Granični teorem za procese rizika

master's thesis

• Title (english): The limit theorem for risk processes
• Author: Lana Puščak
• Mentor: Nikola Sandrić (mentor)
• Committee member: Nikola Sandrić (predsjednik povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Committee member: Matej Mihelčić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Glavni cilj ovog rada je predstaviti procese rizika i granično ponašanje njihove distribucije te distribuciju vremena propasti. Na početku uvodimo osnovne pojmove poput metričkih prostora, konvergencije po distribuciji te konvergencije po vjerojatnosti. Zatim razmatramo pojam Brownovog gibanja, odnosno Wienereovog procesa te Wienerove mjere. Također, bavimo se osnovnim konceptima teorije obnove koji su važni za konstruiranje procesa rizika. Nadalje, dajemo neke najpoznatije modele procesa rizika: Cramer-Lundbergov i Iglehartov model rizika te navodimo i dokazujemo teorem o graničnoj distribuciji procesa rizika. Na kraju, u trećem poglavlju dajemo eksplicitne izraze za granično ponašanje distribucije procesa rizika i distribuciju vremena propasti što je ujedno i glavni rezultat teorije kolektivnog rizika.

Abstract (english)

The main goal of this master thesis is to present the risk processes and the limiting behavior of their distribution and the distribution of their ruin time. At the beginning, we recall the notion of basic terms such as metric spaces, convergence in distribution and convergence in probability. We then discuss the concept of Brownian motion, i.e. Wiener process and Wiener measure and recall some standard results from the standard renewal theory. Furthermore, we present some of the most classical risk process models: Cramer-Lundberg and Iglehart’s risk model, and state and prove the limiting distribution theorem for the risk process. Finally, in the third chapter, we give explicit expressions for the limiting behavior of the distribution of the risk process and distribution of the ruin time, which are the main result of the collective risk theory.

• Keywords: procesi rizika; distribucija vremena propasti; Brownovo gibanje; Wienereov proces; Wienerova mjera; Cramer-Lundbergov model rizika; Iglehartov model rizika
• Keywords (english): risk processes; fdistribution of ruin time; Brownian motion; Wienere's process; Wiener measure; Cramer-Lundberg risk model; Iglehart’s risk model
• Language: croatian
• URN:NBN: urn:nbn:hr:217:553348
• 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: 2023-08-18 08:02:11