Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation
type: Item
instance of: scholarly article
title: Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation
zbMATH Open document ID: 1192.91111
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-05-10T00:00:00Z
review text: In a renewal risk model or a Markov-modulated risk model \(\{X_t\}\), the time spent in the negative \(\tau(u,T) = \int_0^T \text{1}_{\{u + X_t < 0\}}\; d t\) and the integrated negative part \(I_T(u) = \int_0^T (-u-X_t)^+ \;d t\) are studied. For a claim size distribution with a regularly varying tail, the asymptotic behavior of \(E[\tau(u,\infty)]\), \(E[I_\infty(u)]\) and \(E[I_T(u)]\) is calculated. For \(E[I_T(u)]\), the result is obtained for subexponential claim size distributions. Then the light-tail case is considered and the asymptotics of \(E[\tau(u,\infty)]\) and \(E[I_\infty(u)]\) are obtained in the case where the Lundberg exponent exists. The main part of the article deals with two independent Markov modulated models. The question is to allocate the initial reserve \(u = u_1 + u_2\), such that \(I_T(u_1,u_2) = E[I_T^1(u_1)] + E[I_T^2(u_2)]\) becomes minimal, where \(I_T^j(u) = \int_0^T (-u-X^j_t)^+ \;d t\) is the time spent in the negative of the \(j\)-th process. The optimal allocation has to fulfill \[ \frac{\partial E[I_T^1(u_1)]}{\partial u_1} = \frac{\partial E[I_T^2(u_2)]}{\partial u_2} \;. \] Writing \(u_2 = \beta(u)\), the asymptotic behavior of \(\beta(u)\) is given in the regular varying case, if the distribution tail of the claim sizes of the first process is heavier. Here, \(T\) can be both finite or infinite. The disadvantage in the infinite horizon case is, that the coefficient of regular variation has to be larger than three. Since in practical situations, the coefficient often lies between one and two, the most interesting case is not covered. For the light-tailed case with \(T = \infty\), it is also shown that \(u_1/u_2\) tends to a constant. The proofs rely on the fact, that in the asymptotic expression for \(E[I_T^j(u_j)]\), limit and derivative can be interchanged.
reviewed by: Hanspeter Schmidli
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 5706224
zbMATH Keywords: ruin theory
zbMATH Keywords: heavy-tailed and light-tailed claim size distribution
zbMATH Keywords: risk measure
zbMATH Keywords: optimal reserve allocation
author: Romain Biard
author: Stéphane Loisel
author: Claudio Macci
author: Noël Veraverbeke
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.01.051
OpenAlex ID: W2049624494
cites work: Differentiation of some functionals of risk processes, and optimal reserve allocation
cites work: Estimates for the probability of ruin with special emphasis on the possibility of large claims
cites work: The probability of ruin in finite time with discrete claim size distribution
cites work: Another look at the Picard--Lefèvre formula for finite-time ruin probabilities
cites work: On finite-time ruin probabilities for classical risk models
cites work: Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed
cites work: Finite-time ruin probabilities for discrete, possibly dependent, claim severities
cites work: Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin
cites work: Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes
cites work: Sensitivity analysis and density estimation for finite-time ruin probabilities
cites work: Mathematical fun with ruin theory
cites work: The surpluses immediately before and at ruin, and the amount of the claim causing ruin
cites work: On some measures of the severity of ruin in the classical Poisson model
cites work: How long is the surplus below zero?
cites work: Q4343010
cites work: Subexponentiality and infinite divisibility
cites work: Large deviations for the time-integrated negative parts of some processes
cites work: Q4391441
DOI: 10.1016/J.JMAA.2010.01.051
description: scientific article; zbMATH DE number 5706224