Representation of stationary and stationary increment processes via Langevin equation and self-similar processes

TL;DR

This paper demonstrates that all continuous stationary processes can be represented via a Langevin equation driven by a specific noise process, extending the classical Ornstein-Uhlenbeck framework.

Contribution

It proves that every continuous stationary process can be derived from a Langevin equation with a particular noise process, establishing a converse to existing constructions.

Findings

All continuous stationary processes can be represented by a Langevin equation.

The paper provides discrete analogies of the continuous results.

Applications of the representation are discussed.

Abstract

Let $W_t$ be a standard Brownian motion. It is well-known that the Langevin equation $d U_t = -\theta U_td t + d W_t$ defines a stationary process called Ornstein-Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian motion $W_t$ with some other process $G$ with stationary increments. In this article we prove that the converse also holds and all continuous stationary processes arise from a Langevin equation with certain noise $G=G_\theta$. Discrete analogies of our results are given and applications are discussed.