On optimal periodic dividend strategies for Lévy risk processes

Abstract

In this paper, we revisit the optimal periodic dividend problem, in which dividend payments can only be made at the jump times of an independent Poisson process. In the dual (spectrally positive Lévy) model, recent results have shown the optimality of a periodic barrier strategy, which pays dividends at Poissonian dividend-decision times, if and only if the surplus is above some level. In this paper, we show the optimality of this strategy for a spectrally negative Lévy process whose dual has a completely monotone Lévy density. The optimal strategies and value functions are concisely written in terms of the scale functions. Numerical results are also provided.AMS 2010 Subject Classifications: 60G51, 93E20, 91B30 JEL Classifications: C44, C61, G24, G32, G35

Optimality of refraction strategies for a constrained dividend problem

“…The Lévy measure Π of the process X has a completely monotone density. That is, Π has a density π whose n th derivative π (n) exists for all n ≥ 1 and satisfies This assumption is known to be a sufficient optimality condition for threshold strategies in the classical spectrally negative case by Loeffen [14], for the absolutely continuous case (with Λ = 0) by Kyprianou et al [11], and for the periodic case by Noba et al [15] (with Λ = 0).…”

American options under periodic exercise opportunities

“…This paper is motivated by recent developments on the optimal dividend problem where one wants to maximize the total discounted dividends until ruin, with an extra restriction that the dividend payment opportunities arrive only periodically. It has recently been shown, for the case of exponential interarrival times, that a periodic barrier strategy is optimal when the underlying process is a spectrally one-sided Lévy process (see [3,12,13]). This current paper can be seen as its optimal stopping version.…”

Optimal Periodic Dividend Strategies for Spectrally Positive Lévy Risk Processes With Fixed Transaction Costs

“…From Noba, Pérez, Yamazaki and Yano (2018), we know that the value function of a periodic barrier strategy at barrier level b ≥ 0, π b is given by…”

“…However, with the additional assumption that the Lévy measure has completely monotonic density, the shape of the scale function is "nice" and barrier type of strategy is optimal, see e.g. Loeffen (2008b), Loeffen (2008a), Noba, Pérez, Yamazaki and Yano (2018).…”

“…In particular, one sufficient condition for the barrier type of strategy to be optimal is that the Lévy measure has a completely monotonic density, as discovered by Loeffen (2008b). Under such assumption, Noba, Pérez, Yamazaki and Yano (2018) recently proved that a periodic barrier stategy is optimal when the surplus is a spectrally negative Lévy process. Extending Loeffen (2008b) and Noba, Pérez, Yamazaki and Yano (2018), we show that a periodic (b u , b l ) strategy is also optimal under SNLP with the same assumption on the Lévy measure, when fixed transaction costs on dividends are present.…”