20.500.12556/RUL-160474 Sestavljeni Poissonov proces in njegova uporaba v financah delo diplomskega seminarja Compound Poisson process and its application in finance V prvem delu diplome najprej definiramo sestavljeno Poissonovo porazdelitev in izpeljemo obliko njenih rodovnih funkcij, obravnavamo njeno povezavo s splošnimi porazdelitvami in izpeljemo Panjerjevo rekurzivno shemo. Nato definiramo sestavljeni Poissonov proces in pokažemo nekaj osnovnih lastnosti kot je neodvisnost in stacionarnost prirastkov. Izpeljemo nekaj rezultatov, ki jih dobimo, ko sestavljeni Poissonov proces markiramo. V drugem delu diplome obravnavamo aplikacijo sestavljenega Poissonovega procesa v Cramér-Lundbergovem modelu. Definiramo verjetnost propada in preživetja ter slednjo izrazimo z defektno prenovitveno enačbo. Dokažemo Lundbergovo neenakost in obravnavamo asimptotično obnašanje verjetnosti propada, ko zahtevke modeliramo z lahkorepimi in težkorepimi porazdelitvami. Obnašanje verjetnosti propada na koncu praktično prikažemo z večkratnim simuliranjem procesa tveganja. In the first half of the diploma, we define the compound Poisson distribution and derive the form of its generating functions. We discuss its connection with general distributions and derive the Panjer recursion scheme. We then define the compound Poisson process and show some basic properties such as the independence and stationarity of increments. We derive some results that follow from a space-time decomposition of the compound Poisson process. In the second half of the diploma, we discuss the application of the compound Poisson process in the Cramér-Lundberg model. We define the probability of ruin and survival and express the latter as a defective renewal equation. We prove the Lundberg inequality and discuss the asymptotic behavior of the probability of ruin when claims are modeled with light-tailed and heavy-tailed distributions. We practically demonstrate the behavior of the probability of ruin by repeatedly simulating the risk process. slučajni proces sestavljena Poissonova porazdelitev Panjerjeva rekurzivna shema sestavljeni Poissonov proces markiranje Cramér-Lundbergov model Verjetnost propada lahkorepa porazdelitev težkorepa porazdelitev stochastic process compound Poisson distribution Panjer recursion scheme compound Poisson process space-time decomposition Cramér-Lundberg model probability of ruin light-tailed distribution heavy-tailed distribution true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2024-08-29 08:15:04 2024-08-29 08:15:07 2024-08-31 03:34:00 0000-00-00 00:00:00 2024 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.2 140897 205692419 12835.pdf 12835.pdf 1 FDFEBAC469A5673C86789B5DDCE954C3 84f980c736922d73f8b51a28dfbdcba9c2a73b64c884fd2f6e90529bdb9ba3b1 cf94ed94-65cd-11ef-8f74-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=189176 12836.R 12836.R 1 4FFBD1C9F14A28A6063B1AD8B4B896EB 9a844e39dd00890e80ab9ace3cd05b8f596f55df67f4ef9bdfd654a2740c7bc4 cf99a4cf-65cd-11ef-8f74-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=189177 Fakulteta za matematiko in fiziko 0 0 0