随机过程

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文件:BMonSphere.jpg
A computer-simulated realization of a Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.[1][2][3]

[[文件:BMonSphere.jpg|thumb |计算机模拟在球体表面实现 WienerBrownian motion过程。Wiener过程被广泛认为是概率论中研究最多、最核心的随机过程[1][2][3]]]


Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

球面上的 Wiener维纳 Brownian布朗 运动过程 维纳过程Wiener process被广泛认为是概率论研究最多和最核心的 随机过程Stochastic processes 随机过程被广泛用作以随机方式变化的系统和现象的数学模型。它们在生物学、化学、生态学、神经科学、物理学、图像处理、信号处理、控制理论、信息理论、计算机科学、密码学和电信学等许多学科都有应用。此外,金融市场表面上的随机变化促进了 随机过程在金融领域的广泛应用。

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.[1][4][5][6] Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines such as biology,[7] chemistry,[8] ecology,[9] neuroscience[10], physics[11], image processing, signal processing,[12] control theory, [13] information theory,[14] computer science,[15] cryptography[16] and telecommunications.[17] Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.[18][19][20]

在[[概率论]及相关领域中,“随机”或“随机过程”是一个数学对象,通常被定义为随机变量索引族,给出对一个随机过程的解释,该过程表示某个系统随机的数值随时间的变化,例如细菌l种群的增长,电流由于热噪声而波动,或者一个气体分子的运动。[1][4][5][6]随机过程被广泛用作以随机方式变化的系统和现象的数学模型。它们在许多学科都有应用,比如生物学[7]化学 [8] 生态学,[9] 神经科学[10], 物理学[11], 图像处理, signal processing,[12] control theory, [13] 信息论,[14] 计算机科学,[15] 密码学[21]电信.[17] 此外,金融市场中看似随机的变化激发了随机过程在金融中的广泛使用。[18][19][20]

Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.

现象的应用和研究反过来激发了新的随机过程的提出。这类随机过程的例子包括路易斯 · 巴舍利耶用来研究巴黎证券交易所价格变化的 维纳过程Wiener process 布朗运动过程Brownian motion process,以及 a · k · 埃尔朗用来研究在一定时期内通话次数的 泊松过程Poisson process。这两个随机过程在随机过程理论中被认为是最重要和最核心的,并且在巴舍利耶和 Erlang 之前和之后,在不同背景和国家多次独立地被发现。


Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process,模板:Efn used by Louis Bachelier to study price changes on the Paris Bourse,[22] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time.[23] These two stochastic processes are considered the most important and central in the theory of stochastic processes,[1][4][24] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.[22][25]

应用和现象研究反过来又启发了新随机过程的提出。这种随机过程的例子包括维纳过程或布朗运动过程,{efn |术语“布朗运动”可以指物理过程,也被称为“布朗运动”,以及随机过程,一个数学对象,但为了避免歧义,本文使用“布朗运动过程”或“维纳过程”来表示后者,其风格类似于,例如,Gikhman和Skorokhod [26] 或Rosenblatt。[27]}} 使用人Louis Bachelier为了研究巴黎证券交易所的价格变化,[22] 以及A.K.Erlang使用的泊松过程来研究某段时间内发生的电话号码。[23]这两个随机过程被认为是随机过程理论中最重要和最核心的,[1][4][28] 并且在Bachelor和Erlang之前之后在不同的环境和国家被多次独立地发现[22][25]

The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

随机函数Random function这个术语也用来指随机或随机过程,因为随机过程也可以被解释为函数空间中的随机元素。随机(stochastic)过程和随机(random)过程这两个术语可以互换使用,通常没有专门的数学空间用于对随机变量进行索引。但是,当随机变量被整数或实线的一个区间索引时,通常使用这两个项。随机过程的值并不总是数字,可以是向量或其他数学对象。 马尔可夫过程Markov processes,列维过程Lévy processes,高斯过程Gaussian processes,随机场random fields,更新过程renewal processes, 分支过程branching processes。随机过程的研究使用了概率微积分线性代数集合论的数学知识和技术,和拓扑学以及数学分析的分支,如[[实分析],测度理论,[[傅立叶分析],和[[泛函分析]。随机过程理论被认为是对数学的一个重要贡献,无论从理论上还是应用上,它都一直是一个活跃的研究课题。

The term random function is also used to refer to a stochastic or random process,[29][30] because a stochastic process can also be interpreted as a random element in a function space.[31][32] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables.[31][33] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line.[5][33] If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.[5][34] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.[5][32]

术语“随机函数”也用于指随机或随机过程,随机过程的值并不总是数字,可以是向量或其他数学对象。[35][36]

Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks,[37] martingales,[38] Markov processes,[39] Lévy processes,[40] Gaussian processes,[41] random fields,[42] renewal processes, and branching processes.[43] The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology[44][45][46] as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.[47][48][49] The theory of stochastic processes is considered to be an important contribution to mathematics[50] and it continues to be an active topic of research for both theoretical reasons and applications.[51][52][53]

根据随机过程的数学性质,随机过程可以分为不同的类别,包括随机游走s,[37] ,[38] 马尔可夫过程es,[39] 莱维过程es,[40] 高斯过程es,[54] 随机场,[42] 更新过程es, 和 分支过程es.[43]。随机过程的研究使用了概率微积分线性代数集合论的数学知识和技术,和拓扑学以及数学分析的分支,如[[实分析],测量理论,[[傅立叶分析],和[[泛函分析]。随机过程理论被认为是对数学的重要贡献[50],不论由于理论还是应用,它都是一个活跃的研究课题。[51][52][53]

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.

一个随机过程可以被定义为一组随机变量的集合,这些随机变量被一些数学集合索引,这意味着随机过程的每个随机变量唯一地与集合中的一个元素相关联。


Introduction简介

When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.

当解释为时间时,如果随机过程的索引集有有限个或可数的元素,例如有限的一组数字、一组整数或自然数,则该随机过程称为离散时间的。如果索引集是实数轴的某个区间,则时间被称为连续的。这两类随机过程分别称为 离散时间随机过程 连续时间随机过程。离散时间随机过程被认为更容易研究,因为连续时间过程需要更先进的数学技术和知识,特别当索引集不可数时。如果索引集是整数或整数的子集,那么随机过程也可以称为 随机序列。Jakob Bernoulli于1713年在拉丁文中首次发表了关于概率的著作Ars Conspectandi,他使用了“Ars consuctandi-sive-Stochastice”一词,该词已被翻译为“推测或随机的艺术”。这个短语是由Ladislaus Bortkiewicz在1917年用德语写下的单词stochastik,意思是随机的。“随机过程”一词最早出现在1934年约瑟夫·杜布的一篇论文中。尽管德语这个词在早些时候就被使用过,例如,安德烈·科尔莫戈洛夫(Andrei Kolmogorov)在1931年就使用过。

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.[4][5] The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time.[1] Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or [math]\displaystyle{ n }[/math]-dimensional Euclidean space.[1][5] An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time.[55][56] A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization.[32][57]

随机(stochastic)或随机(random)过程可以定义为随机变量的集合,这些随机变量由一些数学集合构成索引,这意味着随机过程中的每个随机变量都与集合中的一个元素唯一关联。[58][35]用于索引随机变量的集合称为“索引集”。从历史上看,索引集是实线的一些子集,例如自然数,为索引集提供了对时间的解释。[59]集合中的每个随机变量都从相同的数学空间中获取值,称为“状态空间”。例如,这个状态空间可以是整数、实线或维欧几里德空间。[59]“increment”是随机过程在两个索引值之间变化的量,通常被解释为两个时间点。[60][61]由于随机性,随机过程可以有许多结果,随机过程的单个结果称为其他名称中的一个,“示例函数”或“实现”。[36][62]

According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888.

根据牛津英语词典的研究,英语中随机这个词的早期出现和它现在的意思有关,可以追溯到16世纪,而早期记录的用法开始于14世纪,是一个名词,意思是“浮躁、极速、力量或暴力(在骑马、奔跑、惊人等等)”。这个单词本身来自中世纪法语单词,意思是“速度,匆忙” ,它可能来源于法语动词,意思是“奔跑”或“疾驰”。随机(random)过程这个术语的第一次书面出现早于随机(stochastic)过程,牛津英语词典也把它作为同义词,并在 Francis Edgeworth 1888年发表的一篇文章中使用。

文件:Wiener process 3d.png
A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.

thumb | right |单个计算机模拟时间0≤t≤2的三维Wiener或Brownian运动过程的“样本函数”或“实现”。这个随机过程的指标集是非负数,而其状态空间是三维欧几里德空间

Classifications分类

The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. Both "collection", while instead of "index set", sometimes the terms "parameter set" though sometimes it is only used when the stochastic process takes real values. while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is [math]\displaystyle{ n }[/math]-dimensional Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math] or a manifold. [math]\displaystyle{ \{X(t)\} }[/math] or simply as [math]\displaystyle{ X }[/math] or [math]\displaystyle{ X(t) }[/math], although [math]\displaystyle{ X(t) }[/math] is regarded as an abuse of function notation. For example, [math]\displaystyle{ X(t) }[/math] or [math]\displaystyle{ X_t }[/math] are used to refer to the random variable with the index [math]\displaystyle{ t }[/math], and not the entire stochastic process. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial.

随机过程的定义各不相同,但随机过程通常被定义为由一组随机变量组成的集合。两者都是“集合” ,而不是“索引集合” ,有时使用术语“参数集合” ,但只有在随机过程数据库采用真实值时才使用。当索引集被解释为时间时,通常使用术语随机(stochastic)过程和随机(random)过程,当索引集是 [math]\displaystyle{ n }[/math]-维欧几里得空间 [math]\displaystyle{ \mathbb{R}^n }[/math]或者是流形时,则使用随机场。虽然[math]\displaystyle{ \{X(t)\} }[/math]被认为是对函数表示法的滥用,但[math]\displaystyle{ \{X(t)\} }[/math]还是被简单地称为[math]\displaystyle{ X }[/math][math]\displaystyle{ X(t) }[/math]。例如,[math]\displaystyle{ X(t) }[/math][math]\displaystyle{ X_t }[/math] 用于指代带有索引 [math]\displaystyle{ t }[/math] 的随机变量,而不是整个随机过程。换句话说,伯努利过程是一系列 iid Bernoulli 随机变量,每次抛硬币都是 Bernoulli 试验的一个例子。

A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space.[63][64][65]

随机过程可以用不同的方法进行分类,例如,根据其状态空间、索引集或随机变量之间的相关性。一种常见的分类方法是通过索引集和状态空间的基数进行分类。[63][64][66]

When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time.[67][68] If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes.[55][69][70] Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.[71][72] If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence.[68]

当解释为时间时,如果随机过程的指标集有有限个或可数个元素,例如有限的一组数、一组整数或自然数,那么随机过程被称为“离散时间[67][68] 。如果索引集是实数轴上的某个区间,则时间被称为“'连续时间”。这两类随机过程分别被称为“离散时间”和“连续时间随机过程es”[55][69][70]。离散时间随机过程被认为更容易研究,因为连续时间过程需要更先进的数学技术和知识,特别当索引集不可数时。[71][72] 如果索引集是整数或整数的子集,则随机过程也可以称为“随机序列”。[73]

If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is [math]\displaystyle{ n }[/math]-dimensional Euclidean space, then the stochastic process is called a [math]\displaystyle{ n }[/math]-dimensional vector process or [math]\displaystyle{ n }[/math]-vector process.[63][64]

如果状态空间是整数或自然数,则随机过程称为“离散”或“整值随机过程”。如果状态空间是实数轴,则随机过程被称为“实值随机过程”或“具有连续状态空间的过程”。如果状态空间是[math]\displaystyle{ n }[/math]-维欧几里德空间,则随机过程称为[math]\displaystyle{ n }[/math]-“维向量过程”或[math]\displaystyle{ n }[/math]—“向量过程”。[74][75]

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.

随机游走Random walks是一种随机过程,通常定义为欧氏空间中的等价随机变量或随机向量的和,因此它们是在离散时间中变化的过程。但有些人也用这个词来指连续时间中发生变化的过程,特别是在金融领域使用的维纳过程,这种过程导致了一些混淆,因此招致了批评。还有其他各种类型的 随机游走,定义它们的状态空间可以是其他数学对象,如格子和群。一般来说,它们被高度研究,在不同学科中有许多应用。


Etymology词源学

A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, [math]\displaystyle{ p }[/math], or decreases by one with probability [math]\displaystyle{ 1-p }[/math], so the index set of this random walk is the natural numbers, while its state space is the integers. If the [math]\displaystyle{ p=0.5 }[/math], this random walk is called a symmetric random walk.

一个经典的 随机游走的例子被称为 简单随机游走SimpleRandom walk,这是一个以整数为状态空间的离散时间随机过程,它基于一个伯努利过程Bernoulli process,其中每个 伯努利Bernoulli 变量要么取值为正,要么取值为负。换句话说,简单随机游走发生在整数上,它的值随概率[math]\displaystyle{ p }[/math]的增加而增加1,或随概率[math]\displaystyle{ 1-p }[/math]的减少而减少1,所以这种 随机游走的索引集是自然数,而它的状态空间是整数。如果 [math]\displaystyle{ p=0.5 }[/math],这种随机漫步称为 对称随机游走Symmetric Random walk

The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence.[76] In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics".[77] This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz[78] who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob.[76] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin,[79][80] though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.[81]

英语中,“随机”一词最初用作形容词,其定义是“与推测有关”,源于一个希腊语词,意思是“瞄准一个标记,猜测”,而牛津英语词典将1662年作为最早出现的年份。[82]在他关于概率“Ars conquectandi”的著作中,最初于1713年以拉丁文出版,Jakob Bernoulli使用了“Ars conquectandi istice”这个短语,这本书已经被翻译成“猜想或随机的艺术”。[77]这一短语是拉迪斯劳斯·博特基维茨]在关于伯努利问题中使用,[78]他在1917年用德语写下了“随机”一词。术语“随机过程”最早出现在1934年Joseph Doob的一篇论文中。[83]对于这个术语和一个具体的数学定义,Doob引用了另一篇1934年的论文,其中Aleksandr Khinchin在德语中使用了术语“随机过程”,[79][80]尽管德语这个词在早些时候就被使用过,例如,安德烈·科尔莫戈洛夫(Andrei Kolmogorov)在1931年就使用过。[81]


According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888.[84]

根据《牛津英语词典》,英语中“random”(随机)一词的最早出现时间可追溯到16世纪,而早期有记载的用法则始于14世纪,意思是“急躁、速度快、力量大或暴力(骑马、跑步、击打等)”。这个词本身来自法语中间的一个词,意思是“速度,匆忙”,它可能是从法语动词“奔跑”或“飞奔”衍生而来。术语“随机过程”的首次书面出现是在“随机过程”之前出现的,牛津英语词典也将其作为同义词出现,并被Francis Edgeworth于1888年发表的一篇文章中使用。[85]

Terminology术语

The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. The Wiener process is named after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids.

