Extreme values statistics for Markov chains with applications to Finance and Insurance

Abstract

We review in this paper several statistical methods, specifically tailored for Markov processes with a view towards their extremal behavior. Precisely, this paper proposes some statistical inference tools for extremal events from a regeneration theory angle.Indeed, Harris Markov chains may be decomposed into independent regeneration cycles, namely data segments between consecutive regeneration times tau_1; tau_2 ... (i.e. random times at which the chain forgets its past). Working on this approach, the methodology proposed in this paper boils down to split up the observed sample path into regeneration data blocks (or into data blocks drawn from a distribution approximating the regeneration cycle's distribution, in the general case when regeneration times cannot be observed). Then, the analysis boils down to examining the sequence of maxima over the resulting data segments, as if they were i.i.d. We focus on the estimation of the extremal dependence index and the tail index. We illustrate the method on two examples taken from the insurance and finance literature, ruin models and times series exhibiting smooth threshold and/or strong conditional heteroscedasticity. An illustration of the estimation methods to the CAC40 shows the potential of regenerative tools for real data applications.