Interplay of financial and insurance risks in dependent discrete-time risk models

Abstract

Consider a discrete-time insurance risk model with two kinds of risks, i.e., insurance and financial risks. Within period i, the real-valued net insurance loss caused by traditional claims is regarded as the insurance risk, denoted by Xi, and the positive stochastic discount factor over the same time period is the financial risk, denoted by Yi. Assume that {(X,Y),(Xi,Yi),i≥1} form a sequence of independent and identically distributed random vectors. This work investigates the interplay of the insurance and financial risks, allowing a dependence structure exists between these two kinds of risks. Following the work of Li and Tang (2015), we derive some asymptotic and uniformly asymptotic formulas for the finite-time and infinite-time ruin probabilities, under the assumption that the generic random vector (X,Y) follows a bivariate Sarmanov distribution and every convex combination of the distributions of X and Y is of strongly regular variation. As an extension, we further consider a general pair (X,Y), whose dependence structure is more verifiable than the Sarmanov one to some extent. In such a setting, we study the asymptotic tail behavior of the product XY, which forms an analogue of the well-known Breiman’s theorem in a different and dependent situation.