B—模糊集合代数和广义互信息公式

基于两种概率的区分，推导出了一个广义Shannon熵公式和一个广义互信息公式。后者和模糊性有关，并且柯用于语言和感觉中的信息度量。为了由原子语句为真的条件概率求出合语句为真的条件概率，提出了一个遵循存尔运算的模糊集合代数。所谓的模糊信息被还原为概率信息。新的理论在经典理论－概率论，集合论及Shannon信息论－的基础上容易理解。     

Fuzzy概率空间的Fuzzy格刻划

本文研究了Fuzzy概率空间中Fuzzy事件及概率的代数性质.利用Fuzzy概率空间中的概率为集函数这一特征和Fuzzy格的相关理论，得到了Fuzzy概率空间中的概率是从一个Fuzzy格到某个区间的Fuzzy格模同态，并将概率分解成Fuzzy格同态与Fuzzy格模同态的乘积。     

概率

重点：随机事件的概率，等可能事件的概率，互斥事件有一个发生的概率，相互独立事件同时发生的概率．     

一类双险种风险过程的破产概率的估计

本文研究了一类双险种风险模型，理赔额均服从指数分布，其中一个险种的保费到达为齐次Poisson过程，给出了最终破产概率的上界和t。时刘之间破产概率的一个上界估计。     

概率区间空间中的新型KKM定量及应用

本文建立了要概率区间空间的概念，并在此框架下建立了一个新型的ＫＫＭ定理，作为应用我们得到概率区间空间中的一个新的极大极小定理和截口定理，匹配定理及一些重合点定理。     

保险系统中一类双险种风险模型的破产概率

本研究了一类双险种风险模型，对此模型得到了最终破产概率的一般表达式和破产概率的一个上界估计。     

概率

概率问题与实际问题联系密切，是排列组合的一个重要应用．本章介绍了四种基本的概率模型：等可能事件的概率、互斥事件的概率、相互独立事件的概率和事件在九次独立重复试验中恰好发生k次的概率．解概率题的关键是要搞清楚事件的类型．     

概率约束随机规划的一种近似方法及其它的有效解模式

根据最小风险的投资最优问题，我们给出了一个统一的概率约束随机规划模型。随后我们提出了求解这类概率约束随机规划的一种近似算法，并在一定的条件下证明了算法的收敛性。此外，提出了这种具有概率约束多目标随机规划问题的一种有效解模型。     

Radon概率空间中随机过程到Loeb概率空间中的转换

在Lindstrom于1980年工作的基础上进一步研究了标准概率空间的弱Loeb空间表示的性质，把Radon概率空间中的局部鞅和半鞅转换到了超有限的Loeb概率空间中，为非标准分析方法在随机分析中的应用提供了一个有效的框架。     

用Venn图表示相互独立事件及其概率

在概率论中，常用Ｖｅｎｎ图来表示事件及其概率，但尚未见用它来表示相互独立事件及其概率的报导．本文提出了一个解决这一问题的方法     

The Random Parameters AACD Models and Their Geometric Ergodicity

This paper proposes a new type of random parameters AACD (RPAACD) models, which extends the AACD model. Depending on the state of the price process, the RPAACD models seem to be a valuable alternative to existing approaches and have the better overall performance. We give the transition probability of the process. Moreover by employing the transition probability, we obtain the probability properties of the RPACD model.     

转移概率最优可测耦合的存在性

转移概率最优可测耦合的存在性是耦合理论中的基本问题之一,本文运用随机线性规划和s空间中的技巧,在较一般的条件下,给出转移概率最优可测耦合存在性的构造性证明.     

Hunt过程在Girsanov变换下的转移概率密度的表示公式

当Hunt过程为半鞅时,建立了在Girsanov变换下的转移概率密度的表示公式,并给出了变换后过程的Levy系和无穷小生成元.     

The GI/M/1 queue with exponential vacations

In this paper, we give a detailed analysis of the GI/M/1 queue with exhaustive service and multiple exponential vacation. We express the transition matrix of the imbedded Markov chain as a block-Jacobi form and give a matrix-geometric solution. The probability distribution of the queue length at arrival epochs is derived and is shown to decompose into the distribution of the sum of two independent random variables. In addition, we discuss the limiting behavior of the continuous time queue length processes and obtain the probability distributions for the waiting time and the busy period.     

Holdings of financial assets: A Markov chain analysis

This paper introduces the Markov chain model as a simple tool for analyzing the pattern of financial asset holdings over time. The model is based on transition probabilities which give the probability of switching $1 of wealth from one asset to another. An illustrative application is provided.     

On characterisation of Markov processes via martingale problems

It is well-known that well-posedness of a martingale problem in the class of continuous (or r.c.l.l.) solutions enables one to construct the associated transition probability functions. We extend this result to the case when the martingale problem is well-posed in the class of solutions which are continuous in probability. This extension is used to improve on a criterion for a probability measure to be invariant for the semigroup associated with the Markov process. We also give examples of martingale problems that are well-posed in the class of solutions which are continuous in probability but for which no r.c.l.l. solution exists.     

Conditions for Permanence and Ergodicity of Certain SIR Epidemic Models

In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with Beddington-DeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results.

     

A matrix perturbation method for computing the steady-state probability distributions of probabilistic Boolean networks with gene perturbations

Modeling genetic regulatory interactions is an important issue in systems biology. Probabilistic Boolean networks (PBNs) have been proved to be a useful tool for the task. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution involves the construction of the transition probability matrix of the PBN. The size of the transition probability matrix is 2ⁿ×2ⁿ where n is the number of genes. Although given the number of genes and the perturbation probability in a perturbed PBN, the perturbation matrix is the same for different PBNs, the storage requirement for this matrix is huge if the number of genes is large. Thus an important issue is developing computational methods from the perturbation point of view. In this paper, we analyze and estimate the steady-state probability distribution of a PBN with gene perturbations. We first analyze the perturbation matrix. We then give a perturbation matrix analysis for the captured PBN problem and propose a method for computing the steady-state probability distribution. An approximation method with error analysis is then given for further reducing the computational complexity. Numerical experiments are given to demonstrate the efficiency of the proposed methods.     

The bilateral birth–death chain generated by the associated Jacobi polynomials

We give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three-term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one-step transition probability matrix of a discrete-time bilateral birth–death chain with state space on $\mathbb{Z}$. We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.     

Metastability in stochastic dynamics of disordered mean-field models

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of “admissible transitions”. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor. The distributions of the rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model. Received: 26 November 1998 / Revised version: 21 March 2000 / Published online: 14 December 2000