维纳过程Wiener process是一个具有平稳和独立增量的随机过程,这些增量是基于增量大小的正态分布。维纳过程是以诺伯特 · 维纳的名字命名的,他证明了维纳过程的数学存在性。

The definition of a stochastic process varies,[86] but a stochastic process is traditionally defined as a collection of random variables indexed by some set.[87][88] The terms random process and stochastic process are considered synonyms and are used interchangeably, without the index set being precisely specified.[31][33][34][89][90][91] Both "collection",[32][89] or "family" are used[4][92] while instead of "index set", sometimes the terms "parameter set"[32] or "parameter space"[34] are used.

随机过程的定义是不同的,[86] 但是随机过程传统上被定义为一组随机变量的集合[87][88]。术语“随机random过程”和“随机stochastic过程”被视为同义词,可以互换使用,而无需精确指定索引集。[31][33][34][89][90][91]。两个“集合”[32][89],或“家庭”使用[4]引用错误:没有找到与</ref>对应的<ref>标签[93] though sometimes it is only used when the stochastic process takes real values.[32][92] This term is also used when the index sets are mathematical spaces other than the real line,[5][94] while the terms stochastic process and random process are usually used when the index set is interpreted as time,[5][94][95] and other terms are used such as random field when the index set is [math]\displaystyle{ n }[/math]-dimensional Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math] or a manifold.[5][32][34]

术语 “随机函数”Random function也用于指随机或随机过程,[35][96][97]尽管有时它只在随机过程取实值时使用。[36][98]当索引集是数学空间而不是实线时,也使用这个术语,[35][99],而术语“随机过程”和“随机过程”通常在指数集被解释为时间时使用,[35][100][101]和其他术语,例如当索引集是[math]\displaystyle{ n }[/math]-维欧几里德空间[math]\displaystyle{ \mathbb{R}^n }[/math]流形.[35][36][35][36]name=“adlertaylor2009第7页”/>

Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state space can be [math]\displaystyle{ n }[/math]-dimensional Euclidean space. If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant [math]\displaystyle{ \mu }[/math], which is a real number, then the resulting stochastic process is said to have drift [math]\displaystyle{ \mu }[/math].

在概率论中起着核心作用的 维纳过程,通常被认为是最重要的和研究过的随机过程,与其他随机过程有联系。它的索引集和状态空间分别为非负数和实数,因此它既有连续索引集又有状态空间。但是这个过程可以定义得更广泛,因此它的状态空间可以是[math]\displaystyle{ n }[/math]维的 欧氏空间Euclidean space。如果增量的平均值为零,那么由此产生的维纳Wiener或布朗Brownian运动过程称为具有零漂移。如果任意两个时间点的增量的平均值等于时间差乘以某个常数[math]\displaystyle{ \mu }[/math],即一个实数,那么得到的随机过程就具有[math]\displaystyle{ \mu }[/math]漂移。

Notation符号

A stochastic process can be denoted, among other ways, by [math]\displaystyle{ \{X(t)\}_{t\in T} }[/math],[69] [math]\displaystyle{ \{X_t\}_{t\in T} }[/math],[88] [math]\displaystyle{ \{X_t\} }[/math][102] [math]\displaystyle{ \{X(t)\} }[/math] or simply as [math]\displaystyle{ X }[/math] or [math]\displaystyle{ X(t) }[/math], although [math]\displaystyle{ X(t) }[/math] is regarded as an abuse of function notation.[103] For example, [math]\displaystyle{ X(t) }[/math] or [math]\displaystyle{ X_t }[/math] are used to refer to the random variable with the index [math]\displaystyle{ t }[/math], and not the entire stochastic process.[102] If the index set is [math]\displaystyle{ T=[0,\infty) }[/math], then one can write, for example, [math]\displaystyle{ (X_t , t \geq 0) }[/math] to denote the stochastic process.[33]

随机过程可以用[math]\displaystyle{ \{X(t)\{t} }[/math][math]\displaystyle{ \{X(t)\} }[/math][69] [math]\displaystyle{ \{X_t\}_{t\in T} }[/math],[88] [math]\displaystyle{ \{X_t\} }[/math][102]或简单地称为[math]\displaystyle{ X }[/math][math]\displaystyle{ X(t) }[/math],尽管[math]\displaystyle{ X(t) }[/math]被视为函数表示法滥用[103] 例如, [math]\displaystyle{ X(t) }[/math][math]\displaystyle{ X_t }[/math]引用具有索引[math]\displaystyle{ t }[/math]的随机变量,而不是整个随机过程。[102]如果索引集是[math]\displaystyle{ T=[0,\infty) }[/math],然后,我们可以写,例如,[math]\displaystyle{ (X_t , t \geq 0) }[/math]来表示随机过程。[104]

Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.

几乎可以肯定, 维纳过程Wiener process的样本路径在任何地方都是连续的,但是没有可微的地方。它可以看作是简单随机游走的连续形式。这个过程作为其他随机过程的数学极限出现,例如某些随机游走的重新标度,这是 Donsker 定理或不变性原理的主题,也被称为 函数中心极限定理

Examples示例

The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes. It plays a central role in quantitative finance, where it is used, for example, in the Black–Scholes–Merton model. The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.

维纳过程Wiener process是马尔可夫过程、 列维Lévy 过程和 高斯Gaussian 过程等重要随机过程的一个成员。它在定量金融学中扮演着核心角色,例如,在 布莱克-斯科尔斯-默顿模型Black–Scholes–Merton model中就使用了它。这个过程也用于不同的领域,包括大多数自然科学和一些社会科学分支,作为各种随机现象的数学模型。


Bernoulli process伯努利过程

One of the simplest stochastic processes is the Bernoulli process,[105] which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability [math]\displaystyle{ p }[/math] and zero with probability [math]\displaystyle{ 1-p }[/math]. This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is [math]\displaystyle{ p }[/math] and its value is one, while the value of a tail is zero.[106] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables,[107] where each coin flip is an example of a Bernoulli trial.[108]

最简单的随机过程之一是伯努利过程[109]它是独立且相同分布(iid)随机变量的序列,其中每个随机变量取1或0,比如概率[math]\displaystyle{ p }[/math]的值为1,概率[math]\displaystyle{ 1-p }[/math]为零。这个过程可以与反复翻动硬币有关,其中获得头部的概率为[math]\displaystyle{ p }[/math],其值为1,而尾部的值为零=https://books.google.com/books?id=z5sebqaaqbaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | page=301}</ref>换句话说,伯努利过程是一个 iid伯努利随机变量的序列,[110]每一次抛硬币都是Bernoulli审判的一个例子。[111]

The Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes. If the parameter constant of the Poisson process is replaced with some non-negative integrable function of [math]\displaystyle{ t }[/math], the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.

泊松过程Poisson process是一个具有不同形式和定义的随机过程。它可以被定义为一个计数过程,这是一个随机过程,代表到某个时间,点或事件的随机数。从零到给定时间区间内的过程点数是泊松随机变量,取决于该时间和某些参数。该过程以自然数为状态空间,非负数为索引集。这个过程也被称为泊松计数过程,因为它可以被解释为计数过程的一个例子。 齐次泊松过程Homogeneous Poisson process是一类重要的随机过程,如马尔可夫过程和 Lévy 过程的成员。如果将泊松过程的参数常数替换为 < math > t </math > 的非负可积函数,则得到的过程称为 非齐次或非齐次泊松过程Inhomogeneous or nonhomogeneous Poisson process,其点的平均密度不再是常数。泊松过程作为排队论中的一个基本过程,是数学模型中的一个重要过程,它在特定时间窗内随机发生的事件模型中找到了应用。


随机游走Random walk

Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the [math]\displaystyle{ n }[/math]-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.

在实际线上定义的泊松过程可以被解释为随机过程过程,以及其他随机对象。但是,它可以定义在维欧氏空间或其他数学空间上,在这些空间中,它通常被解释为一个随机集或随机计数测度,而不是一个随机过程。但是人们注意到泊松过程并没有得到应有的重视,部分原因是泊松过程通常只考虑实线,而不考虑其他数学空间。

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.[112][113][114][115][116] But some also use the term to refer to processes that change in continuous time,[117] particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism.[118] There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.[117][119]

Random walks是随机过程,通常定义为欧几里德空间中iid随机变量或随机向量的和,因此它们是离散时间变化的过程=https://books.google.com/books?id=aqURswEACAAJ | year=2013 | publisher=Springer | isbn=978-1-4471-5362-7 | pages=347}</ref>[120][121][122][123]但是有些人也使用这个术语来指代连续时间变化的过程,[124]尤其是金融中使用的维纳过程,这导致了一些混乱,导致其受到批评。[125]还有其他各种类型的随机游动,它们的状态空间可以是其他数学对象,例如格和群,一般来说,它们都是高度研究的,在不同的学科中有许多应用。[124][126]

A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, [math]\displaystyle{ p }[/math], or decreases by one with probability [math]\displaystyle{ 1-p }[/math], so the index set of this random walk is the natural numbers, while its state space is the integers. If the [math]\displaystyle{ p=0.5 }[/math], this random walk is called a symmetric random walk.[127][128]

随机游走Random walk的一个经典例子被称为“简单随机游动”,它是一个离散时间的随机过程,以整数为状态空间,它基于伯努利过程,其中每个贝努利变量取正值或负值。换言之,简单随机游走发生在整数上,例如其值随概率[math]\displaystyle{ p }[/math]增加1,,或随着概率[math]\displaystyle{ 1-p }[/math]而减小1,因此这种随机游动的指标集是自然数,而其状态空间是整数。如果[math]\displaystyle{ p=0.5 }[/math],这种随机游动称为对称随机游动。[129][130]

A stochastic process is defined as a collection of random variables defined on a common probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math], where [math]\displaystyle{ \Omega }[/math] is a sample space, [math]\displaystyle{ \mathcal{F} }[/math] is a [math]\displaystyle{ \sigma }[/math]-algebra, and [math]\displaystyle{ P }[/math] is a probability measure; and the random variables, indexed by some set [math]\displaystyle{ T }[/math], all take values in the same mathematical space [math]\displaystyle{ S }[/math], which must be measurable with respect to some [math]\displaystyle{ \sigma }[/math]-algebra [math]\displaystyle{ \Sigma }[/math].

随机过程被定义为一系列随机变量的集合,这些随机变量定义在一个普通的概率空间上(Omega,mathcal { f } ,p) </math > ,其中 < math > Omega </math > 是一个样本空间,< math > mathcal { f } </math > 是 < math > sigma </math >-algebra,而 < math > p </math > 是一个机率量测;以及随机变量,用一些集合作为指标,它们都在同一个数学空间中取值,这些值必须是可以测量的。


[math]\displaystyle{ \lt 中心 \gt \lt 数学 \gt ==='''\lt font color="#ff8000"\gt Wiener process维纳过程\lt /font\gt '''=== \{X(t):t\in T \}. { x (t) : t in t }. {{Main|Wiener process}} }[/math]

[数学中心]


The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments.[2][131] The Wiener process is named after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids.[132][133][133][134]

Wiener process维纳过程是一个随机过程,具有平稳的独立的增量并且基于增量的大小是[[正态分布的].[135][136]维纳过程是以Norbert Wiener命名的,他证明了它的数学存在性,但是这个过程也被称为布朗运动过程或仅仅是布朗运动,因为它是液体中布朗运动的模型科学|卷=5 |议题=1 |年份=1968 |页数=1-2 | issn=0003-9519 | doi=10.1007/BF00328110}}</ref>引用错误:无效<ref>标签;name属性非法,可能是内容过长

Historically, in many problems from the natural sciences a point [math]\displaystyle{ t\in T }[/math] had the meaning of time, so [math]\displaystyle{ X(t) }[/math] is a random variable representing a value observed at time [math]\displaystyle{ t }[/math]. A stochastic process can also be written as [math]\displaystyle{ \{X(t,\omega):t\in T \} }[/math] to reflect that it is actually a function of two variables, [math]\displaystyle{ t\in T }[/math] and [math]\displaystyle{ \omega\in \Omega }[/math].

从历史上看,在自然科学的许多问题中,t 中的一个点[math]\displaystyle{ t\in T }[/math]代表时间,所以[math]\displaystyle{ X(t) }[/math]是一个随机变量,代表时间[math]\displaystyle{ t }[/math]观察到的值。一个随机过程也可以写成[math]\displaystyle{ \{X(t,\omega):t\in T \} }[/math] 来反映它实际上是一个双变量的函数,[math]\displaystyle{ t\in T }[/math][math]\displaystyle{ \omega\in \Omega }[/math]


文件:DriftedWienerProcess1D.svg
Realizations of Wiener processes (or Brownian motion processes) with drift (模板:Color) and without drift (模板:Color).

拇指|左|实现维纳Wiener过程(或布朗运动过程),具有漂移({color |蓝色}且不漂移({color |红色}红色})。

There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a [math]\displaystyle{ S^T }[/math]-valued random variable, where [math]\displaystyle{ S^T }[/math] is the space of all the possible [math]\displaystyle{ S }[/math]-valued functions of [math]\displaystyle{ t\in T }[/math] that map from the set [math]\displaystyle{ T }[/math] into the space [math]\displaystyle{ S }[/math]. of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set [math]\displaystyle{ T }[/math] the interpretation of time. such as the Cartesian plane [math]\displaystyle{ R^2 }[/math] or [math]\displaystyle{ n }[/math]-dimensional Euclidean space, where an element [math]\displaystyle{ t\in T }[/math] can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.

还有其他的方法来考虑随机过程,上面的定义被认为是传统的定义。例如,随机过程可以被解释或定义为一个 [math]\displaystyle{ S^T }[/math] 值随机变量,其中 [math]\displaystyle{ S^T }[/math][math]\displaystyle{ t\in T }[/math] 中所有可能的 [math]\displaystyle{ S }[/math]值函数的空间,这些函数从集合 [math]\displaystyle{ T }[/math] 映射到空间 [math]\displaystyle{ S }[/math] 。随机过程。这个集合通常是实数线的一些子集,比如使集合 [math]\displaystyle{ T }[/math] 时间有意义的自然数集或者区间。比如笛卡尔平面 [math]\displaystyle{ R^2 }[/math][math]\displaystyle{ n }[/math] 维欧氏空间,其中的一个元素 [math]\displaystyle{ t\in T }[/math]可以表示空间中的一个点。但是一般来说,当指标集是有序的时候,对于随机过程可能有更多的结果和定理。


Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.[1][2][3][137][138][139][140] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.[141] But the process can be defined more generally so its state space can be [math]\displaystyle{ n }[/math]-dimensional Euclidean space.[119][138][142] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant [math]\displaystyle{ \mu }[/math], which is a real number, then the resulting stochastic process is said to have drift [math]\displaystyle{ \mu }[/math].[143][144][145]

Wiener process维纳过程在概率论中起着中心作用,通常被认为是最重要和研究的随机过程,并与其他随机过程联系在一起微积分与金融应用|网址=https://books.google.com/books?id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=29}</ref>[146][147]引用错误:无效<ref>标签;name属性非法,可能是内容过长其索引集和状态空间分别是非负数和实数,因此它既有连续索引集又有状态空间=https://books.google.com/books?id=am1IDQAAQBAJ | year=2006 | publisher=World Scientific Publishing Co Inc | isbn=978-981-310-165-4 | page=186}</ref>但是过程可以定义得更广泛,这样它的状态空间可以是维欧几里德空间。[148][147][149]如果任何增量的平均值为零,则所得到的维纳或布朗运动过程称为零漂移。如果任意两个时间点的增量的平均值等于时间差乘以某个常数[math]\displaystyle{ \mu }[/math],即实数,由此产生的随机过程被称为漂移[math]\displaystyle{ \mu }[/math][150][151][152]

The mathematical space [math]\displaystyle{ S }[/math] of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, [math]\displaystyle{ n }[/math]-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.

