Modeling Dependencies in Multivariate Counting Processes
Specialeforsvar ved Lars Lynne Hansen.
Abstract:
The motivation for this thesis is to investigate how to model dependencies in multivariate counting processes. The focus is on semi-parametric models and the theory necessary to formulate the models in a probabilistic frame-work. We develop the theory of counting processes and review a recent result giving a criterion for the existence of counting processes through measure change on general ltered spaces. The result builds on a Theorem of Lepingle and Memin on when the exponential martingale is a martingale which we prove in a suitable setting. We discuss various algorithms for simulating counting process and end the thesis by applying the theory to the Hawkes processes and modulated renewal processes.
Motivationen for denne opgave er at unders ge hvordan man kan modellere vekselvirkninger i erdimensionelle tlleprocesser. Fokus er pa semi-parametriske modeller og teorien som er n dvendig for at formulere disse modeller i en sandsynlighedsteoretisk ramme. Vi laver en fremstilling af teorien for tlleprocesser og viser et nyligt resultat som giver et kriterium for eksistensen af tlleprocesser ved malskift pa generelle ltrerede rum.
Resultatet bygger pa et teorem af Lepingle og Memin for hvornar den eksponentielle martingal er en martingal, hvilket vi viser i en passende version. Vi diskuterer algoritmer for simulering af tlleprocesser og anvender den givne teori pa Hawkes processer og modulerede fornyelsesprocesser.
Vejleder: Niels Richard Hansen
Censor: Alexander Sokol