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The time to ruin for a class of Markov additive risk process with two-sided jumps

Part of: Special processes Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Martin Jacobsen
Martin Jacobsen*
Affiliation:
University of Copenhagen
*
∗ Postal address: Department of Applied Mathematics and Statistics, University of Copenhagen, 5, Universitetsparken, DK-2100 Copenhagen Ø, Denmark. Email address: martin@math.ku.dk
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Abstract

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We consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.

Keywords

Probability of ruintime to ruinundershootpassage timemartingaleoptional samplingadditive processRouché's theorem

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60J25: Continuous-time Markov processes on general state spaces 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60G44: Martingales with continuous parameter 60J35: Transition functions, generators and resolvents 60J60: Diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 963 - 992
Copyright
Copyright © Applied Probability Trust 2005 

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