随机过程的数学空间 [math]\displaystyle{ S }[/math]称为状态空间。这个数学空间可以使用整数、实数线、维欧氏空间、复平面或更抽象的数学空间来定义。状态空间使用元素定义,这些元素反映了随机过程可以采用的不同值。

Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk.[56][144] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,[153][154] which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.[155][156][157]

几乎可以肯定 Wiener process维纳过程的样本路径处处连续,但无处可微。它可以看作是简单随机游走的一个连续版本。[61]当其他随机过程(如某些随机游动重新缩放)的数学极限时,该过程出现,[158][159]这是Donsker定理或不变性原理的主题,也被称为函数中心极限定理=https://books.google.com/books?id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | pages=225260}}</ref>引用错误:无效<ref>标签;name属性非法,可能是内容过长

The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.[2][56] The process also has many applications and is the main stochastic process used in stochastic calculus.[160][161] It plays a central role in quantitative finance,[162][163] where it is used, for example, in the Black–Scholes–Merton model.[164] The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.[3][165][166]

Wiener process维纳过程是一些重要的随机过程家族的成员,包括马尔可夫过程,Lévy过程和高斯过程。[135][61]该过程也有许多应用,是随机微积分中使用的主要随机过程。[167][168]它在数量金融中起着核心作用,模板:本刊</ref>引用错误:无效<ref>标签;name属性非法,可能是内容过长该过程也被用于不同的领域,包括大多数自然科学以及社会科学的一些分支,作为各种随机现象的数学模型=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=47}</ref>[169]

A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if [math]\displaystyle{ \{X(t,\omega):t\in T \} }[/math] is a stochastic process, then for any point [math]\displaystyle{ \omega\in\Omega }[/math], the mapping

样本函数是随机过程的单一结果,所以它是由每个随机过程的随机变量的单一可能值构成的。更确切地说,如果 < math > { x (t,Omega) : t in t } </math > 是随机过程,那么对于任意点 < math > Omega </math > ,映射就是


[math]\displaystyle{ \lt 中心 \gt \lt 数学 \gt ===Poisson process泊松过程=== X(\cdot,\omega): T \rightarrow S, X (cdot,omega) : t, {{Main|Poisson process}} }[/math]

[数学中心]


is called a sample function, a realization, or, particularly when [math]\displaystyle{ T }[/math] is interpreted as time, a sample path of the stochastic process [math]\displaystyle{ \{X(t,\omega):t\in T \} }[/math]. This means that for a fixed [math]\displaystyle{ \omega\in\Omega }[/math], there exists a sample function that maps the index set [math]\displaystyle{ T }[/math] to the state space [math]\displaystyle{ S }[/math]. or path.

被称为一个样本函数,一个实现,或者,特别是当 < math > t </math > 被解释为时间,一个随机过程 < math > { x (t,omega) : t in t } </math > 的样本路径。这意味着对于 Omega </math > 中的一个固定的 < math > Omega,存在一个示例函数,它将索引集 < math > t </math > 映射到状态空间 < math > s </math > 。或者路径。

The Poisson process is a stochastic process that has different forms and definitions.[170][171] It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.[170]

{124jms{124tik=随机过程的定义是不同的=https://books.google.com/books?id=eBeNngEACAAJ | year=2003 | publisher=Wiley | isbn=978-0-471-49881-0 | pages=1,2}</ref>[172]它可以定义为一个计数过程,它是一个随机过程,表示某个时间点或事件的随机数量。在从零到某个给定时间区间内的过程点的数目是一个泊松随机变量,它取决于该时间和某个参数。该过程以自然数为状态空间,非负数为索引集。此过程也称为泊松计数过程,因为它可以被解释为计数过程的一个示例。[173]

If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.[170][174] The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.[56]

如果一个泊松过程是用一个正常数定义的,那么这个过程称为齐次泊松过程=https://books.google.com/books?id=PqUmjp7k1kEC | year=2011 | publisher=academical Press | isbn=978-0-12-381416-6 | page=241}</ref>齐次泊松过程是随机过程的一个重要类,如马尔可夫过程和Lévy过程

An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if [math]\displaystyle{ \{X(t):t\in T \} }[/math] is a stochastic process with state space [math]\displaystyle{ S }[/math] and index set [math]\displaystyle{ T=[0,\infty) }[/math], then for any two non-negative numbers [math]\displaystyle{ t_1\in [0,\infty) }[/math] and [math]\displaystyle{ t_2\in [0,\infty) }[/math] such that [math]\displaystyle{ t_1\leq t_2 }[/math], the difference [math]\displaystyle{ X_{t_2}-X_{t_1} }[/math] is a [math]\displaystyle{ S }[/math]-valued random variable known as an increment.

一个随机过程的增量是同一个随机过程的两个随机变量之间的差。对于一个索引集可以被解释为时间的随机过程,增量是随机过程在一定时间段内的变化量。例如,如果 < math > { x (t) : t in t } </math > 是一个状态空间 < math > s </math > 并且索引设置为 < math > t = [0,infty ] </math > ,那么对于[0,infty ] </math > 中的任意两个非负数 t _ 1和[0,infty ] </math > </math > t _ 2在[0,infty ] </math </math > 这样 < t _ 1 leq t _ 2 </math > ,差值 < math > x _ { t _ 2}-x _ { t _ 1} </math > 是一个 < math > s </math >-valued 随机变量,称为递增量。


The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.[175][176] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of [math]\displaystyle{ t }[/math], the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.[177] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.[178][179]

齐次泊松过程可以用不同的方法定义和推广。它的指标集可以定义为实线,这个随机过程也被称为平稳泊松过程=https://books.google.com/books?id=VEiM OtwDHkC | year=1992 | publisher=Clarendon Press | isbn=978-0-19-159124-2 | page=38}</ref>[172]如果泊松过程的参数常数被某个非负可积函数的[math]\displaystyle{ t }[/math]代替,则得到的过程称为非齐次或非齐次Poisson过程,其中过程点的平均密度不再是常数=https://books.google.com/books?id=VEiM OtwDHkC | year=1992 | publisher=Clarendon Press | isbn=978-0-19-159124-2 | page=22}</ref>作为排队论中的一个基本过程,泊松过程是数学模型的一个重要过程,在这里,它找到了在特定时间窗口中随机发生的事件模型的应用程序。[180][181]

For a measurable subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ S^T }[/math], the pre-image of [math]\displaystyle{ X }[/math] gives

对于 < math > s ^ t </math > 的可测子集 < math > b </math > ,< math > x </math > 的前映像给出了


[math]\displaystyle{ \lt 中心 \gt \lt 数学 \gt Defined on the real line, the Poisson process can be interpreted as a stochastic process,\lt ref name="Applebaum2004page1337"/\gt \lt ref name="Rosenblatt1962page94"\gt {{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press|page=[https://archive.org/details/randomprocesses00rose_0/page/94 94]}}\lt /ref\gt among other random objects.\lt ref name="Haenggi2013page10and18"\gt {{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5|pages=10, 18}}\lt /ref\gt \lt ref name="ChiuStoyan2013page41and108"\gt {{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41, 108}}\lt /ref\gt But then it can be defined on the \lt math\gt n }[/math]-dimensional Euclidean space or other mathematical spaces,[182] where it is often interpreted as a random set or a random counting measure, instead of a stochastic process.[183][184] In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons.[23][185] But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.[185][186]

在实线上定义的泊松过程可以解释为一个随机过程,[61][187]等随机变量对象。[188][189]但是它可以定义在[math]\displaystyle{ n }[/math]维欧几里德空间或其他数学空间上,[190]其中它通常被解释为随机集或随机计数度量,而不是随机过程。[188][189]在此设置中,是泊松过程,也称为泊松点过程,是概率论中最重要的研究对象之一,无论是应用还是理论原因=https://books.google.com/books?id=KAWmFYUJ5zsC&pg=PA11 | year=2010 | publisher=Springer Science&Business Media | isbn=978-1-4419-6923-1 | page=1}}</ref>但有人指出,Poisson过程并没有得到应有的重视,部分原因是它经常被认为只是在实线上,而不是在其他数学空间中。[191]<refname=“kingmann1992pagev”>{cite book | author=J.F.C.Kingman | title=Poisson进程| url=https://books.google.com/books?id=VEiM OtwDHkC | year=1992 | publisher=Clarendon Press | isbn=978-0-19-159124-2 | page=v}</ref>

X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},

X ^ {-1}(b) = { Omega: x (Omega) in b } ,


</math>

[数学中心]

Definitions定义

so the law of a [math]\displaystyle{ X }[/math] can be written as:

所以 [math]\displaystyle{ X }[/math]的定律可以写成:


Stochastic process随机过程

A stochastic process is defined as a collection of random variables defined on a common probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math], where [math]\displaystyle{ \Omega }[/math] is a sample space, [math]\displaystyle{ \mathcal{F} }[/math] is a [math]\displaystyle{ \sigma }[/math]-algebra, and [math]\displaystyle{ P }[/math] is a probability measure; and the random variables, indexed by some set [math]\displaystyle{ T }[/math], all take values in the same mathematical space [math]\displaystyle{ S }[/math], which must be measurable with respect to some [math]\displaystyle{ \sigma }[/math]-algebra [math]\displaystyle{ \Sigma }[/math].[32]

随机过程被定义为在一个公共概率空间[math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math]上定义的随机变量集合,其中[math]\displaystyle{ \Omega }[/math]样本空间[math]\displaystyle{ \mathcal{F} }[/math]是一个[math]\displaystyle{ \sigma }[/math]-代数[math]\displaystyle{ P }[/math]概率测度;而随机变量,由某个集合[math]\displaystyle{ T }[/math]索引,所有值都取同一个数学空间[math]\displaystyle{ S }[/math],对于某些[math]\displaystyle{ \sigma }[/math]-代数[math]\displaystyle{ \sigma }[/math][36]

For a stochastic process [math]\displaystyle{ X }[/math] with law [math]\displaystyle{ \mu }[/math], its finite-dimensional distributions are defined as:

对于一个随机过程,其有限维分布被定义为:

In other words, for a given probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math] and a measurable space [math]\displaystyle{ (S,\Sigma) }[/math], a stochastic process is a collection of [math]\displaystyle{ S }[/math]-valued random variables, which can be written as:[105]

换言之,对于给定的概率空间[math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math]和可测空间[math]\displaystyle{ (S,Sigma) }[/math],随机过程是一个值为[math]\displaystyle{ S }[/math]的随机变量的集合,可以写成:[105]

[math]\displaystyle{ \lt center\gt \lt math\gt \mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1}, Mu _ { t _ 1,dots,t _ n } = p circ (x ({ t _ 1}) ,dots,x ({ t _ n })) ^ {-1} , \{X(t):t\in T \}. }[/math]


</math>

where [math]\displaystyle{ n\geq 1 }[/math] is a counting number and each set [math]\displaystyle{ t_i }[/math] is a non-empty finite subset of the index set [math]\displaystyle{ T }[/math], so each [math]\displaystyle{ t_i\subset T }[/math], which means that [math]\displaystyle{ t_1,\dots,t_n }[/math] is any finite collection of subsets of the index set [math]\displaystyle{ T }[/math].

其中 [math]\displaystyle{ n\geq 1 }[/math] 是一个计数数字,每个集 [math]\displaystyle{ t_i }[/math] 是指数集 [math]\displaystyle{ T }[/math] 的非空有限子集,因此每个 [math]\displaystyle{ t_i\subset T }[/math] ,这意味着 [math]\displaystyle{ t_1,\dots,t_n }[/math]是指数集 [math]\displaystyle{ T }[/math] 的任何有限子集。


Historically, in many problems from the natural sciences a point [math]\displaystyle{ t\in T }[/math] had the meaning of time, so [math]\displaystyle{ X(t) }[/math] is a random variable representing a value observed at time [math]\displaystyle{ t }[/math].[192] A stochastic process can also be written as [math]\displaystyle{ \{X(t,\omega):t\in T \} }[/math] to reflect that it is actually a function of two variables, [math]\displaystyle{ t\in T }[/math] and [math]\displaystyle{ \omega\in \Omega }[/math].[32][193]

历史上,在许多自然科学问题中,一个点[math]\displaystyle{ t\in T }[/math] 具有时间的意义,因此,[math]\displaystyle{ X(t) }[/math]表示是一个在时间[math]\displaystyle{ t }[/math]的随机变量。[194]随机过程也可以写成[math]\displaystyle{ \{X(t,omega):t\ in t\} }[/math]来反映它实际上是两个变量的函数,[math]\displaystyle{ t\in t }[/math][math]\displaystyle{ \omega\in\omega }[/math][36][195]

For any measurable subset [math]\displaystyle{ C }[/math] of the [math]\displaystyle{ n }[/math]-fold Cartesian power [math]\displaystyle{ S^n=S\times\dots \times S }[/math], the finite-dimensional distributions of a stochastic process [math]\displaystyle{ X }[/math] can be written as: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.

对于任何可测量的子集 < math > c </math > n </math >-fold 笛卡尔幂 < math > s ^ n = s 乘以 s </math > ,一个随机过程 < math > x </math > 的有限维分布可以写成: 但是平稳性的概念也存在于点过程和随机场,其中指数集不被解释为时间。


There are other ways to consider a stochastic process, with the above definition being considered the traditional one.[87][88] For example, a stochastic process can be interpreted or defined as a [math]\displaystyle{ S^T }[/math]-valued random variable, where [math]\displaystyle{ S^T }[/math] is the space of all the possible [math]\displaystyle{ S }[/math]-valued functions of [math]\displaystyle{ t\in T }[/math] that map from the set [math]\displaystyle{ T }[/math] into the space [math]\displaystyle{ S }[/math].[31][87]

还有其他方法可以考虑随机过程,上面的定义被认为是传统的=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1 | year=2000 | publisher=Cambridge University Press | isbn=978-1-107-71749-7 | pages=121,122}</ref>[196]例如,一个随机过程可以解释或定义为一个[math]\displaystyle{ S^T }[/math]值的随机变量,其中[math]\displaystyle{ S^T }[/math]是所有可能的[math]\displaystyle{ S }[/math]-值函数的空间T</math>从集合[math]\displaystyle{ T }[/math]到空间[math]\displaystyle{ S }[/math][197][198]

When the index set [math]\displaystyle{ T }[/math] can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.

