生灭链和生灭过程的观测与统计

本文证明了具有反射壁的生灭链（生灭过程）在一个反射壁的击中时分布能够完全唯一决定整个生灭链（生灭过程）。并将该结果应用到了生物学上细胞膜离子单通道门控动力学分析，产生了与实验一致的结果。     

确定环形Markov链的Q-矩阵

给出了Markov链中任一状态集的逗留时间或击中时间的分布(混合指数分布),以及其分布的各阶微分与Q-矩阵之间的约束方程组.利用该约束关系及环形链结构的先验信息,采用Markov链反演方法证明了:对于有限状态环形Markov链,其Q-矩阵能由其中任意两个相邻状态的逗留时间和击中时间分布唯一决定,并给出了相应的算法.     

非时齐复合泊松过程下对偶风险模型的破产特征量研究

将经典的对偶风险模型中的收益到达过程推广为非时齐的泊松过程.运用经典方法和时变方法,计算了该模型下的破产概率,并定义了时变后相应模型的广义期望折罚函数,验证了时变方法对非时齐泊松风险模型的有效性,最后又考虑了该模型在带壁分红策略下的情形,当单次索赔额服从指数分布时,得到了它的期望折罚函数以及期望折现分红函数.     

基于HPM的高中数学概念课教学设计与应用研究——以“函数的概念”教学为例

从数学史的角度看,函数概念经历了“萌芽”“解析定义”“对应定义”“集合定义”四个时期.本文中通过若干情境,结合数学史对函数概念的教学进行了重构,加强函数概念发展史内容的渗透,促使学生更好地掌握所学内容,提升学生的人文情怀,提高学生数学核心素养.     

OPTIMAL PROPORTIONAL REINSURANCE AND INVESTMENT FOR A CONSTANT ELASTICITY OF VARIANCE MODEL UNDER VARIANCE PRINCIPLE

This article studies the optimal proportional reinsurance and investment problem under a constant elasticity of variance(CEV) model.Assume that the insurer’s surplus process follows a jump-diffusion process,the insurer can purchase proportional reinsurance from the reinsurer via the variance principle and invest in a risk-free asset and a risky asset whose price is modeled by a CEV model.The diffusion term can explain the uncertainty associated with the surplus of the insurer or the additional small claims.The objective of the insurer is to maximize the expected exponential utility of terminal wealth.This optimization problem is studied in two cases depending on the diffusion term’s explanation.In all cases,by using techniques of stochastic control theory,closed-form expressions for the value functions and optimal strategies are obtained.     

MARKOV CHAIN INVERSION APPROACH TO IDENTIFY THE TRANSITION RATES OF ION CHANNELS

We consider how to identify the transition rates of ion channels with the underlying scheme which is kinetically modelled as time-homogeneous Markov chain.A Markov chain inversion approach is developed to perform a difficult inversion to identify the transition rates from the parameters characterizing the lifetime distributions at a small number of states,although it is straightforward to derive the lifetime distribution.The general explicit equations relating the parameters of the lifetime distribution to the transition rates are derived and transition rates are then obtained as roots to this system of equations.The concrete solutions are proposed to the basic and regular schemes such as linear,star-graph branch and loop.Useful conclusions and solutions to realistic schemes are also included to show its efficiency.     

基于Gibbs抽样的贝叶斯稳健ARMA模型研究

针对ARMA模型建模过程中模型识别和参数估计易受观测值异常点影响问题,构建了同时考虑加性异常点和更新性异常点的ARMA模型．运用基于Gibbs抽样的Markov Chain Monte Carlo贝叶斯方法,估计稳健ARMA模型参数,同步确定观测值中异常点的位置,辨别异常点类型．并利用我国人口自然增长数据进行仿真分析,研究结果表明：贝叶斯方法能够有效地识别ARMA序列的异常点．     

风险相依下再保险双方的联合最优再保险问题

结合保险人和再保险人的共同利益，研究了具有两类相依险种风险模型下的最优再保险问题.假定再保险公司采用方差保费原理收取保费，利用复合Poisson模型和扩散逼近模型两种方式去刻画保险公司和再保险公司的资本盈余过程，在期望效用最大准则下，证明了最优再保险策略的存在性和唯一性，通过求解Hamilton-Jacobi-Bellman（HJB）方程，得到了两种模型下相应的最优再保险策略及值函数的明晰解答，并给出了数值算例及分析.     

层次模型Markov链的观测与统计

对于连续时间的层次模型M arkov链,所有的转移速率都可以由最底层状态的逗留时间和击中时间分布惟一决定,因而整个M arkov链的统计性质由它们的统计所决定.并给出了相应的算法和数例.