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Martingales and insurance risk. (English) Zbl 0676.62083
This paper describes the applicability of the piecewise deterministic Markov process set-up of M. H. A. Davis [J. R. Stat. Soc., Ser. B 46, 353-388 (1984; Zbl 0565.60070)] to risk theory. The purpose is to provide a unified setting more than to simplify the proofs of known results or to obtain new ones (there are some, nevertheless!) which cannot be derived by classical tools. However, formulating a risk process within this set-up permits to spot martingales by looking for solutions of \(Af=0\) where A is the generator. Suitable versions of Dynkin’s identity and the optional stopping theorem yield identities containing information on the probability of ruin \(\Psi\) (u) with initial reserve u, so that \(\psi\) (u) can either be calculated explicitly in some special cases or (more typically) upper and lower bounds of Lundberg type can be derived.
Beyond standard risk processes, the paper discusses periodical claim arrival intensities, premiums which depend on the size of the current reserve, renewal-type arrivals with some dependency on the claim sizes, and optimal barrier strategies.
Reviewer: S.Asmussen

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G44 Martingales with continuous parameter
60J25 Continuous-time Markov processes on general state spaces
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