engleski

Marked Poisson cluster processes and application

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. We determine the effect of the clustering on the total claim amount 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 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 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.

Point process ; Poisson cluster processes ; limit theorems ; Hawkes process ; total claim amount ; maximal claim size

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