We consider the general class of spectrally positive Lévy risk processes, which are appropriate for businesses with continuous expenses and lump sum gains whose timing and sizes are stochastic. Motivated by the fact that dividends cannot be paid at any time in real life, we study $\textit{periodic}$ dividend strategies whereby dividend decisions are made according to a separate arrival process.
In this paper, we investigate the impact of fixed transaction costs on the optimal periodic dividend strategy, and show that a periodic $(b_u,b_l)$ strategy is optimal when decision times arrive according to an independent Poisson process. Such a strategy leads to lump sum dividends that bring the surplus back to $b_l$ as long as it is no less than $b_u$ at a dividend decision time. The expected present value of dividends (net of transaction costs) is provided explicitly with the help of scale functions. Results are illustrated.
Maximising dividends is one classical stability criterion in actuarial risk theory. Motivated by the fact that dividends are paid periodically in real life, $\textit{periodic}$ dividend strategies were recently introduced (Albrecher, Gerber and Shiu, 2011). In this paper, we incorporate fixed transaction costs into the model and study the optimal periodic dividend strategy with fixed transaction costs for spectrally negative Lévy processes. The value function of a periodic $(b_u,b_l)$ strategy is calculated by means of exiting identities and Itô's excusion when the surplus process is of unboun…
This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative \lev process with a final payoff at an exponential terminal time…
For an insurance company with reserve modeled by the spectrally negative Lévy process, we study the optimal impulse dividend maximizing the expected accumulated net dividend payment subtracted by the accumulated cost of injecting capital. In this setting, the beneficiary of the dividends injects capital to ensure a non-negative risk process so that the insurer never goes bankrupt. The optimal impulse dividend and capital injection strategy together with its value function are obtained.
In this paper, we investigate the problem of optimal strategies of dividend and reinsurance under the Cramér-Lundberg risk model embedded with the thinning-dependence structure which was firstly introduced by Wang and Yuen (2005), subject to the optimality criteria of maximizing the expected accumulated discounted dividends paid until ruin. To enhance the practical relevance of the optimal dividend and reinsurance problem, non-cheap reinsurance is considered and transaction costs and taxes are imposed on dividends, which converts our optimization problem into a mixed classical-impulse control …
In this paper, we model the cash surplus (or equity) of a risky business with a Brownian motion. Owners can take cash out of the surplus in the form of "dividends", subject to transaction costs. However, if the surplus hits 0 then ruin occurs and the business cannot operate any more. We consider two types of dividend distributions: (i) periodic, regular ones (that is, dividends can be paid only at countable many points in time, according to a specific arrival process); and (ii) extraordinary dividend payments that can be made immediately at any time (that is, the dividend decision time space i…
On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs
在Bai和Paulsen（SIAM J. Control optim. 48，2010）交易费用下的最优股息问题进行了分析，一类相当普遍的扩散过程。它被分成几个子类，对于大多数子类，最优策略是一个简单的障碍策略;每当过程遇到一个上障碍$\bar{u}^*$时，通过股息支付将其减少到$\bar{u}^*-ξ$。扣除交易成本后，股东将获得$kξ-K$。证明了简单的屏障策略并不总是最优的，这里解决了这些更困难的情况。最优解是相当复杂的，但有趣的。
Solving Optimal Dividend Problems via Phase-type Fitting Approximation of Scale Functions
De Finetti（1957）的最优分红问题最近被推广到谱负Lévy模型，其中最优策略的实现依赖于尺度函数及其导数的计算。本文提出了一种相位型拟合逼近的最优策略。我们考虑具有相型跳跃的谱负Lévy过程以及亚纯Lévy过程（Kuznetsov et al.，2010 a），并使用其尺度函数来近似一般谱负莱维过程的尺度函数。我们得到了解析的收敛结果和数值说明的有效性的逼近方法使用的例子与谱负莱维过程与i.i.d.威布尔分布跳跃，β族和CGMY过程。
This paper studies the bail-out optimal dividend problem with regime switching under the constraint that the cumulative dividend strategy is absolutely continuous. We confirm the optimality of the regime-modulated refraction-reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. To verify the conjecture of a barrier type optimal control, we first introduce and study an auxiliary problem with the final payoff at an exponential terminal time and characterize the optimal threshold explicitly using fluctuation identities of the refracted-r…
Optimal Reinsurance-Dividend Strategy with Fixed Transaction Costs in a Regime-Switching Brownian Risk Model: A Viscosity Solution to the Impulse Control Problem
Fujian Normal University（福建师范大学）；The Hong Kong Polytechnic University（香港理工大学）；Xi’an Jiaotong University（西安交通大学）
The first motivation of our paper is to explore further the idea that, in risk control problems, it may be profitable to base decisions both on the position of the underlying process Xt and on its supremum Xt := sup 0$\le$s$\le$t Xs. Strongly connected to Azema-Yor/generalized draw-down/trailing stop time (see [AY79]), this framework provides a natural unification of draw-down and classic first passage times. We illustrate here the potential of this unified framework by solving a variation of the De Finetti problem of maximizing expected discounted cumulative dividends/consumption gained under…
We consider a modification of the dividend maximization problem from ruin theory. Based on a classical risk process we maximize the difference of expected cumulated discounted dividends and total expected discounted additional funding (subject to some proportional transaction costs). For modelling dividends we use the common approach whereas for the funding opportunity we use the jump times of another independent Poisson process at which we choose an appropriate funding height. In case of exponentially distributed claims we are able to determine an explicit solution to the problem and derive a…
In this paper we consider the De Finetti's optimal dividend and capital injection problem under a Markov additive model. We assume that the surplus process before dividends and capital injections follows a spectrally positive Markov additive process. Dividend payments are made only at the jump times of an independent Poisson process. Capitals are required to be injected whenever needed to ensure a non-negative surplus process to avoid bankruptcy. Our purpose is to characterize the optimal periodic dividend and capital injection strategy that maximizes the expected total discounted dividends su…
This paper studies de Finetti's optimal dividend problem with capital injection. We confirm the optimality of a double barrier strategy when the underlying risk model follows a Lévy process that may have positive and negative jumps. The main result in this paper is a generalization of Theorem 3 in Avram et al.(2007), which is the spectrally negative case, and Theorem 3.1 in Bayraktar et al.(2013), which is the spectrally positive case. In contrast with the spectrally one-sided cases, double barrier strategies cannot be handled by using scale functions to obtain some properties of the expected …
In this paper, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction--reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes.
