Large claims approximations for risk processes in a Markovian environment
type: Item
instance of: scholarly article
title: Large claims approximations for risk processes in a Markovian environment
zbMATH Open document ID: 0814.60067
DOI: 10.1016/0304-4149(93)00003-X
author: Lotte Fløe Henriksen
author: Claudia Klüppelberg
author: Søren Asmussen
published in: Stochastic Processes and their Applications
publication date: 1995-06-18T00:00:00Z
review text: The paper provides some results concerning a risk process with initial reserve \(u\), which evolves in a finite Markovian environment with initial state \(i\). Denoting by \(\beta_ j\) the arrival intensity and by \(B_ j\) the claim size distribution when the environment is in the state \(j( \in E)\), the authors suppose that there is a subset \(E^{(1)}\) of \(E\) such that, for every \(j \in E^{(1)}\), \(B_ j\) verify the asymptotic condition \(1-B_ j(x) \sim b_ j (1 - H(x))\), as \(x \to \infty\), for some \(b_ j \in (0, + \infty)\) and some p.d.f. \(H\), whose tail is a subexponential density, and, for every \(j \in E \backslash E^{(1)}\), \(1 - B_ j(x) = o(1 - H(x))\). Under these hypotheses, denoting by \(\psi_ i (u)\) the probability of ruin for the above mentioned risk process, the authors prove that \(\psi_ i (u) \sim c_ i \int^ \infty_ u (1 - H(x))dx\), for some explicitly computed constant \(c_ i\). In the last part of the paper it is proved that similar results hold for the tail of the waiting time in a Markov-modulated \(M/G/1\) queue whenever the service times satisfy similar hypotheses.
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 60K25
zbMATH DE Number: 713915
zbMATH Keywords: Markov process
zbMATH Keywords: asymptotic ruin probability
zbMATH Keywords: non-Cramér case
zbMATH Keywords: subexponential distributions
zbMATH Keywords: probability of ruin
zbMATH Keywords: risk process
reviewed by: Emilia Di Lorenzo
MaRDI profile type: Publication
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full work available at URL: 0304-4149(93)00003-x
OpenAlex ID: W1998015333
description: scientific article; zbMATH DE number 713915