Brownovo gibanje

master's thesis

• Author: Marija Šuker
• Mentor: Ante Mimica (mentor)
• Committee member: Ante Mimica (predsjednik povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu smo definirali i opisali Brownovo gibanje, njegovu konstrukciju, neka zanimljiva svojstva i načine simuliranja. Za početak smo definirali Brownovo gibanje i opisali ga kao Gaussovski proces. Za taj opis su nam bile potrebne definicije standardnih normalnih slučajnih vektora, normalnih slučajnih vektora i Gaussovskog procesa. Koristeći te definicije, opisali smo Brownovo gibanje kao Gaussovski proces s neprekidnim trajektorijama, funkcijom očekivanja $m=0$ i... More kovarijacijskom funkcijom $\gamma_{st} = s \wedge t$. U nastavku smo dokazali još neka svojstva standardnih normalnih slučajnih vektora. Zatim smo opisali Lévyjevu konstrukciju Brownovog gibanja, tj. konstruirali smo Brownovo gibanje kao uniformni limes neprekidnih funkcija, što automatski povlači da taj proces ima neprekidne trajektorije. U drugom poglavlju opisali smo i dokazali razna zanimljiva svojstva Brownovog gibanja. Neka jednostavnija svojstva su invarijantnost na skaliranje, na vremensku inverziju, na simetriju i na obnavljanje. Svojstvo invarijantnosti Brownovog gibanja na obnavljanje smo koristili u dokazu Markovljevog svojstva Brownovog gibanja koje zapravo znači da Brownovo gibanje nema memoriju ni povijest. Nadalje, dokazali smo i jako Markovljevo svojstvo Brownovog gibanja koje smo primijenili da bismo pokazali princip refleksije Brownovog gibanja. Princip refleksije je svojstvo koje kaže da je Brownovo gibanje, koje se u nekom vremenu zaustavljanja krene reflektirati, i dalje Brownovo gibanje. Na kraju drugog poglavlja pokazali smo i da je Brownovo gibanje martingal. U trećem poglavlju ispitivali smo regularnost trajektorija Brownovog gibanja. Poglavlje smo podijelili na dva dijela. U prvom dijelu smo ispitivali i pokazivali svojstva neprekidnosti Brownovog gibanja preko Lévyjevog modula neprekidnosti, a u drugom dijelu smo, koristeći razne rezultate poput Hewitt-Savageovog 0-1 zakona i Fatouove leme, dokazali da Brownovo gibanje gotovo sigurno nije nigdje diferencijabilno. Za kraj smo pokazali najjednostavnije načine za simuliranje Brownovog gibanja. Simulacija Brownovog gibanja se temelji na nezavisnosti i normalnoj distribuciji prirasta Brownovog gibanja. Izveli smo nekoliko algoritama i primjera za simuliranje uniformne i normalne distribucije, a na kraju smo simulirali Brownovo gibanje koristeći nezavisnost i normalnu distribuiranost njegovih prirasta. Less

Abstract (english)

In this thesis we have defined and described Brownian motion, its construction, some interesting properties and methods for its simulation. At the beginning we have defined Brownian motion and described it as a Gaussian process. For this description we needed definitions of standard normal random vectors, normal random vectors and Gaussian processes. By using these definitions, we could describe Brownian motion as Gaussian process with continuous sample paths, with mean $m=0$... More and covariance function $\gamma_{st} = s \wedge t$. In addition, we have proven another properties of standard normal random vectors. Then, we have described Lévy’s construction of Brownian motion, i.e. we have constructed Brownian motion as a uniform limit of continuous functions, to ensure that it automatically has continuous paths. In the second chapter, we have described and proven various interesting properties of Brownian motion. Some of the simplest properties are scaling invariance property, time inversion, symmetry invariance property and renewal invariance property. We have used the renewal invariance property of Brownian motion in the proof of Markov property of Brownian motion. The fact that Brownian motion has Markov property means that Brownian motion does not have memory or history. Furthermore, we have proven strong Markov property of Brownian motion, which we have used to prove the reflection principle of Brownian motion. The reflection principle is property which states that Brownian motion reflected at some stopping time is still a Brownian motion. At the end of the second chapter we have shown that Brownian motion is also a martingale. In the third chapter, we have examined the regularity of Brownian motion. We have divided the chapter into two parts. In the first part we have examined and proven some continuity properties of Brownian motion by using the Lévy’s modul of continuity. In the second part, we have proven, by using results such as Hewitt-Savage 0-1 Law and Fatou’s lemma, that Brownian motion is, almost surely, nowhere differentiable. In the end, we have shown the simplest ways to simulate Brownian motion. The simulation of Brownian motion is based on the independence and normal distribution of increments of Brownian motion. We have shown several examples and algorithms for simulating uniform and normal distribution, and finally we have simulated Brownian motion by using independence and normal distribution of its increments. Less

• Keywords: Brownovo gibanje; Gaussovski proces s neprekidnim trajektorijama; Lévyjeva konstrukcija Brownovog gibanja; svojstvo invarijantnosti Brownovog gibanja; refleksije Brownovog gibanja; martingal; Hewitt-Savageov 0-1 zakon; Fatouova lema; simulacija Brownovog gibanja
• Keywords (english): Brownian motion; Gaussian process with continuous sample path; Lévy’s construction of Brownian motion; invariance property of Brownian motion; reflection principle of Brownian motion; Hewitt-Savage 0-1 Law; Fatou’s lemma; simulation of Brownian motion
• Language: croatian
• URN:NBN: urn:nbn:hr:217:636410
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-14 11:16:23