Title Marked Poisson cluster processes and applications Title (croatian) Označeni Poissonovi procesi s klasterima i primjene Author Petra Žugec Mentor Bojan Basrak (mentor)Committee member Nikola Sandrić (predsjednik povjerenstva)Committee member Bojan Basrak (član povjerenstva)Committee member Olivier Wintenberger (član povjerenstva)194001109 Committee member Azra Tafro (član povjerenstva)Granter University of Zagreb Defense date and country 2019-09-26, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 51 - Mathematics Abstract The theory of point processes constitutes an important part of modern stochastic process theory and is widely recognized as a useful and elegant tool for modelling. Point processes are well understood models and have applications in a wide range of applied probability areas, especially in risk theory which is important for understanding non-life insurance mathematics. It deals with the modelling of claims and gives answers on premium amount. Elegant mathematical analysis of the classical Cramér - Lundberg risk model has an important place in non-life insurance theory. The theory yields precise or approximate computations of the ruin probabilities, appropriate reserves, distribution of the total claim amount and other properties of an idealized insurance portfolio. In recent years, some special models have been proposed to account for the possibility of clustering of some events, for instance the Hawkes processes. We study asymptotic distribution of the total claim amount in the setting where Cramér - Lundberg risk model is augmented with a marked Poisson cluster structure. Marked Hawkes processes are then a special case and have an important role as the key example in our analysis. To make this more precise, we model arrival of claims in an insurance portfolio by a marked point process, say \[N = \sum_{k=1}^{\infty}\delta_{\tau_k,A^k}\] where \(\tau_k\)'s are non-negative random variables representing arrival times with some degree of clustering and \(A^k\)'s represent corresponding marks in a rather general metric space \(\mathbb{M}\). For each marked event, the claim size can be calculated using a measurable mapping of marks to non-negative real numbers, \(f: \mathbb{M}\) \rightarrow \mathbb{R}_{\geq 0}\) say. So that the total claim amount in the time interval \([0, t]\) can be calculated as \[S(t) = \sum_{\tau_k \leq t}f(A^k).\] We determine the effect of the clustering on the quantity \(S(t)\), as \(t\to \infty \), even in the case when the distribution of the individual claims does not satisfy assumptions of the classical central limit theorem. Besides new results regarding the case when second moments do not exist, we use different approach based on the limit theory for two dimensional random walks which stems from the classical Anscombe's theorem and not on martingale central limit theorem which was commonly used. We present the central limit theorem for the total claim amount \(S(t)\) in our setting under appropriate second moment conditions and prove a functional limit theorem concerning the sums of regularly varying non-negative random variables when subordinated to an independent renewal process. Based on this, we prove the limit theorem for the total claim amount \(S(t)\) in cases when individual claims have infinite variance. Moreover, we apply these results to three special models. In particular, we give a detailed analysis of the marked Hawkes processes which are extensively studied in recent years. In the last chapter we move our attention to the maximal claim size and present our results regarding limiting behaviour of maximum when claims belong to the maximum domain of attraction of one of the three extreme value distributions (Fréchet, Weibull and Gumbel). We also apply those results to three special models which we studied in previous chapter. Besides that, we try to clarify the notion of stochastic intensity which can be described in several different ways. The understanding of the stochastic intensity is important because of it's usage in the implicit definition of Hawkes processes.
