Asymptotic results for perturbed risk processes with delayed claims

The object of this paper is the study of some asymptotic properties of the perturbed risk process with delayed claims (Xt), 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 (Xt) satisfies a large deviation principle, and we give asymptotic estimates of the corresponding ruin probabilities. Moreover, we introduce two suitable processes (Lt(X)) and (Rt(X)), which can be seen as simplified versions of (Xt), and we show some inequalities between the rate function and the Lundberg parameter concerning (Xt), and the rate functions and the Lundberg parameters concerning (Lt(X)) and (Rt(X)), respectively.