Robust reinsurance contracts with risk constraint

This thesis investigates a class of optimal reinsurance contract problems in continuous time. We use the principal-agent framework to incorporate the bargaining between the insurer and the reinsurer. To this end, we extend the reinsurer's relative safety loading factor which is usually a pre-specified constant in the traditional expected value principle to be time-varying and to represent the reinsurance premium. Since the insurance companies should satisfy the regulators' capital requirements and the computation of capital requirements is based on Value-at-Risk (VaR) under Solvency II regime, we introduce the dynamic version of VaR andimpose a dynamic VaR constraint on the insurer. As for the reinsurer, we assumethat she is ambiguity-averse and aims to maximize the expected utility of herterminal wealth under the worst-case scenario of the alternative measures. The dynamic programming technique is applied to derive the principal's Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation and the agent's Hamilton-Jacobi-Bellman(HJB) equation. Additionally, Karush-Kuhn-Tucker (KKT) conditions are utilized to settle the constrained optimization problem of the agent. Explicit expressions for the optimal retained proportional of the claims, the optimal reinsurance premium and the corresponding value functions of the insurer and the reinsurer are derived. Finally, we analyze several numerical examples to illustrate economic intuition. Our results show that the reinsurer's ambiguity aversion and the insurer's risk constraint increase the optimal reinsurance premium, which decreases the optimal reinsurance demand of the insurer.