Struktura Levyjevih procesa, subordinatori i primjene

master's thesis

• Author: Tomislav Kralj
• Mentor: Zoran Vondraček (mentor)
• Committee member: Zoran Vondraček (predsjednik povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Snježana Lubura Strunjak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatramo Lévyjeve procese, subordinatore te primjene u regenerativnim strukturama. U prvom poglavlju uvodimo formalnu definiciju Lévyjevih procesa te s njima povezanih beskonačno djeljivih distribucija. Uvodimo karakteristične funkcije te iskazujemo osnovni teorem o karakterističnom eksponentu beskonačno djeljive distribucije. Konstruiramo i dajemo tri najvažnija primjera Lévyjevih procesa: Poissonov točkovni proces, složen Poissonov proces te Brownovo gibanje. U drugom poglavlju iskazujemo i dokazujemo fundamentalan teorem Lévy-Itôve dekompozicije. To činimo konstrukcijom apstraktne Poissonove slučajne mjere te integracijom odgovarajućih determinstičkih funkcija s obzirom na Poissonovu slučajnu mjeru. U trećem poglavlju promatramo neopadajuće Lévyjeve procese, tzv. subordinatore. Uvodimo pojmove potencijalnih mjera i mjera obnavljanja. Promatramo vezu subordinatora i procesa obnavljanja te dokazujemo asimptotiku relativnog prebačaja i podbačaja. U četvrtom poglavlju promatramo slučajne kompozicije i particije nenegativnih cijelih brojeva, tzv. kompozicijske strukture te koristimo teoriju subordinatora kako bismo izgradili takve strukture.

Abstract (english)

In this master’s thesis, we study Lévy processes, subordinators and their applications in random combinatorial structures. In the first chapter we introduce formal definition of Lévy processes and infinitely divisible distributions that are related to them. Characteristic functions are widely implemented in this thesis. Therefore, we state basic theorem about the characteristic exponent of infinitely divisible distribution. Three most important examples of Lévy processes are given: Poisson point process, compound Poisson process and Brownian motion. Lévy-Itô decomposition is the essential topic of second chapter. We prove that theorem using construction that involves Poisson random measures and integration of non-random functions with respect to Poisson random measure. In third chapter we investigate non-decreasing Lévy processes, subordinators. Introducing potential measures and renewal measures, we examine the connection between subordinators and renewal processes and we prove the asymptotic behaviour of relative overshoot and undershoot. Random composition and partition structures represent key concepts of the concluding chapter. Subordinator theory is used to construct regenerative composition structures.

• Keywords: Lévyjevi procesi; beskonačno djeljive distribucije; Poissonov točkovni proces; složen Poissonov proces; Brownovo gibanje
• Keywords (english): Lévy processes; infinitely divisible distributions; Poisson point process; compound Poisson process; Brownian motion
• Language: croatian
• URN:NBN: urn:nbn:hr:217:379822
• 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: 2020-11-25 13:36:54