Minimizing the probability of absolute ruin under the mean-variance premium principle

Abstract

In this article, we assume that the insurer can purchase per-loss reinsurance and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. It is also assumed that the investment amount of the risky asset is capped at a fixed level and that short-selling is prohibited. Our objective is to minimize the probability of absolute ruin, and the reinsurance premium is computed according to the mean-variance premium principle, that is, a combination of the expected-value and variance premium principles. By solving the corresponding Hamilton–Jacobi–Bellman equation, we derive explicit expressions for the S-shaped minimum absolute ruin function and its associated optimal reinsurance-investment strategy. We further study the same optimization problem for a slightly modified version of absolute ruin, and the corresponding optimal results are obtained as well. To gain insights into the optimal problems, we investigate the reason for the S-shaped value function and discover that the constraint on investment control can result in the kink of minimum absolute ruin function. Finally, some properties and numerical examples are presented to show the impact of model parameters on the optimal results.