Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes

Abstract

In this paper, we investigate an optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. We assume that the periodic dividend strategy has exponential inter-dividend-decision times and continuous monitoring of solvency. Both proportional and fixed transaction costs from capital injection are considered. The objective is to maximize the total value of the expected discounted dividends and the penalized discounted capital injections until the time of ruin. By the fluctuation theory of Lévy processes in Albrecher et al. (2016), the optimal periodic dividend and capital injection strategies are derived. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Finally, numerical examples are studied to illustrate our results.

On the optimality of periodic barrier strategies for a spectrally positive Lévy process

“…Before closing the introduction, we discuss here the connections with the results in Zhao et al [24]. The first problem considered in [24] is the special case of our first problem with no terminal payoff/cost at ruin.…”

“…Before closing the introduction, we discuss here the connections with the results in Zhao et al [24]. The first problem considered in [24] is the special case of our first problem with no terminal payoff/cost at ruin. While our paper directly uses the results of Avram et al [8] to derive the expected NPV of dividends under the periodic barrier strategy, they obtained it in a different way using the results by Albrecher et al [1], which gives the identities for spectrally negative Lévy processes observed at Poisson arrival times.…”

On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model

“…Yang and Deng [25] study the discounted Gerber-Shiu type function for a perturbed risk model with interest and periodic dividend strategy. Other recent articles on risk models with dividend strategy and capital injection involving periodic observations can be found in Zhang and Liu [26], Zhang [27], Zhang and Han [28], Zhao et al [29], Pérez and Yamazaki [30], Noba et al [31], Dong and Zhou [32], Xu et al [33], Zhang et al [34], Liu and Yu [35], Zhang and Cheung [36], Yu et al [37] and Liu and Zhang [38].…”

On optimal periodic dividend strategies for Lévy risk processes

“…In this paper, we focus on the periodic barrier strategy and its optimality when dividend payments can only be made at the jump times of an independent Poisson process. In this context, Avanzi et al [5] solved the case with positive hyperexponential jumps; this case was later generalized by Pérez and Yamazaki [25] and Zhao et al [31] for a general spectrally positive Lévy process. By assuming that the intervals are independent exponential random variables, we can still formulate it as a one-dimensional Markovian problem.…”