Optimal Per-Loss Reinsurance and Investment Problem Under Negative Correlation Assumption

In this paper, we consider the optimal reinsurance and investment problem for the insurance company, where the insurer can purchase per-loss reinsurance and invest the surplus in a financial market, and the insurer’s claim liabilities and capital gains in financial market are negatively correlated which makes our problems more complicated and challenging. We study two optimization problems: minimizing the probability of absolute ruin and maximizing the expected utility of the terminal wealth. By using the dynamic programming and Lagrange multiplier methods, we reduce the infinite-dimensional problem to the one-dimensional problem, and obtain the explicit form of the optimal reinsurance and investment strategies as well as the corresponding value functions. Furthermore, we investigate the impacts of the negative correlation on the optimal strategies, and find that they are totally different in these two problems. For the former, it strongly depends on the wealth level, which means that with the different wealth level, the negative correlation has different influence on the optimal results; but for the latter, both the retained claims and the amount invested in the risky asset increase as the negative correlation becomes stronger