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Let $W_t$ be a standard Brownian motion. It is well-known that the Langevinequation $d U_t = -\theta U_td t + d W_t$ defines a stationary process calledOrnstein-Uhlenbeck process. Furthermore, Langevin equation can be used toconstruct other stationary processes by replacing Brownian motion $W_t$ withsome other process $G$ with stationary increments. In this article we provethat the converse also holds and all continuous stationary processes arise froma Langevin equation with certain noise $G=G_\theta$. Discrete analogies of ourresults are given and applications are discussed.

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