Abstract
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.
Original language | English |
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Pages (from-to) | 45-53 |
Number of pages | 9 |
Journal | Statistics and Probability Letters |
Volume | 115 |
DOIs | |
Publication status | Published - 1 Aug 2016 |
MoE publication type | A1 Journal article-refereed |
Keywords
- Lamperti transform
- Langevin equation
- Self-similar processes
- Stationary increment processes
- Stationary processes