摘要
The authors consider a compound Cox model of insurance risk with the additional economic assumption of a positive interest rate. As the authors note a duality result relating a compound Cox model of insurance risk with a positive interest rate and a double shot noise process, the authors analyze a double shot noise process systematically for its theoretical distributional properties, based on the piecewise deterministic Markov process theory, and the martingale methodology. The authors also obtain the moments of aggregate accumulated/discounted claims where the claim arrival process follows a Cox process with shot noise intensity. Removing the parameters in a double shot noise process gradually, the authors show that it becomes a compound Cox process with shot noise intensity, a single shot noise process and a compound Poisson process. Numerical comparisons are shown between the moments （i.e. means and variances） of a compound Poisson model and their counterparts of a compound Cox model with/without considering a positive interest rate. For that purpose, the authors assume that claim sizes and primary event sizes follow an exponential distribution, respectively.
