Modelling Stochastic Bivariate Mortality

Abstract

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to represent mortality risk. This paper represents a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. On the theoretical side, we extend to couples the Cox processes set up, i.e. the idea that mortality is driven by a jump process whose intensity is itself a stochastic process, proper of a particular generation within each gender. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. On the calibration side, we fit the joint survival function by calibrating separately the (analytical) copula and the (analytical) margins. First, we select the best fit copula according to the methodology of Wang and Wells (2000) for censored data. Then, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the analytical marginal survival functions. Coupling the best fit copula with the calibrated margins we obtain, on a sample generation, a joint survival function which incorporates the stochastic nature of mortality improvements and is far from representing independency.On the contrary, since the best fit copula turns out to be a Nelsen one, dependency is increasing with age and long-term dependence exists.