Limit theorems for generalized risk processes
type: Item
instance of: scholarly article
zbMATH Open document ID: 0988.60008
author: V. Yu. Korolev
author: Lixinb Liu
author: Vladimir E. Bening
publication date: 2002-03-13T00:00:00Z
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 1718878
zbMATH Keywords: limit theorems
zbMATH Keywords: limit law
zbMATH Keywords: Cox process
zbMATH Keywords: risk process
MaRDI profile type: Publication
title: Limit theorems for generalized risk processes
published in: Theory of Stochastic Processes
review text: Consider the classical risk process \(R_0(t)=ct-\sum_{j=1}^{N_1(t)}X_j, t\geq 0,\) where \(c>0\) is the premium rate, \(\{X_j\}_{j\geq 1}\) are insurance claims (payments) which are assumed to be independent identically distributed random variables (r.v.'s) with mean value \(a\), \(N_1(t)\) is the number of claims up to time \(t\) which is assumed to be the standard Poisson process (homogeneous Poisson process with unit intensity) independent of \(\{X_j\}_{j\geq 1}\). Let \(\Lambda(t),t>0,\) be a random process independent of \(N_1(t)\) with infinitely increasing almost surely finite continuous trajectories starting from the origin. A Cox process controlled by the process \(\Lambda(t)\) is defined as \(N(t)=N_1(\Lambda(t))\). The authors consider a natural generalization of the classical risk process \(R(t)=c\Lambda(t)-\sum_{j=1}^{N(t)}X_j, t\geq 0,\) which is a more flexible mathematical model for the surplus of an insurance company since it takes into account both risk and portfolio fluctuations. In the present paper the authors give a new general criterion for the weak convergence of one-dimensional distributions of the generalized risk processes \(R(t)\) and describe the class of possible limit laws as \(\Lambda(t)\) infinitely grows, which makes it possible to construct asymptotic approximations to distributions of these generalized risk processes.
reviewed by: M. P. Moklyachuk
description: scientific article; zbMATH DE number 1718878