An Optimal Dividends Problem For A General Spectrally Negative Lévy Process With Capital Injections

In this paper, we consider the optimal dividends problem with capital injections, inwhich the underlying surplus is modeled by a spectrally negative lévy process. In thismodel, the lévy measure is limited into a finite interval, and the dividends process isassumed to be a impulse process. Thus the problem becomes a mixed singular-regularimpulse control problem. The company is supposed to go bankruptcy under some con-ditions, i.e., if the surplus of the company is lower than the given bankruptcy line, thecompany will go bankruptcy directly.The objective of the manager of the company is to maximize the cumulative ex-pected discounted dividends minus the equity issuance until the time of bankruptcy bycontrolling the dividends payout and the capital injections, i.e., to obtain the optimal re-turn function and the associated optimal strategy. The mixed singular-regular impulsecontrol problem is solved by constructing two categories of suboptimal control models,one is the model without equity insurance, the other is the model with equity insurance.At last, the influences of the key model parameters in the optimal return function and op-timal control policy of the company are discussed by some numerical calculations. Theinnovative points in this paper are as follows:Firstly, the company never goes bankruptcy in former literatures considering bothdividends and capital injections. However, the company is assumed to go bankruptcy inthis model.Secondly, the lévy measure is defined in (0,∞) in former literatures, but in thismodel, it is limited to a finite interval.Finally, the force is assumed to be a constant in former literatures. In this model, weconsider the stochastic force, which is more realistic than the constant ones.