Abstract (croatian) Teorija točkovnih procesa utemeljuje važan dio moderne teorije stohastičkih procesa i široko je prepoznata kao koristan i elegantan alat za modeliranje. Točkovni procesi su model koji se dobro razumije i koriste u mnogim područjima primijenjene vjerojatnosti, posebno u teoriji rizika (matematika neživotnih osiguranja). Teorija rizika se bavi modeliranjem zahtjeva za isplatom u svrhu određivanja visine premije. Elegantna matematička analiza Cramér-Lundbergovog modela rizika ima važnu ulogu u teoriji neživotnih osiguranja. Spomenuta teorija nam daje precizne ili aproksimativne izračune vjerojatnosti propasti, odgovarajućih rezervi, distribuciju sume zahtjeva za isplatama i druga svojstva idealiziranog portfelja osiguravatelja. Posljednjih godina predloženi su neki specijalni modeli koji uključuju mogućnost klasteriranja događaja, na primjer Hawkesovi procesi. Proučavamo asimptotske distribucije ukupnog iznosa zahtjeva za isplatom u označenim Poissonovim procesima s klasterima u kojima oznake određuju visinu, ali i druge karakteristike pojedinih zahtjeva te potencijalno utječu na stopu dolazaka budućih zahtjeva. Označeni Hawkesovi procesi u tom slučaju postaju specijani slučaj općenitog modela označenih Poissonovih procesa s klasterima. Malo preciznije, dolaske zahtjeva za isplatom u promatranom portfelju modeliramo označenim točkovnim procesom, npr. oblika \[N = \sum_{k=1}^{\infty}\delta_{\tau_k,A^k}\] pri čemu su \(\tau_k\) nenegativne slučajne varijable koje predstavljaju vremena dolazaka s nekim stupnjem klasteriranja, a \(A^k\) pripadne oznake u nekom metričkom prostoru \(\mathbb{M}\). Visina zahtjeva za isplatom u svakom označenom događaju može se izračunati upotrebom izmjerivog preslikavanja \(f(A^k)\) iz prostora oznaka u nenegativne realne brojeve. Tada se suma zahtjeva za isplatom u intervalu \([0, t]\) može izraziti kao \[S(t) = \sum_{\tau_k \leq t}f(A^k).\] Promatramo učinak klasteriranja na \(S(t),\) kada \(t\to \infty \) čak i u slučaju kada distribucija individualnih zahtjeva ne zadovoljava pretpostavke klasičnog centralnog graničnog teorema. Osim novih rezultata u slučaju kada drugi moment nije konačan, u izračunima koristimo drugačiji pristup koji se temelji na graničnim teoremima za zaustavljene dvodimenzionalne slučajne šetnje (koji proizlaze iz Anscombeovog teorema), a ne na martingalnom centralnom graničnom teoremu koji je često korišten. Prezentirat ćemo dovoljne uvjete uz koje ukupan iznos zahtjeva zadovoljava centralni granični teorem ili alternativno teži po distribuciji stabilnoj slučajnoj varijabli s beskonačnom varijancom. Diskutirat ćemo nekoliko Poissonnovih modela s klasterima, pri čemu će označeni Hawkesovi procesi biti naš ključni primjer. U posljednjem poglavlju fokus prebacujemo na maksimalan iznos zahtjeva za isplatom u intervalu \([0,t].\) Prezentirat ćemo rezultate vezane uz granično ponašanje maksimuma u slučaju kada pojedinačni zahtjevi za isplatama pripadaju Fréchetovoj, Weibullovoj ili Gumbelovoj maksimalnoj domeni privlačnosti. Ponovo primjenjujemo dobivene rezultate na tri specijalna modela (s posebnim naglaskom na Hawkesove procese). Osim graničnog ponašanja suma i maksimuma, pokušali smo razjasniti pojam stohastičkog intenziteta, posebno jer se u literaturi može pronaći nekoliko različitih definicija spomenutog stohastičkog intenziteta. Razumijevanje stohastičkog intenziteta nam je važno jer se koristi prilikom definiranja Hawkesovih procesa.
Keywords
Point process
Poisson cluster processes
limit theorems
Hawkes process
total claim amount
maximal claim size
Keywords (croatian)
Točkovni procesi
Poissonovi procesi s klasterima
granični teoremi
Hawkesovi procesi
suma zahtjeva za isplatom
maksimalan zahtjev za isplatom
Language english URN:NBN urn:nbn:hr:217:746225 Study programme Title: Mathematics Type of resource Text Extent xvi, 73 str. File origin Born digital Access conditions Open access Terms of use Created on 2019-12-06 10:46:13
