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The Time to Ruin in Some Additive Risk Models with Random Premium Rates

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Martin Jacobsen
Martin Jacobsen*
Affiliation:
University of Copenhagen
*
∗ Postal address: Department of Mathematical Sciences, University of Copenhagen, 5, Universitetsparken, DK-2100 Copenhagen Ø, Denmark. Email address: [email protected]
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Abstract

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The risk processes considered in this paper are generated by an underlying Markov process with a regenerative structure and an independent sequence of independent and identically distributed claims. Between the arrivals of claims the process increases at a rate which is a nonnegative function of the present value of the Markov process. The intensity for a claim to occur is another nonnegative function of the value of the Markov process. The claim arrival times are the regeneration times for the Markov process. Two-sided claims are allowed, but the distribution of the positive claims is assumed to have a Laplace transform that is a rational function. The main results describe the joint Laplace transform of the time at ruin and the deficit at ruin. The method used consists in finding partial eigenfunctions for the generator of the joint process consisting of the Markov process and the accumulated claims process, a joint process which is also Markov. These partial eigenfunctions are then used to find a martingale that directly leads to an expression for the desired Laplace transform. In the final section, three examples are given involving different types of the underlying Markov process.

Keywords

Time to ruindeficit at ruinpartial eigenfunctionmartingaleoptional samplingCramér–Lundberg equationRouché's theorem

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G44: Martingales with continuous parameter 60J35: Transition functions, generators and resolvents
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 915 - 938
Copyright
© Applied Probability Trust 

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