## Journal of Applied Mathematics

### On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance

#### Abstract

We consider an insurance company whose reserves dynamics follow a diffusion-perturbed risk model. To reduce its risk, the company chooses to reinsure using proportional or excess-of-loss reinsurance. Using the Hamilton-Jacobi-Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation (VIDE) which we transform into a linear Volterra integral equation (VIE) of the second kind. We then proceed to solve this linear VIE numerically using the block-by-block method for the optimal reinsurance policy that minimizes the ultimate ruin probability for the chosen parameters. Numerical examples with both light- and heavy-tailed distributions are given. The results show that proportional reinsurance increases the survival of the company in both light- and heavy-tailed distributions for the Cramér-Lundberg and diffusion-perturbed models.

#### Article information

Source
J. Appl. Math., Volume 2018 (2018), Article ID 9180780, 11 pages.

Dates
Received: 28 November 2017
Revised: 19 January 2018
Accepted: 30 January 2018
First available in Project Euclid: 17 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.jam/1521252013

Digital Object Identifier
doi:10.1155/2018/9180780

Mathematical Reviews number (MathSciNet)
MR3770640

#### Citation

Kasumo, Christian; Kasozi, Juma; Kuznetsov, Dmitry. On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance. J. Appl. Math. 2018 (2018), Article ID 9180780, 11 pages. doi:10.1155/2018/9180780. https://projecteuclid.org/euclid.jam/1521252013

#### References

• S. D. Promislow and V. R. Young, “Minimizing the probability of ruin when claims follow Brownian motion with drift,” North American Actuarial Journal, vol. 9, no. 3, pp. 109–128, 2005.
• J. Kasozi, C. W. Mahera, and F. Mayambala, “Controlling ultimate ruin probability by quota-share reinsurance arrangements,” International Journal of Applied Mathematics and Statistics, vol. 49, no. 19, pp. 1–15, 2013.
• C. S. Liu and H. Yang, “Optimal investment for an insurer to minimize its probability of ruin,” North American Actuarial Journal, vol. 8, no. 2, pp. 11–31, 2004.
• C. Hipp and M. Plum, “Optimal investment for insurers,” Insurance: Mathematics & Economics, vol. 27, no. 2, pp. 215–228, 2000.
• H. Schmidli, “On minimizing the ruin probability by investment and reinsurance,” The Annals of Applied Probability, vol. 12, no. 3, pp. 890–907, 2002.
• P. Li, M. Zhou, and C. Yin, “Optimal reinsurance with both proportional and fixed costs,” Statistics & Probability Letters, vol. 106, pp. 134–141, 2015.
• X. Hu and L. Zhang, “Ruin probability in a correlated aggregate claims model with common Poisson shocks: application to reinsurance,” Methodology and Computing in Applied Probability, vol. 18, no. 3, pp. 675–689, 2016.
• X. Zhang and Z. Liang, “Optimal layer reinsurance on the maximization of the adjustment coefficient,” Numerical Algebra, Control and Optimization, vol. 6, no. 1, pp. 21–34, 2016.
• B.-G. Jang and K. T. Kim, “Optimal reinsurance and asset allocation under regime switching,” Journal of Banking & Finance, vol. 56, pp. 37–47, 2015.
• R. Verlaak and J. Beirlant, “Optimal reinsurance programs. An optimal combination of several reinsurance protections on a heterogeneous insurance portfolio,” Insurance: Mathematics & Economics, vol. 33, no. 2, pp. 381–403, 2003.
• P. Linz, Analytical and Numerical Methods for Volterra Equations, vol. 7 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1985.
• L. Centeno, “On combining quota-share and excess of loss,” ASTIN Bulletin, vol. 15, no. 1, pp. 49–63, 1985.
• H. Schmidli, Stochastic Control in Insurance, Probability and Its Applications (New York), Springer, London, UK, 2008.
• J. Paulsen, J. Kasozi, and A. Steigen, “A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments,” Insurance: Mathematics & Economics, vol. 36, no. 3, pp. 399–420, 2005.
• C. Kasumo, Minimizing the probability of ultimate ruin by proportional reinsurance and investment [M.S. thesis], University of Dar es salaam, 2011.
• D. Li, D. Li, and V. R. Young, “Optimality of excess-loss reinsurance under a mean-variance criterion,” Insurance: Mathematics & Economics, vol. 75, pp. 82–89, 2017. \endinput