Mean-variance problem for an insurer with dependent risks and stochastic interest rate in a jump-diffusion market

In this paper, we study an optimal investment and reinsurance problem in which the interest rate is driven by the Vasicek process and two dependent classes of insurance business are correlated through a common shock component. The goal of insurer is to find an optimal investment-reinsurance strategy to minimize the variance of terminal net wealth for a given expected terminal net wealth. By using the linear-quadratic optimal control theory and the corresponding Hamilton-Jacobi-Bellman (HJB) equation, the closed-form expressions for the value function and optimal strategies are obtained under the special case of perfect correlation between the bond and stock processes. We present that the solution of the HJB equation is no longer a classical solution, but a viscosity solution due to the non-negativity constraint of the reinsurance strategy. Furthermore, the efficient strategies and efficient frontier are derived explicitly. Finally, we explore some numerical examples to show the influence of model parameters on the optimal investment and reinsurance strategies.