Small-time ruin for a financial process modulated by a Harris recurrent Markov chain
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.
scientific article; zbMATH DE number 5520640
scientific article