Martingales and insurance risk

Angelos Dassios · Paul Embrechts A

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Probability and Risk Models Probability and Statistical Research Stochastic processes and financial applications

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 claim sizes and investment policies, including borrowing. It should be stressed early on that the piecewise deterministic structure is essentially a vehicle for obtaining interesting martingales, which can then be used to calculate relevant functionals of the process. If the Davis theory is to be used for the analysis of more realistic insurance models, then it is important to understand in detail its scope and versatility. The present paper stresses the request made by E. Arjas [2] : “…I hope to see in the future many worked out examples of how the PD theory can be used as an aid in solving practical OR problems”.

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References

1984Piecewise-Deterministic Markov Processes: A General Class of Non-Diffusion Stochastic Models 1979An introduction to mathematical risk theory 1970Mathematical Methods in Risk Theory 1976The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule

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1998Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities 1993Risk theory in a stochastic economic environment 1987Classical risk theory in an economic environment 1998Ruin probabilities in the presence of heavy-tails and interest rates 1994Ruin estimation for a general insurance risk model 1992Computational methods in risk theory: A matrix-algorithmic approach 2008The Perturbed Compound Poisson Risk Process with Investment and Debit Interest 1998Ruin theory with compounding assets — a survey

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