当指数集 < math > t </math > 可以被解释为时间时,如果一个随机过程的有限维分布在时间平移下是不变的,那么它就是静止的。这种类型的随机过程可以用来描述一个处于稳定状态但仍然经历随机波动的物理系统。只有当随机变量是同分布的时候,一系列随机变量才会形成一个平稳的随机过程。Khinchin 提出了广义平稳性的相关概念,广义的协方差平稳性或平稳性又有其他名称。

Index set索引集

The set [math]\displaystyle{ T }[/math] is called the index set[4][63] or parameter set[32][199] of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set [math]\displaystyle{ T }[/math] the interpretation of time.[1] In addition to these sets, the index set [math]\displaystyle{ T }[/math] can be other linearly ordered sets or more general mathematical sets,[1][67] such as the Cartesian plane [math]\displaystyle{ R^2 }[/math] or [math]\displaystyle{ n }[/math]-dimensional Euclidean space, where an element [math]\displaystyle{ t\in T }[/math] can represent a point in space.[55][200] But in general more results and theorems are possible for stochastic processes when the index set is ordered.[201]

集合[math]\displaystyle{ T }[/math]称为“索引集”[58][202]或“‘参数集’”[36][203]。通常,这个集合是实线的一个子集,例如自然数或一个区间,使集合[math]\displaystyle{ T }[/math]能够解释时间。[59]除了这些集合,索引集[math]\displaystyle{ T }[/math]可以是其他线性有序集或更一般的数学集,[59][204]例如笛卡尔平面[math]\displaystyle{ R^2 }[/math][math]\displaystyle{ n }[/math]维欧几里得空间,其中t中的元素可以表示空间中的一个点=https://books.google.com/books?id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=27}</ref>{cite book | author1=Donald L.Snyder | author2=Michael I.Miller | title=时空中的随机点过程| url=https://books.google.com/books?id=c_3UBwAAQBAJ | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4612-3166-0 | page=25}</ref>但一般情况下,当索引集有序时,随机过程可以得到更多的结果和定理。[205]

A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration [math]\displaystyle{ \{\mathcal{F}_t\}_{t\in T} }[/math], on a probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math] is a family of sigma-algebras such that [math]\displaystyle{ \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} }[/math] for all [math]\displaystyle{ s \leq t }[/math], where [math]\displaystyle{ t, s\in T }[/math] and [math]\displaystyle{ \leq }[/math] denotes the total order of the index set [math]\displaystyle{ T }[/math].

过滤是一个增加序列的 sigma-代数定义关于一些概率空间和一个索引集,有一些总序关系,例如在情况下的索引集是一些子集的实数。更正式的说法是,如果一个随机过程有一个总序的索引集,那么在一个总序的索引集上,对一个概率空间的索引集进行一次过滤,这样的索引集就是一个总序的索引集,这样的索引集的总序就是数学的。


State space 状态空间

The mathematical space [math]\displaystyle{ S }[/math] of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, [math]\displaystyle{ n }[/math]-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.[1][5][32][63][69]

随机过程的数学空间[math]\displaystyle{ S }[/math]称为其“状态空间”。这个数学空间可以用integers、real lines、[math]\displaystyle{ n }[/math]-dimensionalEuclidean spaces、复杂平面或更抽象的数学空间来定义。状态空间是用反映随机过程可以采用的不同值的元素来定义的进程| url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | pages=294295}</ref>[206]

A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process [math]\displaystyle{ X }[/math] that has the same index set [math]\displaystyle{ T }[/math], set space [math]\displaystyle{ S }[/math], and probability space [math]\displaystyle{ (\Omega,{\cal F},P) }[/math] as another stochastic process [math]\displaystyle{ Y }[/math] is said to be a modification of [math]\displaystyle{ Y }[/math] if for all [math]\displaystyle{ t\in T }[/math] the following

一个随机过程的修正是另一个随机过程,它与原始随机过程密切相关。更确切地说,一个随机过程[math]\displaystyle{ X }[/math]具有相同的索引集[math]\displaystyle{ T }[/math]、集空间[math]\displaystyle{ 和概率空间\lt math\gt (\Omega,{\cal F},P) }[/math]作为另一个随机过程[math]\displaystyle{ Y }[/math]的随机过程被称为[math]\displaystyle{ Y }[/math]的修改,如果T</math>中的所有[math]\displaystyle{ T\ \lt center\gt \lt math\gt \lt 中心 \gt \lt 数学 \gt ===Sample function样本函数=== P(X_t=Y_t)=1 , P (x _ t = y _ t) = 1, A '''sample function''' is a single [[Outcome (probability)|outcome]] of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.\lt ref name="Lamperti1977page1"/\gt \lt ref name="Florescu2014page296"\gt {{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=296}}\lt /ref\gt More precisely, if \lt math\gt \{X(t,\omega):t\in T \} }[/math] is a stochastic process, then for any point [math]\displaystyle{ \omega\in\Omega }[/math], the mapping

“样本函数”是随机过程的单个结果,因此,它是由随机过程中每个随机变量的一个可能值构成的=https://books.google.com/books?id=z5sebqaaqbaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | page=296}</ref>更准确地说,如果[math]\displaystyle{ \{X(t,omega):t\in t\} }[/math]是一个随机过程,那么对于任何点[math]\displaystyle{ \omega\in\omega }[/math],则 mapping

</math>

[数学中心]

[math]\displaystyle{ holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be stochastically equivalent or equivalent. 持有。两个相互修正的随机过程具有相同的有限维定律,随机等价或等价。 X(\cdot,\omega): T \rightarrow S, }[/math]

Instead of modification, the term version is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse. The theorem can also be generalized to random fields so the index set is [math]\displaystyle{ n }[/math]-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces. 代替修正,术语版本也被使用,然而当两个随机过程具有相同的有限维分布一些作者使用术语版本,但他们可能被定义在不同的概率空间,因此在后一种意义上,两个相互修改的过程也是彼此的版本,但不是相反。该定理还可以推广到随机域,使指标集是[math]\displaystyle{ n }[/math]维欧氏空间,也可以推广到以度量空间为状态空间的随机过程。

不同的概率空间可以定义不同的两个随机过程,因此两个相互修正的过程,在后一种意义上也是相互修正的过程,但不是相反。这个定理也可以推广到随机场,因此指数集是 < math > n </math > 维欧氏空间,以及以度量空间为状态空间的随机过程。

is called a sample function, a realization, or, particularly when [math]\displaystyle{ T }[/math] is interpreted as time, a sample path of the stochastic process [math]\displaystyle{ \{X(t,\omega):t\in T \} }[/math].[57] This means that for a fixed [math]\displaystyle{ \omega\in\Omega }[/math], there exists a sample function that maps the index set [math]\displaystyle{ T }[/math] to the state space [math]\displaystyle{ S }[/math].[32] Other names for a sample function of a stochastic process include trajectory, path function[207] or path.[208]

称为样本函数,称为“实现”,或者,特别是当[math]\displaystyle{ T }[/math]被解释为时间时,随机过程的“样本路径”[math]\displaystyle{ \{X(T,omega):T\in T\} }[/math]{cite book | author1=L.C.G.Rogers | author2=David Williams | title=扩散,马尔可夫过程,和鞅:第1卷,基金会网址=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1 | year=2000 | publisher=Cambridge University Press | isbn=978-1-107-71749-7 | pages=121–124}</ref>这意味着对于一个固定的[math]\displaystyle{ \omega\in\omega }[/math],存在一个将索引集[math]\displaystyle{ T }[/math]映射到状态空间[math]\displaystyle{ S }[/math][36]的示例函数的其他名称随机过程包括“轨迹”、“路径函数”[209]或“路径”.[210]


Increment增量

Two stochastic processes [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] defined on the same probability space [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math] with the same index set [math]\displaystyle{ T }[/math] and set space [math]\displaystyle{ S }[/math] are said be indistinguishable if the following

两个随机过程 < math > x </math > 和 < math > y </math > 定义在同一个概率空间 < math > (Omega,cal { f } ,p) </math > 具有相同的指数集 < math > t </math > 和集合空间 < math > s </math > 如果下列情况,这两个随机过程是无法区分的

An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if [math]\displaystyle{ \{X(t):t\in T \} }[/math] is a stochastic process with state space [math]\displaystyle{ S }[/math] and index set [math]\displaystyle{ T=[0,\infty) }[/math], then for any two non-negative numbers [math]\displaystyle{ t_1\in [0,\infty) }[/math] and [math]\displaystyle{ t_2\in [0,\infty) }[/math] such that [math]\displaystyle{ t_1\leq t_2 }[/math], the difference [math]\displaystyle{ X_{t_2}-X_{t_1} }[/math] is a [math]\displaystyle{ S }[/math]-valued random variable known as an increment.[55][56] When interested in the increments, often the state space [math]\displaystyle{ S }[/math] is the real line or the natural numbers, but it can be [math]\displaystyle{ n }[/math]-dimensional Euclidean space or more abstract spaces such as Banach spaces.[56]

随机过程的“增量”是同一随机过程的两个随机变量之间的差值。对于一个指数集可以解释为时间的随机过程,增量是随机过程在某个时间段内的变化量。例如,如果t\}</math>中的[math]\displaystyle{ \{X(t):t\in t\} }[/math]是状态空间[math]\displaystyle{ S }[/math]且索引集[math]\displaystyle{ t=[0,infty) }[/math]中的任意两个非负数[math]\displaystyle{ t\u 1\和[0,\infty) }[/math]中的[math]\displaystyle{ t_2\使得\lt math\gt tˉ,差异\lt math\gt X{tu 2}-X{t_1} }[/math]是一个称为增量的[math]\displaystyle{ S }[/math]值随机变量。[60][61]当对增量感兴趣时,通常状态空间[math]\displaystyle{ S }[/math]是实线或自然数,但它可以是[math]\displaystyle{ n }[/math]维欧几里德空间或更抽象的空间,如Banach spaces.[61]

[math]\displaystyle{ \lt 中心 \gt \lt 数学 \gt P(X_t=Y_t \text{ for all } t\in T )=1 , P (x _ t = y _ t text { for all } t in t) = 1, ===Further definitions=== }[/math]

[数学中心]


holds.

持有。

Law定律

For a stochastic process [math]\displaystyle{ X\colon\Omega \rightarrow S^T }[/math] defined on the probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math], the law of stochastic process [math]\displaystyle{ X }[/math] is defined as the image measure:

对于定义在概率空间[math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math]上的随机过程[math]\displaystyle{ X\colon\Omega\rightarrow S^T }[/math],随机过程X</math>的“定律”被定义为[[前推度量|图像度量]:

[math]\displaystyle{ \lt 中心\gt \lt 数学\gt Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space,{{efn|The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic sense), in addition to other conditions. 可分性是随机过程的一个属性,基于它的索引集与机率量测的关系。假设随机过程泛函或具有不可数指标集的随机场泛函可以形成随机变量。对于可分离的随机过程,除了其他条件外,它的索引集必须是可分离的空间。对于一个可分的随机过程集(在概率意义上) ,它的指数集必须是一个可分的空间(在拓扑或分析意义上) ,除了其他条件。 \mu=P\circ X^{-1}, }[/math]

More precisely, a real-valued continuous-time stochastic process [math]\displaystyle{ X }[/math] with a probability space [math]\displaystyle{ (\Omega,{\cal F},P) }[/math] is separable if its index set [math]\displaystyle{ T }[/math] has a dense countable subset [math]\displaystyle{ U\subset T }[/math] and there is a set [math]\displaystyle{ \Omega_0 \subset \Omega }[/math] of probability zero, so [math]\displaystyle{ P(\Omega_0)=0 }[/math], such that for every open set [math]\displaystyle{ G\subset T }[/math] and every closed set [math]\displaystyle{ F\subset \textstyle R =(-\infty,\infty) }[/math], the two events [math]\displaystyle{ \{ X_t \in F \text{ for all } t \in G\cap U\} }[/math] and [math]\displaystyle{ \{ X_t \in F \text{ for all } t \in G\} }[/math] differ from each other at most on a subset of [math]\displaystyle{ \Omega_0 }[/math].

更确切地说,一个带有随机过程的实值连续时间子集 x < math > (Omega,{ cal f } ,p) </math > 是可分的,如果它的指数集 < math > t </math > 有一个稠密的可数子集 < math > u t </math > 并且存在一个集合 < math > Omega 0子集 ω </math > 概率为0,所以 < math > p (Omega _ 0) = 0 </math > ,对于每个开集 < math > g 子集 t </math > 和每个闭集 < math > f 子集文本样式 r = (- infty,infty) </math > ,两个事件 < math > > { x _ t in f text { for all } t in g cap u } </math > 和 < math > { x _ t in f text { for all } t in g } </math > 在 < math > 的子集上最多不同。

where [math]\displaystyle{ P }[/math] is a probability measure, the symbol [math]\displaystyle{ \circ }[/math] denotes function composition and [math]\displaystyle{ X^{-1} }[/math] is the pre-image of the measurable function or, equivalently, the [math]\displaystyle{ S^T }[/math]-valued random variable [math]\displaystyle{ X }[/math], where [math]\displaystyle{ S^T }[/math] is the space of all the possible [math]\displaystyle{ S }[/math]-valued functions of [math]\displaystyle{ t\in T }[/math], so the law of a stochastic process is a probability measure.[31][87][211][212]

其中[math]\displaystyle{ P }[/math]是一个概率度量,符号[math]\displaystyle{ \circ }[/math]表示函数组合,[math]\displaystyle{ X^{-1} }[/math]是可测量函数的前映像,或者等价地,[math]\displaystyle{ S^T }[/math]值随机变量[math]\displaystyle{ X }[/math],其中[math]\displaystyle{ S^T }[/math]是T</math>中所有可能的[math]\displaystyle{ S }[/math]值函数的空间,所以随机过程的规律就是一个概率测度=https://books.google.com/books?id=VQrpBwAAQBAJ |年=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4612-0387-2 |页=40–41}</ref>

The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be [math]\displaystyle{ n }[/math]-dimensional Euclidean space. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

可分性的定义也适用于其他索引集和状态空间,例如在随机场的情况下,索引集和状态空间可以是 < math > n </math >-dimensional Euclidean 空间。Doob 的一个定理,有时也被称为 Doob 的可分性定理,说任何实值连续时间随机过程都有一个可分的修正。这个定理的版本也存在于更一般的索引集和状态空间的随机过程,而不是实线。


For a measurable subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ S^T }[/math], the pre-image of [math]\displaystyle{ X }[/math] gives 对于[math]\displaystyle{ S^T }[/math]的可测子集[math]\displaystyle{ B }[/math],预图像[math]\displaystyle{ X }[/math]给出

[math]\displaystyle{ Two stochastic processes \lt math\gt \left\{X_t\right\} }[/math] and [math]\displaystyle{ \left\{Y_t\right\} }[/math] are called uncorrelated if their cross-covariance [math]\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right] }[/math] is zero for all times. Formally:

两个随机过程 < math > 左{ x _ t 右} </math > 和 < math > 左{ y _ t 右} </math > 如果它们的互协方差 < math > 操作者名{ k }{ mathbf { x }{ y }}(t _ 1,t _ 2) = 操作者名{ e }左(x (t _ 1)-mu _ x (t _ 1)右)左(y (t _ 2)-y _ 2)右] </math > 始终为零,则称为不相关过程。形式上:

X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},

</math>

[math]\displaystyle{ \left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2 }[/math].

