带注资的二维复合泊松模型的最优分红

研究建立两类理赔关系的二维复合泊松模型的最优分红与注资问题,目标为最大化分红减注资的折现. 该问题由随机控制问题刻画, 通过解相应的哈密尔顿-雅克比-贝尔曼(HJB)方程,得到了最优分红策略,并在指数理赔时明确地解决该问题.     

复合Poisson模型带投资-借贷利率和固定交易费用的最优分红策略

本文研究了复合Poisson模型带投资-借贷利率和固定交易费用的最优分红问题。通过控制分红时刻和分红量,最大化直到绝对破产时刻的累积期望折现分红。由于考虑固定交易费用,问题为一个随机脉冲控制问题。首先,本文给出了一个策略是平稳马氏策略的充分必要条件。借助于测度值生成元理论得到测度值动态规划方程(简称测度值DPE),并且在没有任何附加条件下证明了验证定理。通过Lebesgue分解,本文讨论了测度值DPE和拟变分不等式(简称QVI)之间的关系,证明了最优分红策略为具有波段结构的平稳马氏策略。最后,本文给出了求解n-波段策略和相应值函数的算法。当索赔额服从指数分布时,得到了值函数的显示解和最优分红策略。     

跳扩散模型下基金平衡管理的最优脉冲控制

在基金市值波动服从跳扩散过程, 基金持有的罚金成本为当前基金水平的二次函数及存在交易费的假设下研究了无穷时域的基金平衡管理的最小成本模型. 利用随机最优脉冲控制的拟变分不等式理论建立了判定定理,得到了最优脉冲控制策略的存在性, 同时通过构造方法给出了解的数学结构形式.     

离散时间模型下带分红交易费的最优分红

研究离散Sparre-Andersen模型下带分红交易费的最优分红问题.在分红有界的条件下,通过更新初始时间得到最优值函数并证明最优值函数为Hamilton-Jacobi-Bellman方程的唯一有界解.另外,运用Bellman递推算法通过最优值变换获得最优分红.     

扩散风险模型中随机时间区间最优分红和再保险问题

该文在扩散风险模型中研究随机时间区间最优分红和再保险问题.假设应用比例再保险策略,随机时间服从指数分布,若破产时刻先于随机时刻到来,则在破产时刻存在一个固定数额的非负价值;若随机时刻先于破产时刻到来,则在随机时刻存在另一个固定数额的非负价值,得到了最优分红和再保险策略,以及值函数的表达式,并给出一个数值例子.     

复合二项对偶模型的最优分红问题

研究复合二项对偶模型的最优分红问题,通过分析HJB方程得到了最优分红策略和相应的最优值函数之间的关系以及最优值函数的简单计算方法.通过讨论最优红利策略的一些性质得到了最优值函数的可无限逼近的上界和下界.     

带交易费用的最优投资和比例再保险

研究了保险公司的最优投资和再保险问题.保险公司的盈余通过跳-扩散风险模型来模拟,可以把盈余的一部分投资到金融市场,金融市场由一个无风险资产和n个风险资产组成,并且保险公司还可以购买比例再保险;在买卖风险资产时,考虑了交易费用.通过随机控制的理论,获得了最优策略和值函数的显示解.     

基于注资-阀值分红的随机微分投资-再保博弈

为了更好地反映模型风险对保险公司金融策略的影响,考虑了存在模型风险时,保险公司的最优投资-再保-注资-阀值分红策略问题.在分红与注资总量的贴现值之差的期望最大化的准则下,使用零和随机微分博弈理论建立了保险公司的随机微分博弈模型,通过求解HJBI方程得到了最优投资-再保-注资-阀值分红策略的显式解.最后在有模型风险和无模型风险两种不同情形下,通过数值算例分析了保险公司金融策略之间的差异,为保险资金的管理提供了重要的决策指导.     

多用户多准则随机系统最优与最优收费

针对固定交通需求量和出行者的时间价值为离散分布的多准则随机交通均衡,分别研究了依费用度量和依时间度量的多用户多准则随机系统最优和最优收费问题.分别建立了基于费用和基于时间的随机系统最优的最优化模型,阐述了该模型解的唯一性条件及等价的变分不等式问题.运用变分不等式方法,研究了一阶最优收费的可行性,即能否依边际定价原则,通过收取与出行者类别无关的道路收费使多用户多准则随机均衡流与随机系统最优流一致.一阶最优收费不适用于依时间度量的随机系统最优情况,因而建立了一个最优化模型来得到此时的非歧视性道路收费.最后给出了具体算例.     

正相依风险下离散模型的最优分红

将保险公司各期净损失相互独立的假定改进为依随机序正相依.在相依风险下,利用动态规划原理和状态空间约简,刻画了最优分红策略,证明了区域策略最优,同时讨论了值函数的性质,并给出了数值算法.其中,对涉及独立假定的结论,给出了相依条件下的相应结果,对未涉及独立假定的部分结论也做了改进.研究发现,与独立情形不同,在依随机序正相依风险下,保险公司不必以概率1破产.     

