Martingales and insurance risk

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...

[1]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[2]  On risk reserve processes , 1970 .

[3]  Hans U. Gerber,et al.  An introduction to mathematical risk theory , 1982 .

[4]  Mark H. A. Davis.  Piecewise‐Deterministic Markov Processes: A General Class of Non‐Diffusion Stochastic Models , 1984 .

[5]  P. Billingsley,et al.  Probability and Measure , 1980 .

[6]  J. Michael Harrison,et al.  The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule , 1976, Math. Oper. Res..

[7]  Hans Bühlmann,et al.  Mathematical Methods in Risk Theory , 1970 .

[8]  F. Delbaen,et al.  Classical risk theory in an economic environment , 1987 .

[9]  Olof Thorin.  Probabilities of ruin , 1982 .

[10]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[11]  H. Waters,et al.  Martingales in life insurance , 1984 .

[12]  Walter A. Rosenkrantz,et al.  Some martingales associated with queueing and storage processes , 1981 .

[13]  J. Gani,et al.  Birth, immigration and catastrophe processes , 1982, Advances in Applied Probability.

[14]  Samuel Karlin,et al.  ELEMENTS OF STOCHASTIC PROCESSES , 1975 .

[15]  J. Michael Harrison,et al.  The Recurrence Classification of Risk and Storage Processes , 1978, Math. Oper. Res..

[16]  R. Tweedie,et al.  Storage processes with general release rule and additive inputs , 1982, Advances in Applied Probability.

[17]  D. Cox.  The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  Søren Asmussen,et al.  Ruin probabilities expressed in terms of storage processes , 1988, Advances in Applied Probability.