Fractional lower order covariance based-estimator for Ornstein-Uhlenbeck process with stable distribution

• Autorzy: Kruczek Piotr , Żuławiński Wojciech , Pagacz Patrycja , Wyłomańska Agnieszka

Identyfikatory

DOI

10.14708/ma.v47i2.6506

Warianty tytułu

PL

Estymator bazujący na ułamkowych momentach dla procesu Ornsteina-Uhlenbecka z rozkładem stabilnym

• Języki publikacji: EN

Abstrakty

EN

The Ornstein-Uhlenbeck model is one of the most popular stochastic processes. It has found many interesting applications including physical phenomena. However, for many real data, the classical Ornstein-Uhlenbeck process cannot be applied. It is related to the fact that for many phenomena the vectors of observations exhibit the so-called heavy-tailed behaviour. In such cases, the modifications of the classical models need to be used. In this paper, we analyze the Ornstein-Uhlenbeck process based on stable distribution. This distribution is one of the most classical members of the heavy-tailed class of distributions. In the literature, one can find various applications of stable processes. However, the heavy-tailed property implies that the classical methods of estimation and statistical investigation cannot be applied. In this paper, we propose a new method of estimation of the stable Ornstein-Uhlenbeck process. This technique is based on the alternative measure of dependence, called fractional lower order covariance, which replaces the classical covariance for infinite-variance distribution. The proposed research is a continuation of the authors' previous studies, where the measure called covariation was proposed as the base for the estimation technique. We introduce the stable Ornstein-Uhlenbeck process and remind its main properties. In the main part, we define the new estymator of the parameters for discrete representation of the Ornstein-Uhlenbeck process. Its effectiveness is checked by Monte Carlo simulations.

PL

Proces Ornsteina-Uhlenbecka jest jednym z najbardziej popularnych procesów stochastycznych. Znalazł on wiele ciekawych praktycznych zastosowań. Należy jednak zwrócić uwagę, że klasyczny proces Ornsteina-Uhlenbecka nie może być zastosowany dla wielu danych rzeczywistych, ponieważ często pochodzą one z rozkładów ciężko- ogonwych, dla których nie istnieje drugi moment. W takich przypadkach niezbędna jest modyfikacja klasycznego modelu z wykorzystaniem rozkładu stabilnego. Z powodu zastosowania rozkładu stabilnego niezbędne jest użycie innej metody estymacji niż bazującej na autokowariancji. Zaproponowana została nowa metoda bazująca na ułamkowych momentach. Praca jest kontynuacją wcześniej otrzymanych rezultatów dla innej alternatywnej miary zależności, kowariacji. W pracy przypomniana została definicja stabilnego procesu Ornsteina-Uhlenbecka wraz z propozycją nowych estymatorów dla parametrów tego procesu.W celu sprawdzenia ich właściwości wykonane zostały symulacje Monte Carlo.

Słowa kluczowe

EN

Ornstein-Uhlenbeck process; FLOC; estimation; stable distribution

PL

proces Ornsteina-Uhlenbecka; FLOC; estymacja; rozkład stabilny

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2019
• Tom: Vol. 47, no. 2
• Strony: 269--292
• Opis fizyczny: Bibliogr. 47 poz., fot., tab., wykr.

Twórcy

autor

Kruczek Piotr

• Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław

autor

Żuławiński Wojciech

• Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław

autor

Pagacz Patrycja

• Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław

autor

Wyłomańska Agnieszka

• Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław

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Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).