Optimal dividend strategies in discrete risk model with capital injections

In this paper we consider a doubly discrete model used in Dickson and Waters (biASTIN Bulletin 1991; 21 :199–221) to approximate the Cramér–Lundberg model. The company controls the amount of dividends paid out to the shareholders as well as the capital injections which make the company never ruin in order to maximize the cumulative expected discounted dividends minus the penalized discounted capital injections. We show that the optimal value function is the unique solution of a discrete Hamilton–Jacobi–Bellman equation by contraction mapping principle. Moreover, with capital injection, we reduce the optimal dividend strategy from band strategy in the discrete classical risk model without external capital injection into barrier strategy , which is consistent with the result in continuous time. We also give the equivalent condition when the optimal dividend barrier is equal to 0. Although there is no explicit solution to the value function and the optimal dividend barrier, we obtain the optimal dividend barrier and the approximating solution of the value function by Bellman's recursive algorithm. From the numerical calculations, we obtain some relevant economical insights. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal dividend and dynamic reinsurance strategies with capital injections and proportional costs

We consider an optimization problem of an insurance company in the diffusion setting, which controls the dividends payout as well as the capital injections. To maximize the cumulative expected discounted dividends minus the penalized discounted capital injections until the ruin time, there is a possibility of (cheap or non-cheap) proportional reinsurance. We solve the control problems by constructing two categories of suboptimal models, one without capital injections and one with no bankruptcy by capital injection. Then we derive the explicit solutions for the value function and totally characterize the optimal strategies. Particularly, for cheap reinsurance, they are the same as those in the model of no bankruptcy.     

Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs

In this paper we consider the dividend payments and capital injections control problem in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. The objective is to maximize the expected present value of the dividends minus the discounted costs of capital injections. This paper can be considered as an extension of Yao et al. (2010), we include fixed transaction costs incurred by capital injections in this paper. This leads to an impulse control problem. Using the techniques of quasi-variational inequalities (QVI), this optimal control problem is solved. Numerical solutions are provided to illustrate the idea and methodologies, and some interesting economic insights are included.     

Optimal dividend strategy in compound binomial model with bounded dividend rates

We consider the compound binomial model, and assume that dividends are paid to the shareholders according to an admissible strategy with dividend rates bounded by a constant.The company controls the amount of dividends in order to maximize the cumulative expected discounted dividends prior to ruin. We show that the optimal value function is the unique solution of a discrete HJB equation. Moreover, we obtain some properties of the optimal payment strategy, and offer a simple algorithm for obtaining the optimal strategy. The key of our method is to transform the value function. Numerical examples are presented to illustrate the transformation method.     

Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments

In this paper, we study the optimal dividend and capital injection problem with the penalty payment at ruin. The dividend strategy is assumed to be restricted to a small class of absolutely continuous strategies with bounded dividend density. By considering the surplus process killed at the time of ruin, we transform the problem to a combined stochastic and impulse control one up to ruin with a free boundary at zero. We illustrate the theoretical verifications for different types of capital injection strategies comparing to the conventional results given in the literature, where the capital injections are made before the time of ruin. Under the assumption of restricted dividend density, the value function is proved as the unique increasing, bounded, Lipschitz continuous and upper semi-continuous at zero viscosity solution to the corresponding quasi-variational Hamilton–Jacobi–Bellman (HJB) equation. The uniqueness of such class of viscosity solutions is shown by considering its boundary condition at infinity. The optimality of a specific band-type strategy is proved for the case when the premium rate is (i) greater than or (ii) less than the ceiling dividend rate respectively. Some numerical examples are presented under the exponential and gamma claim size assumptions.     

考虑红利支付与提前退休的最优投资组合

研究了在经济代理人通过不可逆退休时间选择来调整劳动时间框架下的最优消费和投资问题,主要考虑风险资产派发红利的情形.运用随机控制方法,求解使得消费-闲暇预期效用最大化的最优策略.最优投资组合及最优退休时刻表明,代理人在为提前退休积累财富的同时,也能最佳享受消费和闲暇所带来的快乐.     

The compound Poisson risk model with dependence under a multi-layer dividend strategy

In this paper, a compound Poisson risk model with time-dependent claims is studied under a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber-Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.     

The perturbed compound Poisson risk model with constant interest and a threshold dividend strategy

In this paper, we consider the compound Poisson risk model perturbed by diffusion with constant interest and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the nth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber-Shiu functions. The special case that the claim size distribution is exponential is considered in some detail.     

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In the classical Cram\'{e}r-Lundberg model in risk theory the problem of finding the optimal dividend strategy and optimal dividend return function is a widely discussed topic. In the present paper, we discuss the problem of maximizing the expected discounted net dividend payments minus the expected discounted costs of injecting new capital, in the Cram\'{e}r-Lundberg model with proportional taxes and fixed transaction costs imposed each time the dividend is paid out and with both fixed and proportional transaction costs incurred each time the capital injection is made. Negative surplus or ruin is not allowed. By solving the corresponding quasi-variational inequality, we obtain the analytical solution of the optimal return function and the optimal joint dividend and capital injection strategy when claims are exponentially distributed.