Restricted Dividends in the Two-dimension Dual Model under Diffusion and Capital Injection

The problem of optimal dividend payment in the two-dimension dual model with diffusion under capital injection and varying dividend discount rates was discussed. The HJB equation in the stochastic control model is used to prove that the optimal strategy is a threshold strategy and the integral-differential equation satisfied by the value function of the cumulative dividend discount expectation is obtained, and the explicit expression of the value function is obtained when the benefit obeys an exponential distribution.