Optimal Control of Investment‐Reinsurance Problem for an Insurer with Jump‐Diffusion Risk Process: Independence of Brownian Motions

This paper investigates the excess‐of‐loss reinsurance and investment problem for a compound Poisson jump‐diffusion risk process, with the risk asset price modeled by a constant elasticity of variance (CEV) model. It aims at obtaining the explicit optimal control strategy and the optimal value function. Applying stochastic control technique of jump diffusion, a Hamilton‐Jacobi‐Bellman (HJB) equation is established. Moreover, we show that a closed‐form solution for the HJB equation can be found by maximizing the insurer’s exponential utility of terminal wealth with the independence of two Brownian motions W(t) and W1(t). A verification theorem is also proved to verify that the solution of HJB equation is indeed a solution of this optimal control problem. Then, we quantitatively analyze the effect of different parameter impacts on optimal control strategy and the optimal value function, which show that optimal control strategy is decreasing with the initial wealth x and decreasing with the volatility rate of risk asset price. However, the optimal value function V(t; x; s) is increasing with the appreciation rate μ of risk asset.