Presjek funkcija propasti procesa rizika

master's thesis

• Author: Ana Findak
• Mentor: Nikola Sandrić (mentor)
• Committee member: Nikola Sandrić (predsjednik povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatramo vjerojatnosne funkcije propasti. U prvom dijelu definirani su Lévyjevi procesi, razmotrena njihova osnovna svojstva te primjeri. Definirani su sljedeći ključni pojmovi: karakteristična funkcija funkcije distribucije, beskonačno djeljive i stabilne distribucije, karakteristični eksponent i Lévyjeva mjera beskonačno djeljive distribucije. Zatim su navedeni primjeri Lévyjevih procesa te izračunate njihove karakteristične funkcije i Lévyjeve trojke. U drugom dijelu rada bavimo se presjecima funkcija propasti. Definirani su pojmovi koji su nam bitni u računanju funkcija propasti: Laplaceov eksponent, Laplaceova transformacija, funkcija skale. Izračunate su funkcije propasti klasičnog Cramer-Lundbergovog procesa i \(\alpha-\)stabilnog spektralno negativnog procesa te razmotren broj sjecišta dvije funkcije propasti. Zatim je promatran klasični model perturbiran \(\alpha-\) stabilnim spektralno negativnim procesom te je na kraju iskazan rezultat koji daje uvjete za proizvoljan broj sjecišta dviju funkcija propasti.

Abstract (english)

In this thesis, we study ruin probability functions. In the first part, we define the notion of a Levy process. Also, we discuss certain properties and provide examples of these processes. In particular, we discuss the notion of characteristic function of a distribution function, infinitely divisible and stable distribution functions, characteristic exponent and Lévy measure of an infinitely divisible distribution function. At the end we provide several classical examples of Lévy processes, and compute their Lévy characteristics. In the second part of the thesis, we deal with intersections of ruin probability functions. We discuss the notion of Laplace exponent, Laplace transform and scale function. In particular, we explicitly compute ruin probability of the classical risk process and spectrally negative \(\alpha-\)stable process and discuss intersections between two ruin probability functions. Then we discuss a classical risk model perturbed by a \(\alpha-\)stable spectrally negative Lévy processes. In the end, we state a result which gives sufficient conditions for arbitrary number of intersections of two ruin probability functions.

• Keywords: funkcije propasti; Lévyevi procesi; Laplaceov eksponent; Laplaceova transformacija; funkcija skale
• Keywords (english): ruin probability functions; Lévy processes; Laplace exponent; Laplace transform; scale function
• Language: croatian
• URN:NBN: urn:nbn:hr:217:163192
• Study programme: Title: Mathematical Statistics; 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: 2019-03-28 13:18:09