Lundberg parameters for non standard risk processes

Abstract

We consider risk processes with delayed claims in a Markovian environment, and \ud we study the asymptotic behaviour of \ud finite and infinite horizon ruin probabilities under the small claim\ud assumption. We also consider multivariate risk processes of the same kind, and we \ud give upper and lower bounds for the \ud Lundberg parameters of the corresponding total reserve. Our results have strong\ud analogies with those one in the paper by Juri (Super modular order and Lundberg exponents, 2002)

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.…”

Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion

“…There is a limited literature on such shot noise processes. In Macci et al (2005), the noises are modulated by a finite-state Markov chain and are conditionally independent with a distribution depending on the state of the chain at the arrival time of the shot noise. In Ramirez-Perez and Sering (2001), a cluster shot noise model is studied where the noises depend on the same "cluster mark" within each cluster.…”