss bridge processes. As an application, we derive a formula for the density function of the supremum location of certain standardized Gauss-Markov processes on compact time intervals. We also present some sufficient conditions under which mean centered Gauss-Markov processes take zero at a fixed time with probability one.