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

Lauri Viitasaari*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)

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 languageEnglish
Pages (from-to)45-53
Number of pages9
JournalStatistics and Probability Letters
Volume115
DOIs
Publication statusPublished - 1 Aug 2016
MoE publication typeA1 Journal article-refereed

Keywords

  • Lamperti transform
  • Langevin equation
  • Self-similar processes
  • Stationary increment processes
  • Stationary processes

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