Let Wt 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 Wt 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=Gθ. Discrete analogies of our results are given and applications are discussed.
- Lamperti transform
- Langevin equation
- Self-similar processes
- Stationary increment processes
- Stationary processes