Precise large deviations of aggregate claims with arbitrary dependence between claim sizes and waiting times

Abstract

Consider a renewal risk model in which claim sizes and interarrival times correspondingly form a sequence of independent, identically distributed, and nonnegative random pairs with a generic pair (X,θ). Chen and Yuen (2012) studied precise large deviations of aggregate claims in this model under the assumption that (X,θ) obeys a dependence structure described via a stochastic boundedness condition on the waiting time θ for a large claim X. That assumption unfortunately leads to asymptotic independence between X and θ and hence considerably limits the usefulness of the result obtained there. In this short paper, we make an effort to avoid that assumption by allowing X and θ to be arbitrarily dependent. As by-products, we propose two novel applications of the main result, one to pricing insurance futures and the other to approximating both the value at risk and expected shortfall of aggregate claims.