Risk and duality in multidimensions

Abstract

We present, in discrete time, general-state-space dualities between content and insurance risk processes that generalize the stationary recursive duality of Asmussen and Sigman (1996, Probab. Eng. Inf. Sci. 10, 1–20) and the Markovian duality of Siegmund (1976, Ann. Probab. 4, 914–924) (both of which are one dimensional). The main idea is to allow a risk process to be set-valued, and to define ruin as the first time that the risk process becomes the whole space. The risk process can also become infinitely rich which means that it eventually takes on the empty set as its value. In the Markovian case, we utilize stochastic geometry tools to construct a Markov transition kernel on the space of closed sets. Our results connect with strong stationary duality of Diaconis and Fill (1990, Ann. Probab. 18, 1483–1522). As a motivating example, in multidimensional Euclidean space our approach yields a dual risk process for Kiefer–Wolfowitz workload in the classic G/G/c queue, and we include a simulation study of this dual to obtain estimates for the ruin probabilities.