Archimedean Survival Processes

TL;DR

This paper introduces Archimedean survival processes, a new class of multivariate Markov processes linked to Archimedean copulas, expanding the tools for multivariate modeling with gamma process-based structures.

Contribution

It defines ASPs, establishes their connection to Archimedean copulas, and provides characterizations and generalizations of these processes.

Findings

ASP is equivalent in law to a multivariate gamma process

Terminal value of ASP has an Archimedean survival copula

Bijective correspondence between ASPs and Archimedean copulas

Abstract

Archimedean copulas are popular in the world of multivariate modelling as a result of their breadth, tractability, and flexibility. A. J. McNeil and J. Ne\v{s}lehov\'a (2009) showed that the class of Archimedean copulas coincides with the class of multivariate $\ell_1$-norm symmetric distributions. Building upon their results, we introduce a class of multivariate Markov processes that we call `Archimedean survival processes' (ASPs). An ASP is defined over a finite time interval, is equivalent in law to a multivariate gamma process, and its terminal value has an Archimedean survival copula. There exists a bijection from the class of ASPs to the class of Archimedean copulas. We provide various characterisations of ASPs, and a generalisation.