## Journal of Applied Mathematics

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

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

Christian Kasumo, Juma Kasozi, and Dmitry Kuznetsov

#### 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