University of Zagreb. Faculty of Science. Department of Mathematics.
Abstract
Cilj diplomskog rada je opisati pojam subordinatora te prikazati osnovne primjere i primjenu subordinatora. Kako bismo mogli opisati teoriju subordinatora, morali smo opširno definirati dio teorije Lévyjevih procesa i Poissonovu točkovnu mjeru. Poissonovu točkovnu mjeru uvodimo kako bi prikazali subordinator kao proces koji odgovara intuitivno složenom Poissonovom procesu s linearnim pomakom. U radu smo opisali svojstva slike subordinatora te prikazali asimptotsko ponašanje puteva pomoću Dynkin-Lampertijevog teorema. Na kraju, kao primjer primjene subordinatora, definiramo teoriju rizika te dajemo primjer direktne primjene subordinatora u osiguranju opisujući složeni Poissonov model (poznatiji kao Crámer-Lundbergov model u aktuarskom kontekstu).The main goal of this thesis is to define and present basics examples and application of subordinators. In order to describe the theory of subordinators, theory of Lévy processes and Poisson random measures are discussed first. We introduce Poisson random measure in order to present subordinator as a process that correspond essentially to compound Poisson process with linear drift. We describe characteristics of the range of subordinator and show asymptotic behaviour of its paths using Dynkin-Lamperti theorem. In the last chapter, to present applications, we define ruin probability and usage of subordinator in insurance by describing compound Poisson model (known as Crámer-Lundberg model in actuarial contest)
