A classical construction principle for dependent failure times is to consider shocks that destroy components within a system. The arrival times of shocks can destroy arbitrary subsets of the system, thus introducing dependence. The seminal model – based on independent and exponentially distributed shocks - was presented by Marshall and Olkin in 1967, various generalizations have been proposed in the literature since then. Such models have applications in non-life insurance, e.g. insurance claims caused by floods, hurricanes, or other natural catastrophes. The simple interpretation of multivariate fatal shock models is clearly appealing, but the number of possible shocks makes them challenging to work with, recall that there are 2^d subsets of a set with d components. In a series of papers we have identified mixture models based on suitable stochastic processes that give rise to a different - and numerically more convenient - stochastic interpretation. This representation is particularly useful for the development of efficient simulation algorithms. Moreover, it helps to define parametric families with a reasonable number of parameters. We review the recent literature on multivariate fatal shock models, extreme-value copulas, and related dependence structures. We also discuss applications and hierarchical structures. Finally, we provide a new characterization of the Marshall-Olkin distribution.
Authors: Mai, J-F.; Scherer, M.;
- Mathematical and Computational Finance Seminar