Dirichlet sub-mixtures: A new reading of multiplicative background risk models with Pareto margins

Multiplicative background risk models (MBRMs) provide a tractable framework for describing dependent risks. We construct a class of MBRMs with Pareto margins by combining Marshall--Olkin exponential and additive gamma risk factor structures, each allowed to incorporate idiosyncratic and systemic background risk factors. Our main result establishes, under an admissibility condition on the risk factor systems, a finite sub-mixture representation of the joint survival function in terms of Pareto survival functions, whose weights are Dirichlet probabilities of orderings induced by the background risk factors and sum to at most one. This yields a new interpretation of such MBRMs, in which dependence is expressed through Pareto survival functions and their Dirichlet weights. The resulting family encompasses many known multivariate Pareto constructions, is closed under marginalization, and admits characterizations of independence and, under a non-degeneracy condition, of its singular and absolutely continuous parts. It is associated, its covariance decomposition separates the contributions of the two risk factor structures and their interaction, and its joint tails are regularly varying along rays. The presence of asymptotic tail dependence and the decay rates of joint exceedance probabilities are governed by the gamma risk factors alone; the exponential risk factors affect only the tail constants.