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

2014/07/24 by Lauri Viitasaari, Viitasaari, Lauri · 2 citations
Economics, Econometrics and Finance · Mathematics · #60G07 #60G10 #60G18 #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1407.6521

openalex publication_date 2014/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let W_t be a standard Brownian motion. It is well-known that the Langevin\nequation d U_t = -\θ U_td t + d W_t defines a stationary process called\nOrnstein-Uhlenbeck process. Furthermore, Langevin equation can be used to\nconstruct other stationary processes by replacing Brownian motion W_t with\nsome other process G with stationary increments. In this article we prove\nthat the converse also holds and all continuous stationary processes arise from\na Langevin equation with certain noise G=G_\θ. Discrete analogies of our\nresults are given and applications are discussed.\n

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