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Ruin Probabilities in a Finite-Horizon Risk Model with Investment and Reinsurance

Part of: Mathematical economics Markov processes Stochastic systems and control

Published online by Cambridge University Press:  30 January 2018

R. Romera  and
W. Runggaldier
R. Romera*
Affiliation:
University Carlos III de Madrid
W. Runggaldier*
Affiliation:
University of Padova
*
∗ Postal address: Department of Statistics, Universidad Carlos III de Madrid, c/Madrid 126, 28903, Getafe, Madrid, Spain. Email address: [email protected]
∗∗ Postal address: Department of Pure and Applied Mathematics, University of Padova, Via Trieste 63, 35121 Padova, Italy.
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Abstract

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A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

Keywords

Risk processsemi-Markov processoptimal reinsurance and investmentLundberg-type bound

MSC classification

Primary: 91B30: Risk theory, insurance 93E20: Optimal stochastic control 60J28: Applications of continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 954 - 966
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by the Spanish ME grant SEJ2007-64500. Part of the contribution by this author was obtained while he was visiting professor 2009 for the chair Quantitative Finance and Insurance at the LMU University in Munich funded by LMU Excellent. Hospitality and financial support are gratefully acknowledged.

References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Chen, S., Gerber, H. U. and Shiu, E. S. W. (2000). Discounted probabilities of ruin in the compound binomial model. Insurance Math. Econom. 26, 239–250.Google Scholar
Diasparra, M. A. and Romera, R. (2009). Bounds for the ruin probability of a discrete-time risk process. J. Appl. Prob. 46, 99–112.CrossRefGoogle Scholar
Diasparra, M. and Romera, R. (2010). Inequalities for the ruin probability in a controlled discrete-time risk process. Europ. J. Operat. Res. 204, 496–504.Google Scholar
Edoli, E. and Runggaldier, W. J. (2010). On optimal investment in a reinsurance context with a point process market model. Insurance Math. Econom. 47, 315–326.CrossRefGoogle Scholar
Eisenberg, J. and Schmidli, H. (2011). Minimising expected discounted capital injections by reinsurance in a classical risk model. Scand. Actuarial J. 2011, 155–176.CrossRefGoogle Scholar
Gaier, J., Grandits, P. and Schachermayer, W. (2003). Asymptotic ruin probabilitites and optimal investment. Ann. Appl. Prob. 13, 1054–1076.CrossRefGoogle Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.Google Scholar
Huang, T., Zhao, R. and Tang, W. (2009). Risk model with fuzzy random individual claim amount. Europ. J. Operat. Res. 192, 879–890.Google Scholar
Hult, H. and Lindskog, F. (2011). Ruin probabilities under general investments and heavy-tailed claims. Finance Stoch. 15, 243–265.Google Scholar
Paulsen, J. (1998). Sharp conditions for certain ruin in a risk process with stochastic return on investment. Stoch. Process. Appl. 75, 135–148.CrossRefGoogle Scholar
Piscitello, C. (2012). Optimal reinsurance and investment in an insurance risk model. Master's Thesis, University of Padova (in Italian).Google Scholar
Schäl, M. (2004). On discrete-time dynamic programming in insurance: exponential utility and minimizing the ruin probability. Scand. Actuarial J. 3, 189–210.Google Scholar
Schäl, M. (2005). Control of ruin probabilities by discrete-time investments. Math. Meth. Operat. Res. 62, 141–158.CrossRefGoogle Scholar
Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Prob. 12, 890–907.CrossRefGoogle Scholar
Schmidli, H. (2008). Stochastic Control in Insurance. Springer, London.Google Scholar
Wang, R., Yang, H. and Wang, H. (2004). On the distribution of surplus inmediately after ruin under interest force and subexponential claims. Insurance Math. Econom. 35, 703–714.CrossRefGoogle Scholar
Willmot, G. E. and Lin, X. S. (2001). Lundberg Approximations for Compound Distributions with Insurance Applications (Lecture Notes Statist. 156). Springer, New York.CrossRefGoogle Scholar
Xiong, S. and Yang, W. S. (2011). Ruin probability in the Cramér-Lundberg model with risky investments. Stoch. Process. Appl. 121, 1125–1137.Google Scholar