Abstract
The object of this paper is the study of some asymptotic properties of the perturbed risk process with delayed claims ( X t ), which is the sum of a Brownian motion with drift and a shot-noise whose underlying point process is a doubly stochastic Poisson process. More in particular, under suitable hypotheses, we show that ( X t ) satisfies a large deviation principle, and we give asymptotic estimates of the corresponding ruin probabilities. Moreover, we introduce two suitable processes ( L t ( X) ) and ( R t ( X) ), which can be seen as simplified versions of ( X t ), and we show some inequalities between the rate function and the Lundberg parameter concerning ( X t ), and the rate functions and the Lundberg parameters concerning ( L t ( X) ) and ( R t ( X) ), respectively.
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