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

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  • Viitasaari, Lauri

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.

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  • Viitasaari, Lauri, 2016. "Representation of stationary and stationary increment processes via Langevin equation and self-similar processes," Statistics & Probability Letters, Elsevier, vol. 115(C), pages 45-53.
    • Handle: RePEc:eee:stapro:v:115:y:2016:i:c:p:45-53
    DOI: 10.1016/j.spl.2016.03.020
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    Cited by:

    1. Tommi Sottinen & Lauri Viitasaari, 2018. "Parameter estimation for the Langevin equation with stationary-increment Gaussian noise," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 569-601, October.
    2. Douissi, Soukaina & Es-Sebaiy, Khalifa & Alshahrani, Fatimah & Viens, Frederi G., 2022. "AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 886-918.
    3. Marko Voutilainen & Lauri Viitasaari & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor, 2022. "Vector‐valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 992-1022, September.

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