The Pólya-Aeppli process and ruin problems
type: Item
instance of: scholarly article
title: The Pólya-Aeppli process and ruin problems
zbMATH Open document ID: 1127.91037
DOI: 10.1155/S1048953304309032
author: L. D. Minkova
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2005-03-21T00:00:00Z
full work available at URL: https://eudml.org/doc/53628
review text: The investigations of this paper are motivated by a standard model of an insurance company described by a counting process \(N\left( t\right) ,t\geq 0,\) interpreted as the number of claims on the company during the interval \(\left[ 0,t\right] ,\) a sequence of mutually i.i.d. random variables \(\left\{ Z_{k}\right\} _{k=1}^{\infty }\) with common distribution function which is independent of \(N\left( t\right) ,\) a positive real constant \(c\) representing the gross premium rate and the risk process \( \left\{ X\left( t\right) ,t\geq 0\right\} \)\ defined on a complete probability space \(\left( \Omega ,\mathcal{F},P\right) \) by: \[ X\left( t\right) =ct-\sum_{k=1}^{N\left( t\right) }Z_{k} \] \(S\left( t\right) =\sum_{k=1}^{N\left( t\right) }Z_{k}\) is the aggregate claim up to time \(t.\) \ If the number of claims \(N\left( t\right) \) forms a renewal process the model is called a \textit{renewal risk model}. In the classical risk model the counting process is a stationary Poisson counting process. The author is interested in the generalization of the counting process to a family of probability distributions, called \textit{inflated parameter distributions}, which she has introduced in earlier papers, generalizing the classical Poisson, binomial, negative binomial and logarithmic distributions by adding a new parameter \(\rho \in [ 0,1).\) An interpretation of \(\rho \) would be probability of 'failure' where there are two types of claims possible on the insurance company, 'success' or 'failure' with the probability of the latter being \(1-\rho .\) \ If the r.v. \(N \) is the number of trials until the \(i^{th}\) successive claim appears, it is distributed according to the negative binomial with parameters \(1-\rho \) and \(i.\) The r.v. \(N\) can be represented as the sum of \(i\) independent identically \(Ge_{1}\left( 1-\rho \right) \) distributed r.v.'s \(\left\{ X_{j}\right\} _{j=1}^{i}.\) \ Supposing that \(i\) is an outcome of the r.v. \(Y\) ,\(\;\)indepen…
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 2148076
zbMATH Keywords: risk process
zbMATH Keywords: Polya-Aeppli distribution
zbMATH Keywords: renewal process
zbMATH Keywords: ruin probability
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2148076