基于进入过程具有常利率和正则变化索赔的风险模型的渐近破产概率(英文)

本文研究了一类带常利率的,并且索赔过程由进入过程驱动的风险保险模型。在进入过程是一般更新过程以及索赔额是正则尾分布的条件下,得到了当初始资本趋于无穷时,破产概率的渐近行为,类似的结论对于进入过程是齐次泊松过程的情形也同样成立。     

两险种广义Erlang（2）风险模型的破产概率

本文考虑一类具有两个独立险种的风险模型的破产概率,假设该模型的两个索赔计数过程是独立的两个广义Erlang(2)过程。利用微分分析和矩阵表示,得到破产概率满足的一个积分-微分方程组及其边界条件。在索赔计数过程是普通Erlang(2)过程的情形下,证明了广义Lundberg方程有且仅有三个正的实数根,由此并结合破产概率满足的积分-微分方程组,给出了破产概率的Laplace变换。     

具有随机投资组合的双复合 Poisson-Geometric 过程保险风险模型的研究

研究了一个双复合Poisson-Geometric过程保险风险模型，其中保费和索赔的发生均服从复合泊松几何过程。通过鞅方法和停时的技巧，得到了关于破产概率的Lundberger不等式，调节系数方程和破产概率的表达式。生存概率可以作为衡量支付能力的指标，文章得到了无限和有限时间生存概率的微积分方程。     

变保费率复合Poisson-Geometric过程风险模型的Gerber-Shiu折现惩罚函数

本文研究了当保费率随时间变化时的复合Poisson-Geometric过程的风险模型.通过无穷小方法,得到了该模型的Gerber-Shiu折现惩罚函数所满足的更新方程.在此基础上,推导出破产概率,破产前瞬时盈余,以及破产时刻赤字分布满足的更新方程.特别地,当个体索赔服从指数分布时,通过求解微分方程,得到了该模型的破产概率的显式表达式和所满足的不等式.最后通过数值模拟和算例分析,提出了保险公司的赔付政策和保费政策对自身风险的影响.     

常利率下的更新风险模型

讨论了常利率下的更新风险模型，证明了索赔时刻Tk的资产余额Uδ（Tk)(k≥0)构成一个齐次马尔科夫链，且给出其转移概率Q（x,B)。利用转移概率得到了风险问题中的几个重要的量和分布；破产概率，破产时余额分布以及破产前瞬间余额分布的级数展开式和积分主程。     

一类马氏调制风险模型的破产概率

本文研究一类马氏调制风险模型的破产概率,在此模型中索赔到达间隔和索赔额都受一外在马氏过程的影响,保费收入则受该外在的马氏过程和公司的储备金水平的影响.本文不仅考虑了随机环境对保险公司的影响,而且考虑了保险公司为了吸引新的顾客,会根据储备金的水平来调整保费收入.因此所考虑的保险模型更加贴近现实,更加易于应用.通过向后微分讨论,根据外在过程的马氏性,严格推导出破产概率所满足的积分方程.进一步,通过拉普拉斯变换的方法,给出了积分方程的解.最后,给出一个例子来展示所得结果的可行性和有效性.     

带延迟索赔和随机收入的离散风险模型的Gerber-Shiu分析（英）

破产理论是保险数学中的重要问题，它可以为保险公司决策者提供一个非常有用的早期风险预警手段．本文研究了一个带潜在延迟索赔和随机保费收入的复合二项风险模型．利用矩母函数的技巧，得到了 Gerber-Shiu 期望折罚函数的递推公式．特别地，还得到了贴现因子为 1 的特殊情形下的 Gerber-Shiu 期望折罚函数的解析表达式．最后还得到了实际应用中的一些重要的破产特征量，包括破产概率，破产时赤字的密度函数，破产前盈余与破产时赤字的联合密度函数，以及导致破产的索赔密度函数等．     

具有线性红利界限的破产理论

本文讨论了存存线性红利界限的带随机干扰的经典风险模型,给出了破产概率的一个上界,并证明了生存概率及红利付款的期望现值分别满足一个积分-微分方程。最后给出了索赔额服从指数分布时生存概率及红利付款的期望现值的确切表达式。     