左{ x _ t 右} ,左{ y _ t 右}文本{ uncorrelated }四匹配四匹配操作器名{ k }{ mathbf { x }{ y }(t _ 1,t _ 2) = 0四匹配所有 t _ 1,t _ 2 </math > 。

so the law of a [math]\displaystyle{ X }[/math] can be written as:[32]

所以a[math]\displaystyle{ X }[/math]定律可以写成:[36]

[math]\displaystyle{ \mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}). If two stochastic processes \lt math\gt X }[/math] and [math]\displaystyle{ Y }[/math] are independent, then they are also uncorrelated. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression continue à droite, limite à gauche, due to the functions being right-continuous with left limits. A Skorokhod function space, introduced by Anatoliy Skorokhod, The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, [math]\displaystyle{ D[0,1] }[/math] denotes the space of càdlàg functions defined on the unit interval [math]\displaystyle{ [0,1] }[/math].

如果两个随机过程[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]是独立的,那么它们也是不相关的。这类函数被称为cádla g或cadlag函数,基于法语表达式continue a droite,limiteégauche的首字母缩略词,因为这些函数是右连左限的。由Anatoliy Skorokod引入的一个Skorokod函数空间,该函数空间的表示法还可以包括定义所有cédlág函数的区间,因此,例如,[math]\displaystyle{ D[0,1] }[/math]表示定义在单位区间[math]\displaystyle{ [0,1] }[/math]上的cádlág函数空间。

如果两个随机过程 < math > x </math > 和 < math > y </math > 是独立的,那么它们也是不相关的。这种函数称为 càdlàg 或 cadlag 函数,由法语表达式 continue à droite,limite à gauche 的首字母缩写而来,因为这些函数是右连续的,有左限制。由 Anatoliy Skorokhod 引入的 Skorokhod 函数空间,这个函数空间的符号也可以包括定义所有函数的区间,因此,例如,< math > d [0,1] </math > 表示在单位区间 < math > [0,1] </math > 上定义的函数的空间。

</math>


Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.

Skorokhod 函数空间是随机过程理论中的常用空间,因为它经常假定连续时间随机过程的样本函数属于 Skorokhod 空间。

The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution.[192][211][213][214][215]

随机过程或随机变量的规律也被称为“概率定律”,“概率分布”,或“分布”.[194][216][217][218][219]


Finite-dimensional probability distributions有限维概率分布

In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues. For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.

在随机过程的数学构造的背景下,当讨论和假设一个随机过程的某些条件来解决可能的构造问题时,使用术语正则性。例如,为了研究具有不可数指标集的随机过程,我们假设随机过程函数遵守某种类型的正则性条件,如样本函数是连续的。

For a stochastic process [math]\displaystyle{ X }[/math] with law [math]\displaystyle{ \mu }[/math], its finite-dimensional distributions are defined as:

对于随机过程[math]\displaystyle{ X }[/math],其“有限维分布”定义为:

[math]\displaystyle{ \mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1}, }[/math]

where [math]\displaystyle{ n\geq 1 }[/math] is a counting number and each set [math]\displaystyle{ t_i }[/math] is a non-empty finite subset of the index set [math]\displaystyle{ T }[/math], so each [math]\displaystyle{ t_i\subset T }[/math], which means that [math]\displaystyle{ t_1,\dots,t_n }[/math] is any finite collection of subsets of the index set [math]\displaystyle{ T }[/math].[31][220]

其中[math]\displaystyle{ n\geq 1 }[/math]是一个计数数,而每个集合[math]\displaystyle{ t_i }[/math]>是索引集[math]\displaystyle{ T }[/math]的一个非空有限子集,因此每个[math]\displaystyle{ t_i\subset T }[/math],这意味着[math]\displaystyle{ t_1,\dots,t_n }[/math]是索引集[math]\displaystyle{ T }[/math]的任何子集的有限集合[31][220]id=W0ydAgAAQBAJ&pg=PA356 | year=2000 | publisher=剑桥大学出版社| isbn=978-1-107-71749-7 | pages=123}</ref>

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.

马尔可夫过程是随机过程,传统上在离散或连续时间,具有马尔可夫性,这意味着马尔可夫过程的下一个值取决于当前值,但它是有条件地独立于以前的价值随机过程。换句话说,考虑到过程的当前状态,过程在未来的行为随机地独立于过去的行为。


For any measurable subset [math]\displaystyle{ C }[/math] of the [math]\displaystyle{ n }[/math]-fold Cartesian power [math]\displaystyle{ S^n=S\times\dots \times S }[/math], the finite-dimensional distributions of a stochastic process [math]\displaystyle{ X }[/math] can be written as:[32]

对于[math]\displaystyle{ n }[/math]笛卡尔幂[math]\displaystyle{ S^n=S\times\dots \times S }[/math]的任何可测子集[math]\displaystyle{ C }[/math][math]\displaystyle{ X }[/math]的有限维分布可以写成:[36]

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.

布朗运动过程和一维泊松过程都是连续时间马氏过程的例子,而整数上的随机游动和赌徒破产问题都是离散时间马氏过程的例子。

[math]\displaystyle{ \mu_{t_1,\dots,t_n}(C) =P \Big(\big\{\omega\in \Omega: \big( X_{t_1}(\omega), \dots, X_{t_n}(\omega) \big) \in C \big\} \Big). A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). 马尔可夫链是一种具有离散状态空间或离散指标集(通常表示时间)的马尔可夫过程,但是马尔可夫链的精确定义是变化的。例如,通常将'''\lt font color="#ff8000"\gt 马尔可夫链Markov chain\lt /font\gt '''定义为离散或连续时间中具有可数状态空间的马尔可夫过程(因此不考虑时间的性质) ,但也通常将'''\lt font color="#ff8000"\gt 马尔可夫链Markov chain\lt /font\gt '''定义为在可数或连续状态空间中具有离散时间的马尔可夫链(因此不考虑状态空间)。 }[/math]

The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.[70] 随机过程的有限维分布满足两个称为一致性条件的数学条件。[221]

Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.

马尔可夫过程Markov processes是一类重要的随机过程,在许多领域有着广泛的应用。例如,它们是一种通用的随机模拟方法的基础,这种方法被称为 马尔科夫蒙特卡洛模拟法Markov chain MonteCarlo,用于模拟具有特定概率分布的随机目标,并已在贝叶斯统计中得到应用。

Stationarity稳定性

The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as [math]\displaystyle{ n }[/math]-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.

马尔可夫性的概念最初是用于连续和离散时间的随机过程,但是这个性质已经适用于其他指标集,如 [math]\displaystyle{ n }[/math]维欧氏空间,这导致了被称为马尔可夫随机场的随机变量集合。

Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if [math]\displaystyle{ X }[/math] is a stationary stochastic process, then for any [math]\displaystyle{ t\in T }[/math] the random variable [math]\displaystyle{ X_t }[/math] has the same distribution, which means that for any set of [math]\displaystyle{ n }[/math] index set values [math]\displaystyle{ t_1,\dots, t_n }[/math], the corresponding [math]\displaystyle{ n }[/math] random variables

“稳定性”是当随机过程的所有随机变量都是相同分布时随机过程所具有的数学性质。换言之,如果[math]\displaystyle{ X }[/math]是一个平稳随机过程,那么对于任何[math]\displaystyle{ t\in T }[/math],随机变量[math]\displaystyle{ X_t }[/math]具有相同的分布,这意味着对于任何一组[math]\displaystyle{ n }[/math]索引集值[math]\displaystyle{ t_1,\dots, t_n }[/math]而言,对应的[math]\displaystyle{ n }[/math]随机变量

[math]\displaystyle{ X_{t_1}, \dots X_{t_n}, A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general. '''\lt font color="#ff8000"\gt 鞅Martingale\lt /font\gt '''是一个离散时间或连续时间的随机过程,其特性是,在给定过程的当前值和所有过去值的任何时刻,每个未来值的条件期望都等于当前值。在离散时间中,如果此属性对下一个值有效,则对所有未来值都有效。'''\lt font color="#ff8000"\gt 鞅Martingale\lt /font\gt '''的精确数学定义需要两个其他条件加上过滤的数学概念,这与随着时间的推移增加可用信息的直觉有关。'''\lt font color="#ff8000"\gt 鞅Martingale\lt /font\gt '''通常被定义为实值的,但是它们也可以取复值,甚至是更一般的值。 }[/math]

all have the same probability distribution. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.[222][223] But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.[222][224][225]

它们都有相同的概率分布。平稳随机过程的指标集通常被解释为时间,因此可以是整数或实线。[222][223] But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.[222][224][225]


A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

在离散时间和连续时间中, 对称随机游动Symmetric random walk和 维纳Wiener 过程(带零漂移)都是 鞅Martingale的例子。在这方面,离散 鞅Martingale推广了独立随机变量部分和的概念。


When the index set [math]\displaystyle{ T }[/math] can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations.[222] The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same.[226] A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.[222]

当指标集[math]\displaystyle{ T }[/math]可以解释为时间时,如果随机过程的有限维分布在时间平移下是不变的,则称其为平稳过程。这种随机过程可以用来描述处于稳态的物理系统,但是仍然会经历随机波动。[227]平稳性背后的直觉是,随着时间的推移,平稳随机过程的分布保持不变。[228]只有当随机变量相同分布时,一系列随机变量才会形成平稳随机过程。[227]

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.

也可以通过适当的变换从随机过程中产生 鞅Martingale,这是齐次泊松过程(在实线上)产生一个被称为补偿泊松过程的 鞅Martingale的情形。


A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process [math]\displaystyle{ X }[/math] is said to be stationary in the wide sense, then the process [math]\displaystyle{ X }[/math] has a finite second moment for all [math]\displaystyle{ t\in T }[/math] and the covariance of the two random variables [math]\displaystyle{ X_t }[/math] and [math]\displaystyle{ X_{t+h} }[/math] depends only on the number [math]\displaystyle{ h }[/math] for all [math]\displaystyle{ t\in T }[/math].[226][229] Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.[229][230]

具有上述平稳性定义的随机过程有时被称为严格平稳的,但也有其他形式的平稳性。一个例子是当离散时间或连续时间随机过程[math]\displaystyle{ X }[/math]被称为广义平稳时,那么,对于t</math>中的所有[math]\displaystyle{ t\n,过程\lt math\gt X }[/math]有一个有限的第二时刻,两个随机变量的协方差只取决于t</math>中所有[math]\displaystyle{ t\t\的数\lt math\gt h }[/math]Florescu | title=概率和随机过程| url=图书https://books.com/?id=z5sebqaaqabaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | pages=298299}}</ref> Khinchin介绍了“广义平稳性”的相关概念,其他名称包括“协方差平稳性”或“广义平稳性”。[231][232]

Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.

数学上的鞅形式化了公平游戏的概念,它们最初是为了证明不可能赢得公平游戏而开发的。通过在问题中找到一个鞅并研究它,已经解决了许多概率问题。由于鞅收敛定理的存在,在给定矩的一些条件下,鞅会收敛,因此常用它们来推导收敛结果。

Filtration过滤

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process [math]\displaystyle{ X }[/math] is a Lévy process if for [math]\displaystyle{ n }[/math] non-negatives numbers, [math]\displaystyle{ 0\leq t_1\leq \dots \leq t_n }[/math], the corresponding [math]\displaystyle{ n-1 }[/math] increments

鞅Martingales在统计学中有许多应用,但有人指出,鞅的使用和应用并不象在统计学领域,特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性,因此它们被称为具有平稳增量和独立增量的过程。换句话说,如果对于 [math]\displaystyle{ n }[/math] 非负数,[math]\displaystyle{ 0\leq t_1\leq \dots \leq t_n }[/math] ,相应的 [math]\displaystyle{ n-1 }[/math] 递增值是一个列维 Lévy 过程

A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration [math]\displaystyle{ \{\mathcal{F}_t\}_{t\in T} }[/math], on a probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math] is a family of sigma-algebras such that [math]\displaystyle{ \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} }[/math] for all [math]\displaystyle{ s \leq t }[/math], where [math]\displaystyle{ t, s\in T }[/math] and [math]\displaystyle{ \leq }[/math] denotes the total order of the index set [math]\displaystyle{ T }[/math].[63] With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process [math]\displaystyle{ X_t }[/math] at [math]\displaystyle{ t\in T }[/math], which can be interpreted as time [math]\displaystyle{ t }[/math].[63][233] The intuition behind a filtration [math]\displaystyle{ \mathcal{F}_t }[/math] is that as time [math]\displaystyle{ t }[/math] passes, more and more information on [math]\displaystyle{ X_t }[/math] is known or available, which is captured in [math]\displaystyle{ \mathcal{F}_t }[/math], resulting in finer and finer partitions of [math]\displaystyle{ \Omega }[/math].[234][235]

过滤是定义在某个概率空间中的sigma代数的递增序列和具有某种总阶关系的索引集,例如在索引集是实数的某个子集的情况下。更为正式的是,如果随机过程有一个指数集总排序的随机过程,则如果随机过程有一个指数集的总序为总序,那么在概率空间上概率空间[math]\displaystyle{ (\Omega,\mathcal{F{F}u t}{t}{math\gt \{\mathcal{F{F},P,P) }[/math]上是一个西格玛代数家族,这样一个西格玛代数家族使得[math]\displaystyle{ \mathcal{mathcal{F{F}mathcal{F{F{F{F{F{F{F{F{F}数学\gt 为所有\lt 数学\gt s\s\s\subteq\mathcal{F}{F}{leq t }[/math],其中,t中的[math]\displaystyle{ t,s\in t }[/math][math]\displaystyle{ \leq }[/math]表示索引集[math]\displaystyle{ t }[/math]的总顺序[202]通过过滤的概念,可以研究t</math>中随机过程[math]\displaystyle{ X\t }[/math]所包含的信息量,这可以解释为时间[math]\displaystyle{ t }[/math][202][236]过滤背后的直觉是,随着时间的流逝,关于[math]\displaystyle{ t }[/math]的更多信息是已知的或可用的,这些信息可以在[math]\displaystyle{ \mathcal{F}t }[/math]中获得,使[math]\displaystyle{ \Omega }[/math]的分区越来越细{cite book | author=Fima C.Klebaner | title=Introduction to Ratical Calculation with Applications |网址=图书https://books.com/?id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | pages=22–23}</ref>[237]

[math]\displaystyle{ X_{t_2}-X_{t_1}, \dots , X_{t_{n-1}}-X_{t_n}, 2}-x _ { t _ 1} ,点,x _ { t _ { n-1}-x _ { t _ n } , ====Modification修正==== }[/math]


A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process [math]\displaystyle{ X }[/math] that has the same index set [math]\displaystyle{ T }[/math], set space [math]\displaystyle{ S }[/math], and probability space [math]\displaystyle{ (\Omega,{\cal F},P) }[/math] as another stochastic process [math]\displaystyle{ Y }[/math] is said to be a modification of [math]\displaystyle{ Y }[/math] if for all [math]\displaystyle{ t\in T }[/math] the following

随机过程的“修正”是另一个随机过程,它与原始随机过程密切相关。更确切地说,一个随机过程[math]\displaystyle{ X }[/math],与另一个随机过程[math]\displaystyle{ Y }[/math] 具有相同的索引集[math]\displaystyle{ T }[/math]、集空间[math]\displaystyle{ S }[/math]和概率空间[math]\displaystyle{ (\Omega,{\cal F},P) }[/math]具有相同的索引集[math]\displaystyle{ T }[/math]、集空间[math]\displaystyle{ S }[/math]和概率空间[math]\displaystyle{ (\Omega,{\cal F},P) }[/math],被称为[math]\displaystyle{ Y }[/math]的修改,如果对所有[math]\displaystyle{ t\in T }[/math]


are all independent of each other, and the distribution of each increment only depends on the difference in time. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.

