Optimal Dividend Control with Transaction Costs under Exponential Parisian Ruin for a Refracted Levy Risk Model

This paper concerns an optimal impulse control problem associated with a refracted L\'{e}vy process, involving the reduction of reserves to a predetermined level whenever they exceed a specified threshold. The ruin time is determined by Parisian exponential delays and limited by a lower ultimate bankrupt barrier. We initially obtained the necessary and sufficient conditions for the value function and the optimal impulse control policy. Given a candidate for the optimal strategy, the corresponding expected discounted dividend function is subsequently formulated in terms of the Parisian refracted scale function, which is employed to measure the expected discounted utility of the impulse control. Then, the optimality of the proposed impulse control is verified using the HJB inequalities, and a monotonicity-based criterion is established to identify the admissible region of optimal thresholds, which serves as the basis for the numerical computation of their optimal levels. Finally, we present applications and numerical examples related to Brownian risk process and Cram\'{e}r-Lundberg process with exponential claims, demonstrating the uniqueness of the optimal impulse strategy and exploring its sensitivity to parameters.