体制转换下等价鞅测度的构造研究

体制转换资产价格模型可以描述宏观经济的影响,但在金融衍生产品定价所涉及的等价鞅测度的构造问题中,利用传统的Esscher变换方法得到的等价鞅测度实质上只考虑了微观市场风险,而没有考虑体制转换所表示的宏观经济风险.此外,经典的几何布朗运动不能刻画资产收益率的尖峰厚尾现象.本文首先利用马尔科夫过程和最大化非广延熵分布建立了一个新的资产价格模型.该模型可以同时描述体制转换和尖峰厚尾现象.然后利用鞅理论,借助微观市场的资产价格过程和宏观经济的马尔科夫过程的乘积给出了一种新的等价鞅测度构造方法,通过该方法构造的等价鞅测度包含了微观市场和宏观经济两种风险.最后,在该等价鞅测度下,给出了资产价格折现过程为鞅的充要条件,为进一步研究金融衍生产品的定价及风险控制提供了理论基础.     

常利率下有阈红利边界的Erlang(2)风险模型的罚金折现期望函数

为了精确地描述风险投资商实际的经营状况,本文将一般的Erlang(2)风险模型推广为常利率下有阈红利边界的Erlang(2)风险模型。首先利用全概率公式对风险过程进行分析,得到了模型的罚金折现期望函数所满足的积分-微分方程及积分方程,然后在不带利率时将积分方程简化为"第二类非其次Volterra积分方程",给出了罚金折现期望函数的确切表达式,最后给出了不带利率时模型的破产概率及破产前瞬时盈余和破产赤字的联合分布的表达式。     

Ruin Probability in a Generalised Risk Process under Rates of Interest with Homogenous Markov Chains

This article explores recursive and integral equations for ruin probabilities of generalised risk processes, under rates of interest with homogenous Markov chain claims and homogenous Markov chain premiums. We assume that claim and premium take a countable number of non-negative values. Generalised Lundberg inequalities for the ruin probabilities of these processes are derived via a recursive technique. Recursive equations for finite time ruin probabilities and an integral equation for the ultimate ruin probability are presented, from which corresponding probability inequalities and upper bounds are obtained. An illustrative numerical example is discussed.     

A modelling procedure to optimize component safety inspection over a finite time horizon

In this paper a model of a safety inspection process is proposed for the expected consequence of inspections over a finite time horizon. A single dominant failure mode is modelled, which has considerable safety or risk consequences assumed measurable either in cost terms or in terms of the probability of failure over the time horizon. The model established extends earlier work assuming an infinite time horizon, and uses the concept of delay time and asymptotic results from the theory of renewal and renewal reward processes. The paper establishes a pragmatic procedure for formulating objective functions which may be optimized to determine the optimal inspection intervals. Merits of both the exact and asymptotic formulations of these objective functions for possible use in the inspection optimization process are considered. Although the procedure for developing an objective function over a finite time zone inspection process assumes perfect inspection, it can be generalized to the imperfect inspection case. Because of the intractability of the mathematics, it is suggested that when optimizing an inspection process over a finite time zone, an asymptotic formulation of the objective function should be optimized, and this solution then checked and if necessary refined, using simulation calculation. A numerical example illustrates the performance of the basic periodic inspection policy over different time horizons using the asymptotic solution. The results are compared with simulations performed to estimate the exact expected cost measure. It is shown that the simpler asymptotic solution is satisfactory in the case considered, especially when the time horizon is relatively long. © 1997 John Wiley & Sons, Ltd.     

两步保费率下的Erlang（2）风险过程

本文研究了两步保费率下Erlang(2)风险过程,给出了Gerber-Shiu折现罚函数的两个微积分方程及其解或更新方程.在索赔额为指数分布条件下得到了两个与破产相关的量并计算出了相应的数值结果.     

Applications to continuous-time processes of computational techniques for discrete-time renewal processes

For optimising maintenance, the total costs should be computed over a bounded or unbounded time horizon. In order to determine the expected costs of maintenance, renewal theory can be applied when we can identify renewals that bring a component back into the as-good-as-new condition. This publication presents useful computational techniques to determine the probabilistic characteristics of a renewal process. Because continuous-time renewal processes can be approximated with discrete-time renewal processes, it focusses on the latter processes. It includes methods to compute the probability distribution, expected value and variance of the number of renewals over a bounded time horizon, the asymptotic expansion for the expected value of the number of renewals over an unbounded time horizon, the approximation of a continuous renewal-time distribution with a discrete renewal-time distribution, and the extension of the discrete-time renewal model with the possibility of zero renewal times (in order to cope with an upper-bound approximation of a continuous-time renewal process).     

Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.