Compound Poisson process with a Poisson subordinator

A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing-time density and survival functions.Comment: 16 pages, 7 figure

Poissonov proces i subordinatori

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)

Limit theorems for the fractional non-homogeneous Poisson process

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation

Poissonov proces i subordinatori

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)

Poissonov proces i subordinatori

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)