马氏环境下带干扰Cox风险模型的罚金折现期望函数

研究了马氏环境下带干扰的Cox风险模型.首先给出了罚金折现期望函数满足的积分方程,然后给出了破产概率,破产前瞬时盈余、破产赤字的分布及各阶矩所满足的积分方程.最后给出当索赔额服从指数分布且理赔强度为两状态时的破产概率的拉普拉斯变换.     

马氏环境中单生链的灭绝概率

利用算子S_θ,分临界和非临界两种情况,讨论了马氏环境中单生链必然灭绝与非必然灭绝的充分条件;给出了随机环境中单生链灭绝概率的一个上界.     

一类马氏调制费率的双险种风险模型的破产概率

给出一类具有费率均为马氏调制的双险种风险模型,对于给定的初始状态,求出了条件破产概率满足的积分方程,并推导出具有平稳初始分布的破产概率的递归不等式和零初始资产时的破产概率的简洁估计式.     

变保费率Cox风险模型的研究

研究了当保费率随理赔强度的变化而变化时C ox风险模型的折现罚金函数,利用后向差分法得到了折现罚金函数所满足的积分方程,进而得到了破产概率,破产前瞬时盈余、破产时赤字的各阶矩所满足的积分方程.最后给出当理赔额服从指数分布,理赔强度为两状态的马氏过程时破产概率的拉普拉斯变换,对一些具体数值计算出了破产概率的表达式.     

流行病灭绝时间和费用估计随机模型

本文用具有吸收状态的生灭马氏过程建立了流行病随机模型,研究了灭绝之前生灭过程的分布,发现初始分布是拟平稳分布时,其灭绝时间服从指数分布,并得到了灭绝时间与状态概率的关系式和费用估计的期望值.应用模型给出了一个固定人口为N的流行病灭绝时间和平均费用的数值模拟结果.     

随机环境中两性分枝过程的马氏性与灭绝

证明了随机环境中的两性分枝过程是随机环境中马氏链,得到了概率母函数的一些关系,并利用这些关系,针对各种不同的具体的配对函数,给出了过程必然灭绝的一些充分条件.     

马氏调制费率的带干扰风险模型

考虑了一类具有马氏调制的带干扰连续时间风险模型,得到了该模型下其条件Gerber-Shiu折现罚金函数所满足的积分方程,Laplace变换及渐近解.在两状态情形下,当索赔额的分布为有理数情况时得到了条件Gerber-Shiu折现罚金函数的具体表达式并给出了数值例子     

马氏环境中马氏链的一个概率不等式及其应用

本文研究了马氏环境中马氏链构成的随机变量之和的概率不等式问题.利用了结尾的方法,获得了马氏环境中马氏链构成的随机变量之和的尾部概率不等式,作为结果的应用,给出了将过程限制在(S,S∩F,PS)上的强大数定律.文中提出的方法和结果对研究独立的随机变量之和的大样本性质是十分有用的.     

马氏环境中的马氏链与马氏双链

本文研究了马氏环境中的马氏链,利用马氏双链的性质,得到了马氏环境中的马氏链回返于小柱集上的概率的若干估计式.     

保费收入为Poisson过程的更新风险模型

对于保费收入为Poisson过程的更新风险模型,利用马氏链的理论,借助转移概率,得出了破产概率和破产赤字的展式及其所满足的积分方程.     

常利率下的Cox模型的破产概率

讨论了常利率下的Cox模型的破产概率 ,分别得到了条件破产概率和最终破产概率所满足的积分方程 .     

Condition number of matrices derived from two classes of integral equations

We investigate some integral equations, i. a. the so-called Kupradze functional equations, where the two variables of the kernel belong to two different point sets. An extensive survey of the literature shows the various applications of these equations. By a discretization of the integral equations they are replaced by systems of linear algebraic equations. The condition number of the corresponding matrices is investigated, analytically and numerically. It is thereby quantitatively found in which way the condition of the matrices deteriorates when the two point sets are moved away from each other.     

Estimation of the number of failures in the Weibull model using the ordinary differential equation

In estimating the number of failures using right truncated grouped data, we often encounter cases that the estimate is smaller than the true one when we use the likelihood principle to conditional probability. In infectious disease spread predictions, the SIR model described by simultaneous ordinary differential equations is commonly used, and it can predict reasonably well the number of infected patients even when the size of observed data is small. We have investigated whether the ordinary differential equation model can estimate the number of failures more accurately than does the likelihood principle under the condition of right truncated grouped data. The positive results are obtained in the Weibull model, similarly to the cases of the SARS, A(H1N1), and FMD.     

Branching Random Walk in a Catalytic Medium. I. Basic Equations

We consider a continuous-time branching random walk on the integer lattice ^d (d 1 ) with a finite number of branching sources, or catalysts. The random walk is assumed to be spatially homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the random walk, is governed by a Markov branching process. The quantities of interest are the local numbers of particles (at each site) and the total population size. In the present paper, we derive and analyze the Kolmogorov type backward equations for the corresponding Laplace generating functions and also for the successive integer moments and the process extinction probability. In particular, existence and uniqueness theorems are proved and the problem of explosion is studied in some detail. We then rewrite these equations in the form of integral equations of renewal type, which may serve as a convenient tool for the study of the process long-time behavior. The paper also provides a technical foundation to some results published before without detailed proofs.     

Ruin problems for an autoregressive risk model with dependent rates of interest

In this paper, we consider a discrete insurance risk model in which the claims, the premiums and the rates of interest are assumed to have dependent autoregressive structures (AR(1)). We derive recursive and integral equations for expected discounted penalty function. By these equations, we obtain generalized Lundberg inequality for the infinite time severity of ruin and hence for the infinite time ruin probability, consider asymptotic formula for the finite time ruin probability when loss distributions have regularly varying tails, and study some probability properties of the duration of ruin.     

On a logistic growth model with predation and a power‐type diffusion coefficient: I. Existence of solutions and extinction criteria

We consider a logistic growth model with a predation term and a stochastic perturbation yielding constant elasticity of variance. The resulting stochastic differential equation does not satisfy the standard assumptions for existence and uniqueness of solutions, namely, linear growth and the Lipschitz condition. Nevertheless, for any positive initial condition, we prove that a solution exists and is unique up to the first time it hits zero. Additionally, we provide alternative criteria for population extinction depending on the choice of parameters. More precisely, we provide criteria that guarantee the following: (i) population extinction with positive probability for a set of initial conditions with positive Lebesgue measure; (ii) exponentially fast population extinction with full probability for any positive initial condition; and (iii) population extinction in finite time with full probability for any positive initial condition. Copyright © 2015 John Wiley & Sons, Ltd.     

On the solution of a class of integral equations

In this paper we consider a class of Fredholm integral equations of the first kind which arise in a large number of problems in applied mathematics. Although only certain special cases of the equations can be solved exactly, it is shown that a constructive method can be developed for reformulating the equations as Fredholm integral equations of the second kind. This approach will be seen to cover and bring together the large number of isolated cases of the equations which have appeared in the literature. Several examples are given to illustrate the general method.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．