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Ruin probabilities in a Markovian shot-noise environment

Part of: Special processes Markov processes

Published online by Cambridge University Press:  11 November 2022

Simon Pojer  and
Stefan Thonhauser
Simon Pojer*
Affiliation:
Graz University of Technology
Stefan Thonhauser*
Affiliation:
Graz University of Technology
*
*Postal address: Institute of Statistics, University of Technology Graz, Kopernikusgasse 24/III, 8010 Graz, Austria.
*Postal address: Institute of Statistics, University of Technology Graz, Kopernikusgasse 24/III, 8010 Graz, Austria.
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Abstract

We consider a risk model with a counting process whose intensity is a Markovian shot-noise process, to resolve one of the disadvantages of the Cramér–Lundberg model, namely the constant intensity of the Poisson process. Due to this structure, we can apply the theory of piecewise deterministic Markov processes on a multivariate process containing the intensity and the reserve process, which allows us to identify a family of martingales. Eventually, we use change of measure techniques to derive an upper bound for the ruin probability in this model. Exploiting a recurrent structure of the shot-noise process, even the asymptotic behaviour of the ruin probability can be determined.

Keywords

Ruin theoryrisk theorypiecewise-deterministic Markov processes (PDMP)Cox processes

MSC classification

Secondary: 60K10: Applications (reliability, demand theory, etc.) 60J25: Continuous-time Markov processes on general state spaces
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 542 - 556
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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