Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T09:00:49.873Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Markov processes Special 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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Albrecher, H. and Boxma, O. J. (2005). On the discounted penalty function in a Markov-dependent risk model. Insurance Math. Econom. 37, 650–672.CrossRefGoogle Scholar
Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, London.Google Scholar
Ganesh, A. Macci, C. and Torrisi, G. L. (2007). A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling. Queueing Systems 55, 83–94.Google Scholar
Jacobsen, M. (2003). Martingales and the distribution of the time to ruin. Stoch. Process. Appl. 107, 29–51.Google Scholar
Jacobsen, M. (2005). The time to ruin for a class of Markov additive risk process with two-sided Jumps. Adv. Appl. Prob. 37, 963–992.Google Scholar
Jacobsen, M. (2005). Point Processes, Theory and Applications. Birkhäuser, Boston, MA.Google Scholar
Jacobsen, M. and Jensen, A. T. (2007). Exit times for a class of piecewise exponential Markov processes with two-sided Jumps. Stoch. Process. Appl. 117, 1330–1356.Google Scholar
Jasiulewicz, H. (2001). Probability of ruin with variable premium rate in a Markovian environment. Insurance Math. Econom. 29, 291–296.Google Scholar
Møller, C. M. (1995). Stochastic differential equations for ruin probabilities. J. Appl. Prob. 32, 74–89.Google Scholar
Paulsen, J. (2008). Ruin models with investment income. Prob. Surveys 5, 416–434.Google Scholar
Paulsen, J. and Gjessing, H. K. (1997). Ruin theory with stochastic return on investments. Adv. Appl. Prob. 29, 965–985.Google Scholar
Wu, R. and Wei, L. (2004). The probability of ruin in a kind of Cox risk model with variable premium rate. Scand. Actuarial J. 2004, 121–132.Google Scholar