Teorija propasti i de Vylderova aproksimacija

master's thesis

• Author: Anamarija Mešić
• Mentor: Hrvoje Šikić (mentor)
• Committee member: Hrvoje Šikić (predsjednik povjerenstva)
• Committee member: Luka Grubišić (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-14, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo promatrali Cramér-Lundbergov model u teoriji rizika. On se sastoji od pretpostavke da je proces isplata šteta složen Poissonov. U prvom poglavlju smo definirali složeni Poissonov proces, te smo dali još neke definicije i izvode koje kasnije koristimo. Definirali smo proces dobitka kao razliku zbroja početnog kapitala i premija te isplata šteta do trenutka t. Izveli smo formulu za vjerojatnost da taj proces bude manji od 0 u nekom trenutku t, te dali gornju ogradu za tu vjerojatnost. Uz vjerojatnost propasti, definirali smo i vjerojatnost preživljenja. S obzirom da se u nekim slučajevima formula za vjerojatnost propasti ne može eksplicitno izračunati, izveli smo i neke aproksimacije vjerojatnosti propasti. U trećem poglavlju smo definirali dvije aproksimacije, De Vylderovu aproksimaciju i 4MGDV aproksimaciju. U slučaju De Vylderove aproksimacije, vidjeli smo da u slučaju eksponencijalno distribuiranih šteta aproksimacija daje egzaktne rezultate. Također smo istaknuli da 4MGDV aproksimacija daje bolje rezultate od De Vylderove aproksimacije.

Abstract (english)

In this study we observed the Cramér-Lundberg model in the theory of risk. It consists of the assumption that the aggregate claims process is a compound Poisson process. In the first chapter we defined a compound Poisson process, and gave relevant definitions and extracts that we would later use. We defined a surplus process as opposed to the sum of the insurer’s surplus at time 0 and the insurer’s rate of premium income per unit time and aggregate claims at time t. We deduced a formula for the probability that the surplus process is less than 0 at some time t, and gave an upper bound for that probability. In addition to the ruin probability, we also defined the survival probability. Since in some cases the formula for the probability of ruin can not be explicitly calculated, we established some approximations thereof. In the third chapter, we define two approximations, the De Vylder approximation and the 4MGDV approximation. In case of exponentially distributed individual claim amounts, we observed that the De Vylder approximation gives exact results. We also concluded that the 4MGDV approximation gives better results than the De Vylder approximations.

• Keywords: Cramér-Lundbergov model; teorija rizika; De Vylderovu aproksimacija; 4MGDV aproksimacija
• Keywords (english): Cramér-Lundberg model; risk theory; De Vylder approximation; 4MGDV approximation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:502204
• 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: 2019-01-23 11:57:49