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 系统科学与数学 , 2010, Abstract: This paper is concerned with the compound binomial model with general premium rate. The linear equations satisfied by the values of the Gerber-Shiu penalty function is given, and an upper bound and a lower bound of the penalty function are obtained.
 系统科学与数学 , 2007, Abstract: his paper considers the compound binomial risk model, studies the defective renewal equation satisfied by the Gerber-Shiu discount penalty function, and obtains the asymptotic relationship of the the Gerber-Shiu discount penalty function basied on the renewal theory. Furthermore, the asymptotic relationships of the joint distributions and the marginal distributions of the deficit at ruin and the surplus immediately prior to ruin are given.These results extend that obtained by Pavlova and Willmot in 2004.
 方世祖,张春梅,赵培臣,孙歆 应用概率统计 , 2011, Abstract: 本文研究复合马尔可夫二项模型的Gerber-Shiu折现罚金函数,得到了有条件和无条件的Gerber-Shiu折现罚金函数所满足的瑕疵更新方程.然后给出这些折现罚金函数的渐近表达式.
 应用概率统计 , 2010, Abstract: 本文研究了一类带扰动风险模型,得到了此过程下Gerber-Shiu函数的微分积分方程,并得到了推广Erlang(2)情形下Gerber-Shiu函数满足的更新方程
 肖菊霞 Pure Mathematics (PM) , 2014, DOI: 10.12677/PM.2014.44022 Abstract: 相位分布的研究在研究正半轴的其他分布中起着重要作用。考虑带常利率的时间间隔为相位分布的更新风险模型。首先推导出Gerber-Shiu期望折现罚金函数满足的积分微分方程，然后经过一系列的推导过程得到Volterra形式的矩阵积分方程，从而得到Gerber-Shiu期望折现罚金函数的一种解法。Research in the phase-type distribution has an important influence for the research of other dis-tributions on the positive real axis. It considers the risk model with the phase-type inter-claim times and for constant interest, it first derives the integral-differential equation satisfied by the Gerber-Shiu discounted penalty function. Then through a series of deriving, it obtains the volterra integral equation in a form of matrix. It gets a method of solving the Gerber-Shiu expected penalty function.
 Mathematics , 2014, Abstract: Inspired by works of Landriault et al. \cite{LRZ-0, LRZ}, we study discounted penalties at ruin for surplus dynamics driven by a spectrally negative L\'evy process with Parisian implementation delays. To be specific, we study the so-called Gerber-Shiu functional for a ruin model where at each time the surplus process goes negative, an independent exponential clock with rate $q>0$ is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative L\'evy processes and relies on the theory of the so-called scale functions. In particular, our results extend recent results of Landriault et al. \cite{LRZ-0, LRZ}.
 Quantitative Finance , 2011, DOI: 10.1214/14-AAP1038 Abstract: This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative L\'{e}vy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin, which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. A number of concrete examples are analyzed.
 范庆祝~~尹传存 应用概率统计 , 2009, Abstract: 本文考虑了一类相邻两次索赔的时间间隔服从Erlang($n$)和Erlang($m$)的混合分布的SparreAndersen风险模型.主要目的是研究Gerber-Shiu函数$\phi_\delta(u)$,首先证明了$\phi_\delta(u)$满足一个高阶的积分微分方程,然后讨论了广义Lundberg方程根的性质,在此基础上导出了$\phi_\delta(u)$的拉普拉斯变换并且证明了$\phi_\delta(u)$满足一个更新方程,最后给出了一个例子.
 Journal of Mathematical Finance (JMF) , 2016, DOI: 10.4236/jmf.2016.64039 Abstract: This paper analyses the Gerber-Shiu penalty function of a Markov modulated risk model with delayed by-claims and random incomes. It is assumed that each main claim will also generate a by-claim and the occurrence of the by-claim may be delayed depending on associated main claim amount. We derive the system of integral equations satisfied by the penalty function of the model. Further, assuming that the premium size is exponentially distributed, an explicit expression for the Laplace transform of the expected discounted penalty function is derived. For a two-state model with exponential claim sizes, we present the explicit formula for the probability of ruin. Finally we numerically illustrate the influence of the initial capital on the ruin probabilities of the risk model using a specific example. An example for the risk model without any external environment is also provided with numerical results.
 Ying Shen Applied Mathematics (AM) , 2018, DOI: 10.4236/am.2018.911084 Abstract: In this paper, we focus on the perturbed risk model with dependent relation and consider the relevance from two aspects. For one side, we use copula function to model the structure of the claim size and interclaim time, and on the other side, we establish the change of premium rat depending on the random thresholds. At last, we obtain the Integro-differential equations and its Laplace transforms of the Gerber-Shiu functions for the new risk model.
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