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

Abstract

Let $W_t$ be a standard Brownian motion. It is well-known that the Langevin equation $d U_t = -θU_td t + d W_t$ 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 $G=G_θ$. Discrete analogies of our results are given and applications are discussed.

13 pages. this version: some errors corrected and proof shortened