Small-time ruin for a financial process modulated by a Harris recurrent Markov chain
type: Item
instance of: scholarly article
title: Small-time ruin for a financial process modulated by a Harris recurrent Markov chain
zbMATH Open document ID: 1164.60036
author: Jeffrey F. Collamore
author: Andrea Höing
published in: Finance and Stochastics
publication date: 2009-02-28T00:00:00Z
full work available at URL: smalltime-ruin-for-a-financial-process-modulated-by-a-harris-recurrent-markov-chain(edba3bc0-960d-11de-8bc9-000ea68e967b).html
review text: The article deals with Markov-modulated risk processes of the form \(S_n=\zeta_1+ \dots +\zeta_n\), where the random variables (claims) \(\{\zeta_i\}\) are driven by a general Harris recurrent Markov chain \(\{X_n\}\). The main objective is to study the small time ruin probabilities (large exceedance probabilities), more precisely, \(P_n(u)=P(\sup_{n\leq \delta u}S_n>u)\) as \(u\to\infty\) and \(0<\delta<1/\mu\), where \(\mu\) is the mean drift of \(S_n\). The authors present a number of results which characterize the rate of decay of \(P_n(u)\) under the assumption that \(Q(u)=P(\zeta_1\leq u| X_1\sim \pi)\) is heavy-tailed, where \(\pi\) denotes the stationary distribution of \(\{X_n\}\). In particular, \(Q\) may belong to regularly varying, lognormal, or Weibull classes of distribution functions. The obtained results are used to derive the risk estimates which quantify, e.g., repetitive operational risk losses or the extremal behavior for a \(GARCH(1,1)\) process.
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 5520640
zbMATH Keywords: ruin probability
zbMATH Keywords: large deviation
zbMATH Keywords: Markov chain
zbMATH Keywords: subexponential distributions
zbMATH Keywords: GARCH processes
zbMATH Keywords: respective operation risk model
MaRDI profile type: Publication
OpenAlex ID: W2143422360
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DOI: 10.1007/S00780-007-0044-6
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description: scientific article; zbMATH DE number 5520640