Explosive Poisson shot noise processes with applications to risk reserves
type: Item
instance of: scholarly article
title: Explosive Poisson shot noise processes with applications to risk reserves
zbMATH Open document ID: 0842.60030
DOI: 10.2307/3318683
author: T. Mikosch
author: Claudia Klüppelberg
published in: Bernoulli
publication date: 1995-12-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1186078364
review text: A shot-noise process is defined as \(S(t)= \sum X_n (t- T_n)\) where the \(T_n\) are the epochs of a Poisson process and the \(X_n\) are i.i.d. non-negative random functions. In contrast to most of the literature where the \(X_n\) are assumed to be non-increasing with limit 0, this paper considers the non-decreasing case. Laws of large numbers and central limit theorems for \(S(t)\) are derived. Assuming regular variation of \(\text{Cov} (X (s), X(t))\), it is shown that the shot-noise process when properly scaled and normalised has a self-similar Gaussian limit with continuous sample paths. Conditions for the limit to be Brownian motion are investigated in the case \(\lim_{t\to \infty} X_n (t)< \infty\) a.s. As a corollary, ruin probability approximations for an insurance risk model are derived for the case where \(X_n (t)\) is the part of the \(n\)th claim which has been settled before \(t\) time units after its occurrence.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 62P05
zbMATH DE Number: 825067
zbMATH Keywords: functional central limit theorem
zbMATH Keywords: regular variation
zbMATH Keywords: ruin probability
zbMATH Keywords: self-similar process
reviewed by: Søren Asmussen
MaRDI profile type: Publication
OpenAlex ID: W2010463282
Recommended article: Tail behavior of Poisson shot noise processes under heavy-tailed shocks and actuarial applications
Recommended article: Q4627220
Recommended article: Exponential martingale and large deviations for a Cox risk process with Poisson shot noise intensity
Recommended article: Large deviations of Poisson shot noise processes under heavy tail semi-exponential conditions
Recommended article: Poissonian potential measures for Lévy risk models
Recommended article: Risk processes with shot noise Cox claim number process and reserve dependent premium rate
Recommended article: Q5468186
Recommended article: Asymptotic analysis of Poisson shot noise processes, and applications
Recommended article: Moment explosions in stochastic volatility models
description: scientific article; zbMATH DE number 825067