Dividend and Capital Injection Optimization with Transaction Cost for Spectrally Negative L\'{e}vy Risk Processes

For an insurance company with reserve modeled by the spectrally negative L'{e}vy process, we study the optimal impulse dividend maximizing the expected accumulated net dividend payment subtracted by the accumulated cost of injecting capital. In this setting, the beneficiary of the dividends injects capital to ensure a non-negative risk process so that the insurer never goes bankrupt. The optimal impulse dividend and capital injection strategy together with its value function are obtained.