New approaches for modeling and estimation of discrete and continuous time stationary processes

Marko Voutilainen

Uusia menetelmiä diskreetti- ja jatkuva-aikaisten stationaaristen prosessien mallintamiseksi ja estimoimiseksi

Marko Voutilainen

Matematiikan ja systeemianalyysin laitos

Tutkimustuotos: Doctoral Thesis › Collection of Articles

Abstrakti

Stationaariset prosessit muodostavat merkittävän stokastisten prosessien luokan, jota on tutkittu laajalti ja jolle löytyy sovelluksia monilta tieteen eri osa-alueilta. Sovelluskohteita ovat esimerkiksi monien reaalimaailman ilmiöiden mallintaminen ja niiden ennustaminen, kuten pörssikurssit, yrityksen liikevaihto, luonnonkatastroﬁt ja liikevastuksen vaikutuksen alaisen Brownin hiukkasen nopeus. Tässä väitöskirjassa esitellään uusia menetelmiä diskreetti- ja jatkuva-aikaisten stationaaristen prosessien mallintamiseksi ja estimoimiseksi. Diskreetti- ja jatkuva- aikaiset vahvasti stationaariset prosessit karakterisoidaan AR(1) ja Langevin yhtälöiden avulla. Kyseisten karakterisaatioiden pohjalta johdetaan stationaarisen prosessin autokovarianssin avulla ilmaistavat toisen asteen (matriisi)yhtälöt mallin (matriisi)parametrille. Perustuen näihin yhtälöihin, mallin parametrille määritellään estimaattori. Estimaattorin tarkentuvuuden ja suppenemisnopeuden osoitetaan seuraavan suoraan valittujen autokovarianssiestimaattoreiden vastaavista ominaisuuksista. Tämän lisäksi estimaattorin rajajakauma voidaan esittää lineaarisen funktion avulla autokovarianssiestimaattoreiden rajajakaumasta. Esitettyä yleistäteoriaa sovelletaan myös ARCH-mallin erään yleistyksen estimoimiseksi.

Julkaisun otsikon käännös	Uusia menetelmiä diskreetti- ja jatkuva-aikaisten stationaaristen prosessien mallintamiseksi ja estimoimiseksi
Alkuperäiskieli	Englanti
Pätevyys	Tohtorintutkinto
Myöntävä instituutio	Aalto-yliopisto
Valvoja/neuvonantaja	Ilmonen, Pauliina, Vastuuprofessori Viitasaari, Lauri, Ohjaaja
Kustantaja	Aalto University
Painoksen ISBN	978-952-64-0182-9
Sähköinen ISBN	978-952-64-0183-6
Tila	Julkaistu - 2020
OKM-julkaisutyyppi	G5 Tohtorinväitöskirja (artikkeli)

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http://urn.fi/URN:ISBN:978-952-64-0183-6

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@phdthesis{fcca61dd0bbe46a4b60673d27a29e386,

title = "New approaches for modeling and estimation of discrete and continuous time stationary processes",

abstract = "Stationary processes form an important class of stochastic processes that has been extensively studied in the literature, and widely applied in many ﬁelds of science. Applications include modeling and forecasting various real-life phenomena such as stock market behavior, sales of a company, natural disasters and velocity of a Brownian particle under the inﬂuence of friction, to mention a few. In this dissertation, we consider new methods for modeling and estimation of discrete and continuous time stationary processes. We characterize discrete and continuous time strictly stationary processes by AR(1) and Langevin equations, respectively. From these characterizations, we derive quadratic (matrix) equations for the corresponding model parameter (matrix) in terms of autocovariance of the stationary process. Based on the equations, we construct an estimator for the model parameter. Furthermore, we show that the estimator inherits consistency and the rate of convergence from the chosen autocovariance estimators. Moreover, its limiting distribution is given by a linear function of the limiting distribution of the autocovariance estimators. In addition, we apply the presented general theory in modeling and estimationof a generalization of the ARCH model with stationary liquidity.",

keywords = "stationary processes, stochastic processes, stationary processes, stochastic processes",

author = "Marko Voutilainen",

year = "2020",

language = "English",

isbn = "978-952-64-0182-9",

series = "Aalto University publication series DOCTORAL DISSERTATIONS",

publisher = "Aalto University",

number = "209",

school = "Aalto University",

}

TY - THES

T1 - New approaches for modeling and estimation of discrete and continuous time stationary processes

AU - Voutilainen, Marko

PY - 2020

Y1 - 2020

N2 - Stationary processes form an important class of stochastic processes that has been extensively studied in the literature, and widely applied in many ﬁelds of science. Applications include modeling and forecasting various real-life phenomena such as stock market behavior, sales of a company, natural disasters and velocity of a Brownian particle under the inﬂuence of friction, to mention a few. In this dissertation, we consider new methods for modeling and estimation of discrete and continuous time stationary processes. We characterize discrete and continuous time strictly stationary processes by AR(1) and Langevin equations, respectively. From these characterizations, we derive quadratic (matrix) equations for the corresponding model parameter (matrix) in terms of autocovariance of the stationary process. Based on the equations, we construct an estimator for the model parameter. Furthermore, we show that the estimator inherits consistency and the rate of convergence from the chosen autocovariance estimators. Moreover, its limiting distribution is given by a linear function of the limiting distribution of the autocovariance estimators. In addition, we apply the presented general theory in modeling and estimationof a generalization of the ARCH model with stationary liquidity.

AB - Stationary processes form an important class of stochastic processes that has been extensively studied in the literature, and widely applied in many ﬁelds of science. Applications include modeling and forecasting various real-life phenomena such as stock market behavior, sales of a company, natural disasters and velocity of a Brownian particle under the inﬂuence of friction, to mention a few. In this dissertation, we consider new methods for modeling and estimation of discrete and continuous time stationary processes. We characterize discrete and continuous time strictly stationary processes by AR(1) and Langevin equations, respectively. From these characterizations, we derive quadratic (matrix) equations for the corresponding model parameter (matrix) in terms of autocovariance of the stationary process. Based on the equations, we construct an estimator for the model parameter. Furthermore, we show that the estimator inherits consistency and the rate of convergence from the chosen autocovariance estimators. Moreover, its limiting distribution is given by a linear function of the limiting distribution of the autocovariance estimators. In addition, we apply the presented general theory in modeling and estimationof a generalization of the ARCH model with stationary liquidity.

KW - stationary processes

KW - stochastic processes

KW - stationary processes

KW - stochastic processes

M3 - Doctoral Thesis

SN - 978-952-64-0182-9

T3 - Aalto University publication series DOCTORAL DISSERTATIONS

PB - Aalto University

ER -

Uusia menetelmiä diskreetti- ja jatkuva-aikaisten stationaaristen prosessien mallintamiseksi ja estimoimiseksi

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