都是相互独立的,每个增量的分布只取决于时间的差异。如果随机过程的具体定义要求索引集是实线的一个子集,那么随机场可以被认为是随机过程的推广。

[math]\displaystyle{ P(X_t=Y_t)=1 , }[/math]

holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law[238] and they are said to be stochastically equivalent or equivalent.[239]

注意。两个相互修正的随机过程具有相同的有限维法则[238]它们被称为“随机等价”或“等价物”[239]


A point process is a collection of points randomly located on some mathematical space such as the real line, [math]\displaystyle{ n }[/math]-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or [math]\displaystyle{ n }[/math]-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.

点过程是一个点的集合,这些点随机地分布在一些数学空间上,比如实数直线、 n 维欧氏空间或者更多的抽象空间。有时,词汇点过程并不是首选,因为历史上词汇过程表示某个系统在时间上的演变,所以点过程也称为随机点场。一个点过程有不同的解释,比如随机计数测度或随机集合。有些作者把点过程和随机过程过程看作是两个不同的对象,例如,点过程是一个随机的对象,它起源于或与随机过程过程相关联,尽管有人指出点过程和随机过程之间的区别并不清楚。它对应于随机过程术语中的索引集。}它被定义在其上,例如实线或者[math]\displaystyle{ n }[/math] 维欧氏空间。在点过程理论中研究了更新和计数过程等其他随机过程。


Instead of modification, the term version is also used,[224][240][241][242] however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.[243][211]

除了修改,还使用了“版本”一词,[244][245][246][247]然而,当两个随机过程具有相同的有限维分布,但它们可能定义在不同的概率空间上,因此两个过程是相互修改的,在后一种意义上,它们也是彼此的版本,但不是相反=图书https://books.com/?id=Oybncaaqbaj |年份=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-06400-9 | pages=18–19}</ref>引用错误:无效<ref>标签;name属性非法,可能是内容过长[242][248] The theorem can also be generalized to random fields so the index set is [math]\displaystyle{ n }[/math]-dimensional Euclidean space[249] as well as to stochastic processes with metric spaces as their state spaces.[250]

如果一个连续时间的实值随机过程在其增量上满足一定的矩条件,则Kolmogorov连续性定理指出,该过程存在一个修正,其具有概率为1的连续样本路径,因此随机过程有一个连续的修改或版本=图书https://books.com/?id=q7eDUjdJxIkC | year=2004 | publisher=Cambridge University Press | isbn=978-0-521-83263-2 | page=20}</ref>该定理也可以推广到随机域,因此索引集是[math]\displaystyle{ n }[/math]-维欧几里德空间[251]以及以度量空间为状态空间的随机过程=图书https://books.com/?id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | page=35}</ref>


Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago, but very little analysis on them was done in terms of probability. The year 1654 is often considered the birth of probability theory when French mathematicians Pierre Fermat and Blaise Pascal had a written correspondence on probability, motivated by a gambling problem. But there was earlier mathematical work done on the probability of gambling games such as Liber de Ludo Aleae by Gerolamo Cardano, written in the 16th century but posthumously published later in 1663.

概率论游戏起源于机会游戏,这种游戏有着悠久的历史,有些游戏在几千年前就已经开始玩了,但是很少从概率的角度对它们进行分析。1654年通常被认为是概率论的诞生,当时法国数学家 Pierre Fermat 和 Blaise Pascal 因为一个赌博问题写了一封关于概率的信。但是在赌博游戏的可能性方面,早期的数学工作已经完成,比如吉罗拉莫·卡尔达诺的 Liber de Ludo Aleae,写于16世纪,但死后于1663年出版。

Indistinguishable无法识别

Two stochastic processes [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] defined on the same probability space [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math] with the same index set [math]\displaystyle{ T }[/math] and set space [math]\displaystyle{ S }[/math] are said be indistinguishable if the following

两个随机过程[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]定义在同一概率空间[math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math]上,具有相同的索引集[math]\displaystyle{ T }[/math]和集空间[math]\displaystyle{ S }[/math]上的两个随机过程称为“不可识别的”,如果

After Cardano, Jakob Bernoulli wrote Ars Conjectandi, which is considered a significant event in the history of probability theory. But despite some renowned mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev, most of the mathematical community did not consider probability theory to be part of mathematics until the 20th century.

在 Cardano 之后,Jakob Bernoulli 写了 Ars Conjectandi,这被认为是概率论历史上的一个重大事件。但是,尽管一些著名的数学家为概率论做出了贡献,比如皮埃尔-西蒙·拉普拉斯,亚伯拉罕·棣莫弗,Carl Gauss,Siméon Poisson 和巴夫尼提·列波维奇·切比雪夫,大多数数学界直到20世纪才认为概率论是数学的一部分。

[math]\displaystyle{ P(X_t=Y_t \text{ for all } t\in T )=1 , }[/math]

In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius, most of the work had little or no randomness.

在物理科学领域,科学家们在19世纪发展了统计力学学科,在这个学科中,物理系统,例如装满气体的容器,可以被看作或者从数学上被当作许多运动粒子的集合。尽管有一些科学家,比如鲁道夫 · 克劳修斯,试图将随机性纳入统计物理学,但大多数工作几乎没有或根本没有随机性。

holds.[211][238] If two [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are modifications of each other and are almost surely continuous, then [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are indistinguishable.[252]

保留。[216][253]如果两个[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]是相互修改的,几乎肯定是连续的,那么[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]是无法区分的。[254]

This changed in 1859 when James Clerk Maxwell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities. The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement.

这种情况在1859年发生了改变,当时詹姆斯·克拉克·麦克斯韦对这个领域做出了重大贡献,更具体地说,他提出了假设气体粒子以随机速度向随机方向运动的工作,这对分子运动论研究有重大贡献。分子运动论和统计物理学在19世纪下半叶继续发展,主要由克劳修斯、路德维希·玻尔兹曼和约西亚吉布斯完成的工作,后来对阿尔伯特爱因斯坦的布朗运动的数学模型产生了影响。

Separability可分性

Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space,模板:Efn which means that the index set has a dense countable subset.[224][255]

“可分离性”是随机过程的一种性质,它基于与概率测度有关的指标集。假设随机过程或具有不可数指标集的随机场的泛函可以形成随机变量。对于一个随机过程是可分离的,除了其他条件外,它的指标集必须是一个可分离空间,{efn |术语“可分离”在这里出现了两次,有两种不同的含义,第一种含义来自概率,第二种含义来自拓扑和分析。对于一个随机过程是可分的(概率意义上),它的指标集必须是一个可分空间(在拓扑或分析意义上),除了其他条件。[203]}},这意味着索引集有一个稠密的可数子集。[244][256]

At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms.}} and Andrei Kolmogorov. In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.

1900年在巴黎的国际数学家大会,David Hilbert 展示了一系列数学问题,其中他的第六个问题要求对物理学和涉及公理的概率进行数学处理和安德烈 · 科尔莫戈罗夫。在20世纪30年代早期,钦钦和科尔莫戈罗夫设立了概率研讨会,参加研讨会的研究人员有 Eugene Slutsky 和 Nikolai Smirnov,钦钦给出了第一个数学定义,随机过程是一组由实数线索引的随机变量。


More precisely, a real-valued continuous-time stochastic process [math]\displaystyle{ X }[/math] with a probability space [math]\displaystyle{ (\Omega,{\cal F},P) }[/math] is separable if its index set [math]\displaystyle{ T }[/math] has a dense countable subset [math]\displaystyle{ U\subset T }[/math] and there is a set [math]\displaystyle{ \Omega_0 \subset \Omega }[/math] of probability zero, so [math]\displaystyle{ P(\Omega_0)=0 }[/math], such that for every open set [math]\displaystyle{ G\subset T }[/math] and every closed set [math]\displaystyle{ F\subset \textstyle R =(-\infty,\infty) }[/math], the two events [math]\displaystyle{ \{ X_t \in F \text{ for all } t \in G\cap U\} }[/math] and [math]\displaystyle{ \{ X_t \in F \text{ for all } t \in G\} }[/math] differ from each other at most on a subset of [math]\displaystyle{ \Omega_0 }[/math].[257][258][259]

更精确地说,具有概率空间[math]\displaystyle{ (\Omega,{\calf},P) }[/math]的实值连续时间随机过程[math]\displaystyle{ X }[/math]是可分离的,如果它的指数集[math]\displaystyle{ T }[/math]有一个稠密的可数子集[math]\displaystyle{ U\subset\Omega }[/math],因此[math]\displaystyle{ P(\Omega_0)=0 }[/math],这样对于每个开集[math]\displaystyle{ G\subset T }[/math]和每个闭集[math]\displaystyle{ F\subset\textstyle R=(-\infty,\infty) }[/math],在F\text{FORALALL}t\in G\cap U\}</math>和F\text{FORALALL}\t\G\cap U\}</math>和F\text{FORALALL}t\in G\}</math>这两个事件在[math]\displaystyle{ \Omega }[/math][math]\displaystyle{ \lt ref name=“Gikhmankorokod1969Page150”\gt {(引自《引证图书| author1=IOSIIF IlichiGikhman | author2=Anatoly Vladimirovich Vladimirovich author2=Anatol2=AnatolyVladimal斯科罗霍德| title=介绍随机过程理论=图书https://books.com/?id=yJyLzG7N7r8C&pg=PR2 | year=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=150}\lt /ref\gt \lt ref name=“Todorovic2012page19”\gt {cite book | author=Petar Todorovic | title=随机过程及其应用简介| url=图书https://books.com/?第1249页{jqbn=1240页{jqbn=1240第1页{jqbn=1240第1页,第1页,第1页=图书https://books.com/?id=kWEwk1UL42AC |年份=2005 | publisher=Springer Science&Business Media | isbn=978-1-85233-892-3 | page=340}\lt /ref\gt The definition of separability{{efn|The definition of separability for a continuous-time real-valued stochastic process can be stated in other ways.\lt ref name="Billingsley2008page526"\gt {{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=526–527}}\lt /ref\gt \lt ref name="Borovkov2013page535"\gt {{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=535}}\lt /ref\gt }} can also be stated for other index sets and state spaces,\lt ref name="GusakKukush2010page22"\gt {{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 22\lt /ref\gt such as in the case of random fields, where the index set as well as the state space can be \lt math\gt n }[/math]-dimensional Euclidean space.[34][224]

可分离性的定义{efn |连续时间实值随机过程的可分性定义可以用其他方式表述。[260][261]}}}也可以为其他索引集和状态空间而声明,引用错误:没有找到与</ref>对应的<ref>标签 A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.[255][258][262] Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.[199]

随机过程可分性的概念是由Joseph Doob[256]提出的。可分性的基本思想是使指标集的可数点集决定随机过程的性质,因此离散时间随机过程总是可分离的=图书https://books.com/?id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=56}</ref>Doob的一个定理,有时被称为Doob的可分性定理,表示任何实值连续时间随机过程都有一个可分离的修改。[256][263][264]该定理的版本也适用于具有索引集和状态空间而非实线的更一般的随机过程。[265]

Mathematician [[Joseph Doob did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales. Starting in the 1940s, Kiyosi Itô published papers developing the field of stochastic calculus, which involves stochastic integrals and stochastic differential equations based on the Wiener or Brownian motion process.

数学家[约瑟夫 · 杜布在随机过程理论方面做了早期的工作,作出了基本的贡献,尤其是在鞅理论方面。从20世纪40年代开始,Kiyosi itô 发表了论文,拓展了随机分析的研究领域,包括随机积分和基于 Wiener 或 Brownian 运动过程的随机微分方程。


Independence独立性

Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theory, with early ideas by Shizuo Kakutani and then later work by Joseph Doob.

同样从20世纪40年代开始,随机过程,特别是鞅,和势场理论的数学领域之间建立了联系,早期的思想由 Shizuo Kakutani 提出,后来由 Joseph Doob 提出。

Two stochastic processes [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] defined on the same probability space [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math] with the same index set [math]\displaystyle{ T }[/math] are said be independent if for all [math]\displaystyle{ n \in \mathbb{N} }[/math] and for every choice of epochs [math]\displaystyle{ t_1,\ldots,t_n \in T }[/math], the random vectors [math]\displaystyle{ \left( X(t_1),\ldots,X(t_n) \right) }[/math] and [math]\displaystyle{ \left( Y(t_1),\ldots,Y(t_n) \right) }[/math] are independent.[266]:p. 515

两个在相同的概率空间[math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math]上定义,具有相同索引集[math]\displaystyle{ T }[/math]的随机过程[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]被称为“相互独立”,如果对于所有[math]\displaystyle{ n \in \mathbb{N} }[/math],以及每个特定的[math]\displaystyle{ t_1,\ldots,t_n \in T }[/math],随机向量[math]\displaystyle{ \left( X(t_1),\ldots,X(t_n) \right) }[/math][math]\displaystyle{ \left( Y(t_1),\ldots,Y(t_n) \right) }[/math]是独立的。[266]{rp | p.515}}

In 1953 Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-André Meyer. Earlier work had been carried out by Sergei Bernstein, Paul Lévy and Jean Ville, the latter adopting the term martingale for the stochastic process. Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes. which would later result in Varadhan winning the 2007 Abel Prize. In the 1990s and 2000s the theories of Schramm–Loewner evolution and rough paths were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin Werner in 2008 and to Martin Hairer in 2014.

1953年杜布出版了《随机过程》一书,该书对随机过程理论产生了重大影响,并强调了概率测度理论的重要性。Doob 还主要发展了鞅理论,后来保罗-安德烈 · 迈耶做出了重大贡献。早期的工作是由 Sergei Bernstein,Paul Lévy 和 Jean Ville 完成的,Jean Ville 采用了鞅这个术语来称呼随机过程。从鞅理论开始,解决各种概率问题的方法变得流行起来。研究马尔可夫过程的技术和理论得到了发展,并应用于鞅。相反,从鞅理论中建立了处理马尔可夫过程的方法。后来 Varadhan 赢得了2007年的阿贝尔奖。20世纪90年代和21世纪初,Schramm-Loewner 进化理论和粗糙路径理论被引入并发展起来,用于研究21概率论的随机过程和其他数学对象,结果分别在2008年和2014年分别授予 Wendelin Werner 和 Martin Hairer 菲尔兹奖。

Uncorrelatedness不相关

Two stochastic processes [math]\displaystyle{ \left\{X_t\right\} }[/math] and [math]\displaystyle{ \left\{Y_t\right\} }[/math] are called uncorrelated if their cross-covariance [math]\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right] }[/math] is zero for all times.[267]:p. 142 Formally:

两个随机过程[math]\displaystyle{ \left\{X_t\right\} }[/math][math]\displaystyle{ \left\{Y_t\right\} }[/math] 称为“不相关的”的,如果它们的互协方差[math]\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right] }[/math]始终为零。[267]:p. 142 最后:

The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.

