Optimal control with performance dependent dividend and capital injection strategies for spectrally negative Lévy risk processes

Assume that the surplus process without dividend and capital injection for an insurance company evolves as a spectrally negative Lévy process (SNLP). Given 0<ell_1<ell_2<1, dividends are distributed at some restricted fraction ell(·)∈[\ell_{1},~\ell_{2}] of the company's net income only when the company is in a profitable situation, i.e., the surplus process is at its running maximum. Meanwhile, the beneficiary of the dividends injects capital to ensure a non-negative risk process, so that the insurer never goes bankrupt. We consider De Finetti's dividend problem of maximizing the difference between the expected discounted dividends and the expected discounted capital injection. The optimal value function and the optimal dividend strategy are obtained. It turns out that, corresponding to two opposite scenarios, the optimal dividend distribution rate remains at ell_2or switches from ell_1to ell_2once the surplus process hits some critical level and stays in the profitable situation. Some numerical examples are also provided.