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 Lévy 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 Cramér-Lundberg process with exponential claims, demonstrating the uniqueness of the optimal impulse strategy and exploring its sensitivity to parameters.

💡 Research Summary

This paper studies an optimal impulse‑control (dividend) problem for a refracted spectrally negative Lévy risk model in which ruin is defined by a Parisian exponential implementation delay together with a lower ultimate bankruptcy barrier. The underlying Lévy process (X) (drift (\gamma), diffusion (\sigma), jump measure (\nu)) is “refracted’’: when the surplus (R_t) exceeds a preset level (b) a constant linear drift (\delta) is subtracted, i.e. the surplus evolves as (R_t = X_t - \delta\int_0^t \mathbf 1_{{R_s\ge b}}ds). This captures a regime‑switching capital‑management policy.

Ruin occurs at the first of two events: (i) the surplus becomes negative and stays below zero for a random exponential time (\xi\sim\mathrm{Exp}(m)) (Parisian delay); (ii) the surplus hits a hard lower barrier (-l). The ruin time is therefore (T=\tau^{\mathrm{Paris}} \wedge \tau_{-l}).

Dividends are paid only by discrete jumps (impulse control). Each dividend payment incurs a fixed transaction cost (\beta>0); consequently a dividend jump (\Delta D_t) must satisfy (\Delta D_t>\beta) and the post‑payment surplus must remain non‑negative. The controlled surplus is (U_t^\pi = R_t - D_t^\pi). The performance criterion is the expected discounted dividend net of transaction costs up to ruin: \