随机过程理论仍然是研究的焦点,每年都有关于随机过程的国际会议。


[math]\displaystyle{ \left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2 }[/math].


The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. Bernoulli's work, including the Bernoulli process, were published in his book Ars Conjectandi in 1713.

伯努利过程可以作为一个数学模型来抛出一个有偏见的硬币,它可能是第一个被研究的随机过程。伯努利的著作,包括《伯努利过程,于1713年在他的书《猜测》中出版。

Independence implies uncorrelatedness独立意味着不相关

If two stochastic processes [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are independent, then they are also uncorrelated.[267]:p. 151

如果两个随机过程[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]是独立的,那么它们也是不相关的

In 1905 Karl Pearson coined the term random walk while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. and is an example of a random walk with absorbing barriers. Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods, and then more detailed solutions were presented by Jakob Bernoulli and Abraham de Moivre.

1905年,卡尔 · 皮尔森在提出一个描述平面上随机漫步的问题时,创造了 随机漫步Random walk这个术语,这个问题的动机是生物学中的一个应用,但是这种涉及随机漫步的问题已经在其他领域得到了研究。几个世纪前研究过的某些赌博问题可以被认为是涉及随机漫步的问题。这是一个带有吸收屏障的随机漫步的例子。和 Huyens 都给出了这个问题的数值解,但没有详细介绍他们的方法,然后 Jakob Bernoulli 和亚伯拉罕·棣莫弗提供了更详细的解。

Orthogonality正交性

Two stochastic processes [math]\displaystyle{ \left\{X_t\right\} }[/math] and [math]\displaystyle{ \left\{Y_t\right\} }[/math] are called orthogonal if their cross-correlation [math]\displaystyle{ \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E}[X(t_1) \overline{Y(t_2)}] }[/math] is zero for all times.[267]:p. 142 Formally:

由</math>和[math]\displaystyle{ \left\{Y Y{R}{mathbf{math\gt \ left\{YY{right\} }[/math][math]\displaystyle{ \left\{Y Y{right\} }[/math]两个随机过程<数学>\operatorname{R{mathbf{mathbf{X}\mathbf{Y}}(t 1、t U2)的=\operatorname{E{E}[X(t(t U1)1)\顶顶天{Y(t〈2)右}}</math>如果它们的相互关联<正交{正交ref name=KunIlPark/>: p.142形式上:

For random walks in [math]\displaystyle{ n }[/math]-dimensional integer lattices, George Pólya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.

对于 < math > n </math > 维整数格中的随机游动,George Pólya 在1919年和1921年发表的工作中,他研究了对称随机游动回到格中以前位置的概率。Pólya 证明了对称随机游动,它在格子中向任何方向前进的概率相等,将无限次地回到格子中的一个先前的位置,概率为1在一维和2维,但概率为0在三维或更高维。


[math]\displaystyle{ \left\{X_t\right\},\left\{Y_t\right\} \text{ orthogonal} \quad \iff \quad \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2 }[/math].


The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics. The work is now considered as an early discovery of the statistical method known as Kalman filtering, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time. in order to model price changes on the Paris Bourse, a stock exchange, without knowing the work of Thiele. and Bachelier's thesis is now considered pioneering in the field of financial mathematics.

维纳过程或布朗运动过程起源于不同的领域,包括统计学、金融学和物理学。这项工作现在被认为是卡尔曼滤波统计方法的早期发现,但是这项工作在很大程度上被忽视了。人们认为,蒂勒论文中的观点太过先进,当时更广泛的数学和统计学界无法理解。为了模拟巴黎证券交易所的价格变化,不知道蒂勒的工作。巴切利耶的论文现在被认为是金融数学领域的先驱。

Skorokhod space斯科罗霍德空间

Einstein's work, as well as experimental results obtained by Jean Perrin, later inspired Norbert Wiener in the 1920s to use a type of measure theory, developed by Percy Daniell, and Fourier analysis to prove the existence of the Wiener process as a mathematical object. There are a number of claims for early uses or discoveries of the Poisson

爱因斯坦的工作,以及 Jean Perrin 获得的实验结果,后来激发了 Norbert Wiener 在20世纪20年代使用一种由 Percy Daniell 和傅立叶变换家族中的关系提出的测量理论来证明 Wiener 过程作为一个数学对象的存在。关于泊松鱼的早期用途和发现,有许多说法

A Skorokhod space, also written as Skorohod space, is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as [math]\displaystyle{ [0,1] }[/math] or [math]\displaystyle{ [0,\infty) }[/math], and take values on the real line or on some metric space.[268][269][270] Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression continue à droite, limite à gauche, due to the functions being right-continuous with left limits.[268][271] A Skorokhod function space, introduced by Anatoliy Skorokhod,[270] is often denoted with the letter [math]\displaystyle{ D }[/math],[268][269][270][271] so the function space is also referred to as space [math]\displaystyle{ D }[/math].[268][272][273] The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, [math]\displaystyle{ D[0,1] }[/math] denotes the space of càdlàg functions defined on the unit interval [math]\displaystyle{ [0,1] }[/math].[271][273][274]

skorokod space也写为Skorohod space,是所有右连续左极限的函数的数学空间,定义在实线的某个区间上,例如[math]\displaystyle{ [0,1] }[/math][math]\displaystyle{ [0,\infty) }[/math],取实线或度量空间上的值=图书https://books.com/?id=LkQOBwAAQBAJ&pg=PR5 | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21748-2 |页=78–79}</ref>[275][276]这些函数被称为cádLag或cadlag函数,这是基于法语表达式“continue a droite,limiteégauche”的首字母缩略词,因为这些函数是右连续的,具有左极限。[277]<refname=“Klebaner2005page4”>{cite book | author=Fima C.Klebaner | title=随机微积分及其应用简介|网址=图书https://books.com/?id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=4}</ref>由Anatoliy Skorokod引入的Skorokod函数空间,[276]通常用字母[math]\displaystyle{ D }[/math]表示,[277][278][276][279]因此函数空间也被称为空间[math]\displaystyle{ D }[/math][277][280][281]此函数空间的表示法还可以包括定义所有cádlág函数的间隔,因此,例如,[math]\displaystyle{ D[0,1] }[/math]表示在单位间隔[math]\displaystyle{ [0上定义的c|dla g函数的空间,1] }[/math][279][281][282]

process.

过程。


Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.[270][272] Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.[273][283]

在随机过程理论中,由于通常假定连续时间随机过程的样本函数属于一个Skorokod空间,因此经常使用Skorokod函数空间,对应于Wiener过程的样本函数。但是空间也有间断函数,这意味着随机过程的样本函数具有跳跃性,例如泊松过程(在实线上),同时也是这一领域的成员。[281][284]


Another discovery occurred in Denmark in 1909 when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution. Markov was interested in studying an extension of independent random sequences. which had been commonly regarded as a requirement for such mathematical laws to hold. Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.

另一个发现发生在1909年的丹麦。在开发一个有限时间间隔内接听电话数量的数学模型时,Erlang 得出了这个泊松分佈。当时 Erlang 并不知道 Poisson 的早期工作,并且假设每个时间间隔内到达的号码电话是相互独立的。然后他发现了极限情况,这是有效地重铸泊松分佈作为一个二项分布的限制。马尔科夫对研究独立随机序列的推广很感兴趣。这被普遍认为是这样的数学定律的一个必要条件。从1928年开始,莫里斯 · 弗雷切特对马尔可夫链产生了兴趣,最终导致他在1938年发表了一篇关于马尔可夫链的详细研究。

Regularity规律性

Andrei Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s. In addition to Lévy, Khinchin and Kolomogrov, early fundamental contributions to the theory of Lévy processes were made by Bruno de Finetti and Kiyosi Itô.

安德烈 · 科尔莫戈罗夫在1931年的一篇论文中发展了早期连续时间马尔可夫过程理论的很大一部分。他介绍并研究了一组特殊的马尔可夫过程,称为扩散过程,在这组过程中他推导出了一组描述这些过程的微分方程。在研究布朗运动时,Sydney Chapman 在1928年的一篇论文中,独立于 Kolmogorov 的工作,用一种比 Kolmogorov 更不严密的数学方法,推导出了一个方程,现在称为 Chapman-Kolmogorov 方程。这些微分方程现在被称为 Kolmogorov 方程或 Kolmogorov-Chapman 方程。其他对马尔可夫过程的基础做出了重大贡献的数学家包括威廉 · 费勒,从20世纪30年代开始,然后是尤金 · 戴金,从20世纪50年代开始。除了 Lévy,Khinchin 和 Kolomogrov,早期对 Lévy 过程理论的根本性贡献是由德福内梯和 Kiyosi itô。

In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.[285][286] For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.[287][288]

在随机过程的数学构造中,当讨论和假设随机过程的某些条件以解决可能的构造问题时,使用术语“正则性”。[289][290]例如,研究具有不可数索引集的随机过程,假设随机过程服从某种正则条件,例如样本函数是连续的=图书https://books.com/?id=XpjqBwAAQBAJ&pg=PP5 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4613-9742-7 | page=22}</ref>[291]


Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem to prove a corresponding stochastic process exists. says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions. This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.

另一种方法是定义一组具有特定有限维分布的随机变量,然后用 Kolmogorov 的存在性定理证明相应的随机过程存在。他说,如果任何有限维分布满足两个条件,也就是所谓的一致性条件,那么就存在这些有限维分布的随机过程。这意味着随机过程的分布并不一定唯一地指定随机过程的样本函数的属性。

Further examples更多示例

Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined. Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set. Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied. for a continuous-time stochastic process with any metric space as its state space. For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption, but such a stochastic process based on this approach will be automatically separable.

另一个问题是,连续时间过程的泛函依赖于指数集中无法计算的点数,因此某些事件的概率可能无法很好地定义。可分性保证了无穷维分布决定样本函数的性质,它要求样本函数本质上是由指数集中的稠密可数点集上的值决定的。此外,如果随机过程是可分的,那么指数集上不可数个点的泛函是可测的,并且可以研究它们的概率。对于任意度量空间作为状态空间的连续时间随机过程。为了构造这样一个随机过程,我们假设随机过程的样本函数属于某个适当的函数空间,这个空间通常是由所有右连续函数和左极限组成的 Skorokhod 空间。这种方法现在比可分离性假设更常用,但是基于这种方法的随机过程可自动分离。

马尔可夫过程与链Markov processes and chains

Although less used, the separability assumption is considered more general because every stochastic process has a separable version. For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as [math]\displaystyle{ n }[/math]-dimensional Euclidean space.

尽管很少使用,但是可分性假设被认为是更一般的,因为每个随机过程都有一个可分离的版本。例如,在构造和研究随机场时假设可分性,其中随机变量的集合现在由实线以外的集合索引,如 [math]\displaystyle{ n }[/math] 维欧氏空间。

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.[292][293]

马尔可夫过程Markov processes 是一种随机过程,传统上在离散或连续时间中,具有马尔可夫特性,即马尔可夫过程的下一个值取决于当前值,但它与随机过程的先前值条件无关。换句话说,给定进程的当前状态,进程在未来的行为与它过去的行为是随机独立的。[294][295]

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes[296] in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.[297][298]

布朗运动过程和泊松过程(一维)都是马尔可夫过程的例子[296],整数上的随机游走赌徒破产问题是离散时间中马尔可夫过程的例子[297][299]

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A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies.[300] For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time),[301][302][303][304] but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).[300] It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like Joseph Doob and Kai Lai Chung.[305]

马尔可夫链是一种具有离散状态空间或离散索引集(通常表示时间)的马尔可夫过程,但马尔可夫链的精确定义各不相同[300]例如,通常将马尔可夫链定义为具有可数状态空间的离散或连续时间中的马尔可夫过程(因此不管时间的性质),[306][307][308][309]但将马尔可夫链定义为在可数状态空间或连续状态空间(因此与状态空间无关)中具有离散时间也是常见的马尔可夫链的第一个定义,它有离散时间,现在倾向于使用,尽管第二个定义已经被Joseph DoobKai Lai Chung等研究人员所使用。[310]


Markov processes form an important class of stochastic processes and have applications in many areas.[45][311] For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.[312][313]

马尔可夫过程是一类重要的随机过程,在许多领域有着广泛的应用=https://books.google.com/books?id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=47}</ref>例如,它们是一种称为Markov chain Monte Carlo的一般随机模拟方法的基础,该方法用于模拟具有特定概率分布的随机对象,并在Bayesian statistics.[314][315]

The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as [math]\displaystyle{ n }[/math]-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.[316][317][318]

马尔可夫特性的概念最初是针对连续和离散时间的随机过程,但它也适用于其它指标集,如[math]\displaystyle{ n }[/math]维欧氏空间,这导致随机变量的集合被称为马尔可夫随机场。[319][320][321]

鞅Martingale

A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,[322][323][233] but they can also be complex-valued[324] or even more general.[325]

鞅Martingale是一个离散时间或连续时间的随机过程,其性质是在给定过程的当前值和所有过去值的情况下,每个未来值的条件期望值等于当前值。在离散时间中,如果此属性适用于下一个值,则它适用于所有未来值。鞅Martingale的精确数学定义需要另外两个条件与过滤的数学概念相结合,这与随时间推移增加可用信息的直觉有关。鞅Martingale通常被定义为实值,[326][327][236]但是它们也可以是复杂值[328]或更一般的。[329]


A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.[322][323] For a sequence of independent and identically distributed random variables [math]\displaystyle{ X_1, X_2, X_3, \dots }[/math] with zero mean, the stochastic process formed from the successive partial sums [math]\displaystyle{ X_1,X_1+ X_2, X_1+ X_2+X_3, \dots }[/math] is a discrete-time martingale.[330] In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.[331]

对称随机游动和Wiener过程(具有零漂移)分别是离散时间和连续时间的鞅Martingale的例子。[326][327]对于一个独立且同分布随机变量的序列[math]\displaystyle{ X_1, X_2, X_3, \dots }[/math] 且平均值为零,由连续部分和[math]\displaystyle{ X_1,X_1+ X_2, X_1+ X_2+X_3, \dots }[/math] 构成的随机过程是一个离散时间鞅Martingale[330],离散时间鞅推广了独立随机变量的部分和的概念。[331]

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.[323] Martingales can also be built from other martingales.[330] For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.[322][332]

通过应用适当的变换,也可以从随机过程中产生鞅Martingale,这就是齐次泊松过程(在实线上)的情形,其结果是一个称为“补偿泊松过程”的鞅。[333]也可以从其他鞅中构建鞅。[334]例如,有基于鞅的鞅Wiener过程,形成连续时间鞅。[326][335]

