Aspects of risk theory
type: Item
instance of: scholarly article
title: Aspects of risk theory
zbMATH Open document ID: 0717.62100
author: Jan Grandell
published in: Springer Series in Statistics
publication date: 1992-09-17T00:00:00Z
review text: In collective risk theory, a class of stochastic processes is considered which can describe the actual capital that an insurance company has at its disposal and which is governed by random arrival epochs of claims and by radom amounts of money to be paid by the company at these arrival epochs. The classical risk model assumes that the claims arrive according to a homogeneous Poisson process and that the costs of the claims can be described by i.i.d. random variables which are independent of the arrival epochs. Between consecutive arrival epochs of claims the capital is assumed to increase with a constant (and deterministic) rate. An important characteristic of these so-called risk processes is the ruin probability \(\Psi\) (0), i.e. the probability that at any time the actual capital of the company will be less than the initial capital. Furthermore, one is interested in the ruin probability \(\Psi\) (u) that at any time the actual capital undercuts the initial capital by more then u units. The ruin probability \(\Psi\) (0) can be easily expressed by the intensity of the underlying Poisson process describing the arrival epochs of claims, by the expected costs of a single claim, and by the rate of income. In distinction to this, for \(u>0\) the determination of \(\Psi\) (u) is in general much more complicated because, for \(u>0\), \(\Psi\) (u) depends on the cost distribution not only via its expectation. As a consequence of this, bounds and approximation formulas for \(\Psi\) (u) have been derived in the literature, e.g. the Cramér-Lundberg approximation \[ (1)\quad \min_{u\to \infty}e^{Ru}\Psi (u)=C \] where the Lundberg exponent R is given by a certain functional equation, and the Lundberg inequality \[ (2)\quad \Psi (u)\leq e^{-Ru}\text{ for } every\quad u\geq 0. \] Besides a comprehensive explanation of such results for the classical risk model, the present book mainly deals with the question how the consideration of more general sequences of arrival epochs is re…
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60K20
Mathematics Subject Classification ID: 60K10
zbMATH DE Number: 48883
zbMATH Keywords: collective risk theory
zbMATH Keywords: Poisson process
zbMATH Keywords: arrival epochs of claims
zbMATH Keywords: risk processes
zbMATH Keywords: ruin probability
zbMATH Keywords: expected costs
zbMATH Keywords: rate of income
zbMATH Keywords: bounds
zbMATH Keywords: approximation
zbMATH Keywords: Cramér-Lundberg approximation
zbMATH Keywords: Lundberg exponent
zbMATH Keywords: Lundberg inequality
zbMATH Keywords: non-Poisson arrival processes
zbMATH Keywords: insensitivity property
zbMATH Keywords: exponentially distributed costs
zbMATH Keywords: cost distributions
zbMATH Keywords: numerical calculations
zbMATH Keywords: ordinary renewal model
zbMATH Keywords: stationary renewal model
zbMATH Keywords: inter- occurrence time distributions
zbMATH Keywords: Cox arrival process
zbMATH Keywords: risk fluctuations
zbMATH Keywords: Markov process
zbMATH Keywords: Markov renewal process
zbMATH Keywords: general stationary point process
zbMATH Keywords: finite time ruin probabilities
zbMATH Keywords: claim behavior
zbMATH Keywords: martingale techniques
reviewed by: Volker Schmidt
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 48883