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Consistent Single- and Multi-step Sampling of Multivariate Arrival Times: A characterization of self-chaining copulas

by Damiano Brigo of King's College, London, and
Kyriakos Chourdakis of King's College, London

May 1, 2012

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.

JEL Classification: C15, C16.

AMS Classification: 60E07, 62H05, 62H20, 62H99.

Keywords: Dependence Modeling, Arrival Times, Sampling, Archimedean Copula, Gumbel-Hougaard Copula, Marshall-Olkin Copula, Self-Chaining Copula, Multi-Step Simulation, Extreme Value Copulas, Copula Iteration, Copula Chaining.

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