Martingales and insurance risk

Abstract

In [10] M.H.A. Davis introduced a class of non-diffusion models, called piecewise-deterministic Markov processes. As was pointed out by Embrechts [14] in the discussion to Davis's paper, these processes should provide a standard theory for studying applications in insurance risk theory. It is our aim to explain this in more detail by unifying the analysis of stochastic insurance models. Some new results will also be provided. In Section 1 we introduce the mathematics of the basic model together with a formulation of the classical risk processes as piecewise deterministic Markov (PD) processes. We distinguish between two different approaches. The first approach is used in Section 2 to find expressions for the probability of ruin in the classical Andersen model. Some new results are obtained for varying claim arrival rate, e.g. under periodicity assumptions. A general model including service payments is also analysed. Finally in Section 3 we discuss the second approach to establish exact results for Gamma c...