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

Let WtWt be a standard Brownian motion. It is well-known that the Langevin equation dUt=−θUtdt+dWt defines a stationary process called Ornstein–Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian motion WtWt with some other process GG 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=GθG=Gθ. Discrete analogies of our results are given and applications are discussed.