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December 2012 The time to ruin in some additive risk models with random premium rates
Martin Jacobsen
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J. Appl. Probab. 49(4): 915-938 (December 2012). DOI: 10.1239/jap/1354716648


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.


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Martin Jacobsen. "The time to ruin in some additive risk models with random premium rates." J. Appl. Probab. 49 (4) 915 - 938, December 2012.


Published: December 2012
First available in Project Euclid: 5 December 2012

zbMATH: 1264.60067
MathSciNet: MR3058979
Digital Object Identifier: 10.1239/jap/1354716648

Primary: 60K20
Secondary: 60G40 , 60G44 , 60J35

Keywords: Cramér--Lundberg equation , deficit at ruin , martingale , optional sampling , partial eigenfunction , Rouché's theorem , time to ruin

Rights: Copyright © 2012 Applied Probability Trust


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Vol.49 • No. 4 • December 2012
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