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Journal of Applied Mathematics
Volume 2018, Article ID 9180780, 11 pages
Research Article

On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance

1Department of Applied Mathematics and Computational Science, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania
2Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda

Correspondence should be addressed to Christian Kasumo;

Received 28 November 2017; Revised 19 January 2018; Accepted 30 January 2018; Published 22 February 2018

Academic Editor: Saeid Abbasbandy

Copyright © 2018 Christian Kasumo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.