A refracted Lévy process with delayed dividend pullbacks

The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a pre-determined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend threshold. Hence, in this paper, we propose a refracted Lévy risk model with delayed dividend pullbacks triggered by a certain Poissonian observation scheme. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, we obtain explicit expressions for two-sided exit identities of the proposed insurance risk process. Also, penalties are incorporated into the analysis of dividend payouts as a mechanism to penalize for the volatility of the dividend policy and account for an investor's typical preference for more stable cash flows. An explicit expression for the expected (discounted) dividend payouts net of penalties is derived. The criterion for the optimal threshold level that maximizes the expected dividend payouts is also discussed. Finally, several numerical examples are considered to assess the impact of dividend delays on ruin-related quantities. We numerically show that dividend strategies with more steady dividend payouts can be preferred (over the well-known threshold dividend strategy) when penalty fee become too onerous.