Optimal proportional reinsurance and investment with regime-switching for mean–variance insurers

Following the framework of Promislow and Young (2005), this paper considers an optimal investment–reinsurance problem of an insurer facing a claim process modeled by a Brownian motion with drift under the Markowitz mean–variance criterion. The market modes are divided into a finite number of regimes. All the key parameters change according to the value of different market modes. The insurer chooses to purchase proportional reinsurance to reduce the underlying risk. In addition to reinsurance, we suppose that the insurer is allowed to invest its surplus in a financial market consisting of a risk-free asset (bond or bank account) and a risky asset whose price process is modeled by a geometric Brownian motion. We investigate the feasibility of the problem, obtain an analytic expression for the optimal strategy, delineate the efficient frontier and demonstrate our results with numerical examples.