Poissonov proces s klasterima

Cilj je ovog diplomskog rada bio proučiti Poissonov proces s klasterima i njegova osnovna svojstva. Nakon prvoga poglavlja, čitatelju su poznate definicija točkovnih procesa, mjera njihovih momenata te primjene tzv. Laplaceove transformacije i Laplaceova funkcionala točkovnoga procesa. Navedeno je i nekoliko primjera točkovnih procesa te najvažniji točkovni proces, općeniti Poissonov točkovni proces ili Poissonova slučajna mjera (PRM). Pokazano je i kako se dodavanjem nezavisne koordinate točkama poznatoga PRM-a može konstruirati novi PRM. Dalje je navedena definicija i jedna karakterizacija općenitih točkovnih procesa s klasterima. Pretpostavljamo da nam je poznat proces centara klastera. Točke tog procesa generiraju klastere koji su također točkovni procesi. Zbrajanje elemenata po klasterima čini opaženi proces. Ako su središta klastera točke Poissonova procesa, riječ je o Poissonovom procesu s klasterima. U zadnjem poglavlju bavimo se primjenom Poissonova procesa s klasterima u matematici neživotnoga osiguranja. Točku Poissonova procesa interpretiramo kao vrijeme dolaska zahtjeva za isplatu štete, a klaster koji ona uzrokuje opisuje vremena i iznose isplate tog zahtjeva. Proučavamo i chain ladder model. Na kraju se bavimo Poissonovim procesima s Poissonovim klasterima. Analiziramo njihove prve i druge momente kako bismo predvidjeli broj i ukupan iznos isplata šteta. Chain ladder model i Poissnov proces s klasterima često se koriste za procjenu pričuva, što je jedan od najvažnijih praktičnih problema u aktuarstvu.The aim of this thesis was to study the Poisson cluster process as well as its basic properties. After the first chapter the reader is familiar with the definition of point processes, their moment measures, and the application of the so called Laplace transform and the Laplace functional of the point processes. There are several examples of point processes including the most important one, the general Poisson point process or Poisson random measure (PRM). It is also shown how one can adhere an independent coordinate to the points of given PRM to construct a new PRM. In addition, there is a definition and a characterization of general point process with independent clusters. We suppose a point process of cluster centers is known. The points of the center process generate the clusters, which are also point processes. The superposition of the elements within clusters constitutes the observed process. If the cluster centers are the points of a Poisson process, we speak of a Poisson cluster process. In the final chapter we consider the applications of the Poisson cluster process in non-life insurance mathematics. A Poisson process point is interpreted as the arrival time of a claim, and the cluster that point triggers describes the times and amounts of the payment for this particular claim. We also study the chain ladder model. Lastly, we concentrate on the Poisson processes with Poisson clusters. We analyze the first and second moments of those processes in order to predict the claim number and total claim amounts. Chain ladder model and Poisson cluster process are often used to estimate reserves, which is one of the most important insurance practice problems