摘要
综述了杨向群教投在湘潭大学主持的多参数马氏过程讨论班同志们近4年来有关两参数马氏过程方面的一系列研究成果.共分六个部分。1 基本概念及符号。2 各种两参数马氏性间的相互关系.在已有工作的基础上,进一步揭示了各种两参数马氏性间的相互关系,得到了一个比较完美的相互关系图.举出反例,推翻了三个法国学者的结论。3 给出了两参数马氏链的三点转移函数的定义,说明它与单参数马氏链的转移函数之间既存在本质的区别,又有着密切的联系.介绍了可测三点转移函数的概念,并研究了它的正则性,恒正性,状态对空间的分解,极限函数的正则性及参数对称的若干结果.还研究了标准三点转移函数族的状态对空间的分解,微分性及向上、向下、向左、向右偏微分方程组.表明:三点转移函数族的研究不可能是单参数结果的简单推广。4 引入了Poisson单的概念,讨论了它的单跳跃性,鞅性,Kolomogorov 0—1律,刻划,等价条件及其在射线上导出过程的性质.证明了它不存在广义三点转移函数.因而不能根据通常的存在定理证明它的存在性。为此我们引入了扩状Poisson单及其三点转移函数的概念,证明了Poisson单的存在性,作为Poisson单的直接推广,我们给出了两参数随机事件流的定义,讨论了其性质。5 介绍了广义Brownian单的定义,给出了它在增道路下的Levy表现和鞅刻划,积分表现,随机时间变换,概率估计,重对数律,Harness性,广义B—Harness性及表征等.最后指出,用广义Brow-nian单表现的一类指数过程是一强鞅。6 作为随机微分方程解的两参数马氏过程.设B为Browian单,得到了两参数随机微分方程dX(S,t)=α(s,t,X(S,t))dB(s,t)+β(s,t,X(s,t))dsdt X(s,t)|(s,t)∈λ_0= y(s,t)解的存在唯一性,指出此解在一定条件的各种两参数马氏性,旦此解为宽过去规则马氏过程。
A lot of results on two-parameter Markov processes aresummed up in this paper, those results were obtained by comrades of thediscussion class which was taken charge of by Professor Yang Xiangqun.Six sections are included in this paper.1.Basic concepts and symbols.2. All kinds of relations between two-parameter Markov processes, inclu-ding the difference and the connection between them, are all discussed inthis section. A lot of important work has been done by many other'sscholars. Leaving from this work, we found the relations between all two-parameter Markov processes and got a rather beautiful relations graph. Themore important is that we have made an anti-example denying the conclu-sion by three France scholars.3. Three-point translation functions of two-parameter Markov chain.A definition is given first. Then we pointed out there are not only the essentialdifference but also the close connection between two-parameter and one-parameter. We also studied the regularity and the positiveness of themeasurable three-point translation functions, the decomposition of statespace and lots of conclusion on parameter symmetry are also discussed.Another important three-point translation functions are standard three-point translation function, We studied the decompsitioa of their state space and their differentiation and their on-left, on-right differential equa-tion group. From these results, it is very clear that the study of the three-point translation functions collections is not the simple popularization ofthe one-parameter conclusion.4.The Poisson sheet and the two-parameter stationary random eventflows. In this section,we have given the definition of the Poisson sheetsand have studied their properties of one-jump,martingale,Kolmogorov zero-one law. We also have given their descripition, iff-condition and the pro-cesses induced on the rays. In addition,we have proved that the Poissonsheet has no three-point translation functions. So we cann't determine theexistence of the Poisson sheets by the ordinary means. In order to over-come this difficulty we have introduced the expanding-Poisson sheets andtheir three-point translation functions, and based on this work, we haveproved the existence of the Poisson Sheet. As the direct popularizing ofthe Poisson sheets, we have defined the two-parameter stationary rantidomevent flows and have discussed their some properties.5. Generalized Brownian Sheet (G. B. S. ) and a class of two-parameterexponential martingales. In this section, we have given the definition of thegeneralized Brownian Sheet and its Levy representation under the increasingpath and its martingale descripition. We have also studied its integral re-presentations, time changes, probability estirriates, iteiated logarithm laws,Harnesses, generalized B-Harnesses and the characterization of the G.B. S.And finally,we have pointed out that the exponential processes representedby G.B.S.are strong martingales.6. Two-parameter Markov process induced by the solutions to the sto-chastic differential equation. Let B be a Brownian sheet,we have got theexistence and uniqueness of the solution to the two-parameter stochasticdifferential equation:dx(s, t ) = α(s, t ,X(s, t ))dB(s, t ) +β(s, t ,X(s, t )dsdtX(s,t)|(s, t)∈λ_0 =Y(s, t )We have discussed the solutions properties of all kinds under certain con-ditions and also point out that the solutions are rule past-relaxed Markovprocesses.
出处
《湘潭大学自然科学学报》
CAS
CSCD
1991年第2期1-15,共15页
Natural Science Journal of Xiangtan University
基金
国家自然科学基金
关键词
马氏过程
转移函数
泊松单
two-parameter Markov processes
three-point tranaslation functions
Poisson sheets
generalized Brownian sheet
two-parameter sto-chastic differential equation