Martingales mathematically formalize the idea of a fair game,[336] and they were originally developed to show that it is not possible to win a fair game.[337] But now they are used in many areas of probability, which is one of the main reasons for studying them.[233][337][338] Many problems in probability have been solved by finding a martingale in the problem and studying it.[339] Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.[331][340][341]

数学上的鞅形式化了公平博弈的概念,[342]它们最初的开发目的是表明不可能赢得一场公平的比赛。[343]但现在它们被用于许多概率领域,这是研究它们的主要原因之一[236][343]</ref>[121]许多概率问题已经通过在问题中找到鞅并加以研究而得到解决。[344]在给定鞅矩的条件下,鞅会收敛,因此经常使用鞅得到收敛结果,这主要是由于鞅收敛定理s。[331][340][341]

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.[345] They have found applications in areas in probability theory such as queueing theory and Palm calculus[346] and other fields such as economics[347] and finance.[19]

鞅Martingale在统计学中有许多应用,但有人指出,它的使用和应用并不像它在统计学领域那样广泛,尤其是统计推断,293 | issn=0883-4237 | doi=10.1214/088342306000000088 | arxiv=math/0609294 | bibcode=2006math……9294G}</ref>他们在排队论和棕榈微积分等概率论领域找到了应用[348]。以及其他领域,如经济学[349]和金融。[350]

Lévy process莱维过程

Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.[56][351] These processes have many applications in fields such as finance, fluid mechanics, physics and biology.[352][353] The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process [math]\displaystyle{ X }[/math] is a Lévy process if for [math]\displaystyle{ n }[/math] non-negatives numbers, [math]\displaystyle{ 0\leq t_1\leq \dots \leq t_n }[/math], the corresponding [math]\displaystyle{ n-1 }[/math] increments

莱维Lévy过程是随机过程的一种类型,可以看作是连续时间中随机游动的推广[61][354]这些过程在金融、流体力学等领域有着广泛的应用,[352][353] 这些过程和过程的独立性被称为平稳过程的主要特征。换句话说,一个随机过程[math]\displaystyle{ X }[/math]是一个Lévy过程,如果对非负数[math]\displaystyle{ n }[/math][math]\displaystyle{ 0\leq t_1\leq \dots \leq t_n }[/math],当[math]\displaystyle{ n-1 }[/math]递增

[math]\displaystyle{ X_{t_2}-X_{t_1}, \dots , X_{t_{n-1}}-X_{t_n}, }[/math]

are all independent of each other, and the distribution of each increment only depends on the difference in time.[56] 它们彼此独立,每个增量的分布只取决于时间的差异。[61]


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After Cardano, Jakob Bernoulli模板:Efn wrote Ars Conjectandi, which is considered a significant event in the history of probability theory.[355] Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability.[355][356][357] But despite some renowned mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev,[358][359] most of the mathematical community模板:Efn did not consider probability theory to be part of mathematics until the 20th century.[358][360][361][362]

继卡达诺之后,Jakob Bernoulli模板:Efn写了魔术师Ars conjuctandi,在概率论史上被认为是重大事件。[363]伯努利的书出版于1713年,也是在他死后出版的,激发了许多数学家研究概率[363][364][365]。 但尽管一些著名的数学家对概率论做出了贡献,比如[[皮埃尔-西蒙-拉普拉斯]、亚伯拉罕-德-莫伊夫卡尔-高斯西蒙-泊阿松Siméon Poisson帕夫努蒂·切比雪夫Pafnuty Chebyshev[366][367],大多数数学界人士都注意到,一个显著的例外是俄罗斯的圣彼得堡学派,在那里,以切比雪夫为首的数学家研究概率论,[368]}}直到20世纪,概率论才被认为是数学的一部分。[369][370][371][372]

Statistical mechanics统计力学

In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius, most of the work had little or no randomness.[373][374]

在物理科学中,科学家们在19世纪发展了统计力学这门学科,在这个学科中,物理系统,例如装满气体的容器,可以从数学上看作或处理为许多运动粒子的集合。尽管有些科学家试图将随机性纳入统计物理学,比如Rudolf Clausius,大部分工作没有或几乎没有随机性。引用错误:没有找到与</ref>对应的<ref>标签

This changed in 1859 when James Clerk Maxwell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities.[375][376] The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement.[377]

这一点在1859年发生了变化,当时James clark Maxwell对该领域做出了重大贡献,更具体地说,是对气体动力学理论的贡献,通过介绍他的工作,他假设气体粒子以随机速度随机方向移动引用错误:没有找到与</ref>对应的<ref>标签气体动力学理论和统计物理在19世纪后半叶继续发展,主要由克劳修斯,路德维希玻尔兹曼约西亚吉布斯完成,这项工作后来对阿尔伯特爱因斯坦关于布朗运动的数学模型产生了影响。[377]

Measure theory and probability theory测度论与概率论

At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms.[359] Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians, Henri Lebesgue and Émile Borel. In 1925 another French mathematician Paul Lévy published the first probability book that used ideas from measure theory.[359]

1900年在巴黎举行的国际数学家大会上,David Hilbert提出了一份数学问题的清单,其中他的第六个问题要求对涉及公理的物理和概率进行数学处理。[367]大约在20世纪初,数学家发展了测量理论,这是研究数学函数积分的数学分支,其中两位创始人是法国数学家Henri LebesgueÉmile Borel。1925年,另一位法国数学家 Paul Lévy出版了第一本使用测度论思想的概率论书籍

In 1920s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergei Bernstein, Aleksandr Khinchin,模板:Efn and Andrei Kolmogorov.[362] Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory.[378] In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov,[379] and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.[79][380]模板:Efn

20世纪20年代,苏联的数学家们对概率论做出了重大贡献,比如Sergei BernsteinAleksandr Khinchin,{{efn | Khinchin这个名字也用英语写成(或音译成)Khintchine。[381] }和[[Andrei Kolmogorov ] ] < 1929】命名为“CRAME1976”/Kolmogorov于1984年发表了基于测量理论的数学基础的首次尝试。概率论的概率论。[382]在20世纪30年代初,胡仁钦和科尔莫戈罗夫在20世纪30年代初建立了概率研讨会,这些研讨会由研究者参加,如Eugene Slutsky]等和尼古拉·斯米尔诺夫[383]还有金钦给出了第一个随机变量的数学定义,把随机过程作为以实数线索引的一组随机变量。[381][384]模板:Efn

Birth of modern probability theory现代概率论的诞生

In 1933 Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,模板:Efn where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.[359][362]

1933年,Andrei Kolmogorov在德国出版了一本关于概率论基础的书,名为“概率计算的基本概念”,后来翻译成英文,1950年出版,作为概率论的基础。这本书的出版现在被广泛认为是现代概率论的诞生,当时概率论和随机过程理论成为数学的一部分。

After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as Joseph Doob, William Feller, Maurice Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér.[359][362]

在科尔莫戈洛夫的书出版后,钦钦和科尔莫戈洛夫以及其他数学家如Joseph DoobWilliam FellerMaurice Fréchet Paul LévyWolfgang Doeblin等对概率论和随机过程进行了进一步的基础性工作,和Harald Cramér[367][372]

Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".[362] World War II greatly interrupted the development of probability theory, causing, for example, the migration of Feller from Sweden to the United States of America[362] and the death of Doeblin, considered now a pioneer in stochastic processes.[385]

几十年后,克莱姆把20世纪30年代称为“数学概率论的英雄时期”。[372]第二次世界大战极大地中断了概率论的发展,例如,Feller从瑞典迁移到美国[372]以及现在被认为是随机过程先驱的Doeblin之死,1915-1940;journal=The Annals of Probability | volume=19 | issue=3 | year=1991 | pages=929-934 | issn=0091-1798 | doi=10.1214/aop/1176990329 | doi access=free}</ref>

文件:Joseph Doob.jpg
Mathematician Joseph Doob did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales.[386][387] His book Stochastic Processes is considered highly influential in the field of probability theory.[388]

[[文件:Joseph Doob.jpg|thumb |右|数学家Joseph Doob在随机过程理论方面做了早期的工作,做出了基本贡献,尤其是在鞅理论方面。[389][390]他的书《随机过程》被认为在概率论领域具有很高的影响力。<refname=“Bingham2005”/>]]

Stochastic processes after World War II二战后的随机过程

After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas.[362][391] Starting in the 1940s, Kiyosi Itô published papers developing the field of stochastic calculus, which involves stochastic integrals and stochastic differential equations based on the Wiener or Brownian motion process.[392]

第二次世界大战后,概率论和随机过程的研究得到了数学家的更多关注,在概率论和数学的许多领域做出了重大贡献,并开创了新的领域统计学|卷=5 |问题=1 |年=2009 |页=1–42}</ref>从20世纪40年代开始,Kiyosi Itô发表了发展[[随机微积分]领域的论文,它包括基于维纳或布朗运动过程的随机积分和随机微分方程

Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theory, with early ideas by Shizuo Kakutani and then later work by Joseph Doob.[391] Further work, considered pioneering, was done by Gilbert Hunt in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.[22][393][394]

同样从20世纪40年代开始,随机过程(尤其是鞅)与势理论的数学领域之间建立了联系,Shizuo Kakutani的早期思想和Joseph Doob后来的工作。[395]在1950年代Gilbert Hunt完成了被认为是开创性的进一步工作,把马尔可夫过程和势理论联系起来,这对Lévy过程理论产生了重大影响,并使人们对用It开发的方法研究马尔可夫过程产生了更多的兴趣[396][397]引用错误:无效<ref>标签;name属性非法,可能是内容过长 Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-André Meyer. Earlier work had been carried out by Sergei Bernstein, Paul Lévy and Jean Ville, the latter adopting the term martingale for the stochastic process.[398][399] Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.[391]

[388]Doob还主要发展了鞅理论,后来[[保罗.安德烈.梅耶]也作出了重大贡献。早期的研究是由Sergei Bernstein Paul LévyJean Ville进行的,后者采用了随机过程的鞅项。[400][401] [395]鞅理论中的方法已成为解决各种概率问题的常用方法。研究马尔可夫过程的技术和理论发展到鞅上。相反地,从鞅理论中也建立了处理Markov过程的方法。[391]

Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations.[391] The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s fundamental work was done by Alexander Wentzell in the Soviet Union and Monroe D. Donsker and Srinivasa Varadhan in the United States of America,[402] which would later result in Varadhan winning the 2007 Abel Prize.[403] In the 1990s and 2000s the theories of Schramm–Loewner evolution[404] and rough paths[211] were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin Werner[405] in 2008 and to Martin Hairer in 2014.[406]

概率的其他领域也被发展和用于研究随机过程,其中一个主要方法是大偏差理论。[395]该理论在统计物理等领域有许多应用,其核心思想至少可以追溯到20世纪30年代。20世纪60年代和70年代后期,苏联的亚历山大·温策尔和美利坚合众国的Monroe D.DonskerSrinivasa Varadhan完成了基础工作,[407],这将使瓦拉丹获得2007年阿贝尔奖。[408]上世纪90年代和2000年代的理论施拉姆–Loewner演化][409]粗略路径[410]被引入和发展来研究概率论中的随机过程和其他数学对象,分别在2008年和2014年分别授予Wendelin Werner[411]和2014年授予Martin Haierjournal | last1=Quastel | first1=Jeremy | title=2014年菲尔兹奖获得者的作品| journal=AMS的通知|卷=62 |问题=11 |年=2015 |页=1341-1344}</ref>

The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.[51][352] 随机过程理论仍然是研究的焦点,每年都有关于随机过程的国际会议

Category:Stochastic models

类别: 随机模型


Category:Statistical data types

类别: 统计数据类型


This page was moved from wikipedia:en:Stochastic process. Its edit history can be viewed at 随机过程/edithistory

此页摘自维基百科:英语:随机过程。其编辑历史记录可以在随efor过程/edithistory]查阅

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  234. Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. pp. 22–23. ISBN 978-1-86094-555-7. https://books.google.com/books?id=JYzW0uqQxB0C. 
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  243. Daniel Revuz; Marc Yor (2013). Continuous Martingales and Brownian Motion. Springer Science & Business Media. pp. 18–19. ISBN 978-3-662-06400-9. https://books.google.com/books?id=OYbnCAAAQBAJ. 
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  248. David Applebaum (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press. p. 20. ISBN 978-0-521-83263-2. https://books.google.com/books?id=q7eDUjdJxIkC. 
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  266. 266.0 266.1 Lapidoth, Amos, A Foundation in Digital Communication, Cambridge University Press, 2009. 引用错误:无效<ref>标签;name属性“Lapidoth”使用不同内容定义了多次
  267. 267.0 267.1 267.2 267.3 Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3 引用错误:无效<ref>标签;name属性“KunIlPark”使用不同内容定义了多次
  268. 268.0 268.1 268.2 268.3 Ward Whitt (2006). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Science & Business Media. pp. 78–79. ISBN 978-0-387-21748-2. https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5. 
  269. 269.0 269.1 脚本错误:没有“Footnotes”这个模块。, p. 24
  270. 270.0 270.1 270.2 270.3 Vladimir I. Bogachev (2007). Measure Theory (Volume 2). Springer Science & Business Media. p. 53. ISBN 978-3-540-34514-5. https://books.google.com/books?id=CoSIe7h5mTsC. 
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  273. 273.0 273.1 273.2 Patrick Billingsley (2013). Convergence of Probability Measures. John Wiley & Sons. p. 121. ISBN 978-1-118-62596-5. https://books.google.com/books?id=6ItqtwaWZZQC. 
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  275. {harvxt | Gusak | kush | Kulik | Mishura | 2010},p、 24
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  284. {cite book | author1=Nicholas H.Bingham | author2=Rüdiger Kiesel | title=风险中性估值:金融衍生品的定价和对冲| url=图书https://books.com/?id=AOIlBQAAQBAJ |年份=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4471-3856-3 | page=154}
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  286. Davar Khoshnevisan (2006). Multiparameter Processes: An Introduction to Random Fields. Springer Science & Business Media. pp. 148–165. ISBN 978-0-387-21631-7. https://books.google.com/books?id=XADpBwAAQBAJ. 
  287. Petar Todorovic (2012). An Introduction to Stochastic Processes and Their Applications. Springer Science & Business Media. p. 22. ISBN 978-1-4613-9742-7. https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5. 
  288. Ward Whitt (2006). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Science & Business Media. p. 79. ISBN 978-0-387-21748-2. https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5. 
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  291. {cite book | author=Ward Whitt | title=随机过程限制:随机过程限制及其在队列中的应用简介| url=图书https://books.com/?id=LkQOBwAAQBAJ&pg=PR5 | year=2006 | publisher=Springer科学与商业媒体| isbn=978-0-387-21748-2 | page=79}
  292. Richard Serfozo (2009). Basics of Applied Stochastic Processes. Springer Science & Business Media. p. 2. ISBN 978-3-540-89332-5. https://books.google.com/books?id=JBBRiuxTN0QC. 
  293. Y.A. Rozanov (2012). Markov Random Fields. Springer Science & Business Media. p. 58. ISBN 978-1-4613-8190-7. https://books.google.com/books?id=wGUECAAAQBAJ. 
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