Consistent single- and multi-step sampling of multivariate arrival times: A characterization of self-chaining copulas

TL;DR

This paper characterizes self-chaining copulas, linking them to extreme value copulas, and demonstrates their importance for consistent multivariate survival sampling in applications like finance and reliability.

Contribution

It introduces the concept of self-chaining copulas, characterizes them within the Archimedean family, and connects them to extreme value copulas for improved sampling consistency.

Findings

Self-chaining copulas coincide with extreme value copulas.

Gumbel-Hougaard and Marshall-Olkin copulas are self-chaining.

Self-chaining copulas enable consistent multistep survival sampling.

Abstract

This paper deals with dependence across marginally exponentially distributed arrival times, such as default times in financial modeling or inter-failure times in reliability theory. We explore the relationship between dependence and the possibility to sample final multivariate survival in a long time-interval as a sequence of iterations of local multivariate survivals along a partition of the total time interval. We find that this is possible under a form of multivariate lack of memory that is linked to a property of the survival times copula. This property defines a "self-chaining-copula", and we show that this coincides with the extreme value copulas characterization. The self-chaining condition is satisfied by the Gumbel-Hougaard copula, a full characterization of self chaining copulas in the Archimedean family, and by the Marshall-Olkin copula. The result has important practical implications for consistent single-step and multi-step simulation of multivariate arrival times in a way that does not destroy dependency through iterations, as happens when inconsistently iterating a Gaussian copula.