Asymptotic results for perturbed risk processes with delayed claims

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

“…E-valued random variables, independent of (N (t)). This model has been studied by Brémaud (2000), who proved a Lundberg's type inequality and a Cramér-Lundberg type approximation for the corresponding infinite horizon ruin probability ψ (2) (u), by Torrisi (2004), who gave a Monte Carlo algorithm for fast simulation of ψ (2) (u) as u → ∞, under a suitable small claim assumption, and by Macci and Torrisi (2004), and Macci, Stabile and Torrisi (2005). In this paper we combine the ideas underlying the models (2) and (3), considering risk processes which account for reserve-dependent premium rate as well as delay in claim settlement.…”

Ruin probabilities and aggregrate claims distributions for shot noise Cox processes

“…Except for Dassios & Jang [7], who as here consider a Cox process with a shot-noise intensity, the risk process is most often modelled directly as a shot-noise process. In this setting, Klüppelberg & Mikosch [17] gave a diffusion approximation, whereas ruin estimates are in Bremaud [5] and Macci et al [20,21]. Dassios & Jang [7] studied the case g(s) = e −δs of the multiplicative model and used the theory of piecewise deterministic Markov processes (PDMP) to obtain the distribution of the aggregate claim amount under an equivalent Esscher measure.…”

Lundberg parameters for non standard risk processes

“…Proposition 3.1 can be proved by adapting the proofs in Section 2 of Macci and Torrisi [9]; see Appendix for details.…”

“…Define the conditional distributions of H (i ) (, Z k We prove Proposition 3.1 using similar techniques as in Section 2 of Macci and Torrisi [9]. We start with the following Lemma A.1 whose proof is omitted since it is similar to the proof of Lemma 2.3 of Macci and Torrisi [9].…”