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

Lauri Viitasaari

doi:10.1016/j.spl.2016.03.020

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

Lauri Viitasaari^*

^*Corresponding author for this work

Department of Mathematics and Systems Analysis

Research output: Contribution to journal › Article › Scientific › peer-review

5 Citations (Scopus)

Abstract

Let W_t be a standard Brownian motion. It is well-known that the Langevin equation dU_t=-θU_tdt+dW_t defines a stationary process called Ornstein-Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian motion W_t 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=G_θ. Discrete analogies of our results are given and applications are discussed.

Original language	English
Pages (from-to)	45-53
Number of pages	9
Journal	Statistics and Probability Letters
Volume	115
DOIs	https://doi.org/10.1016/j.spl.2016.03.020
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

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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.",

keywords = "Lamperti transform, Langevin equation, Self-similar processes, Stationary increment processes, Stationary processes",

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N2 - 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.

AB - 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.

KW - Lamperti transform

KW - Langevin equation

KW - Self-similar processes

KW - Stationary increment processes

KW - Stationary processes

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