# 随机过程

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.

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]

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.

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]

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.

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]

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]

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.

## 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.

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 $\displaystyle{ n }$-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]

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.

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.

### 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 $\displaystyle{ n }$-dimensional Euclidean space $\displaystyle{ \mathbb{R}^n }$ or a manifold. $\displaystyle{ \{X(t)\} }$ or simply as $\displaystyle{ X }$ or $\displaystyle{ X(t) }$, although $\displaystyle{ X(t) }$ is regarded as an abuse of function notation. For example, $\displaystyle{ X(t) }$ or $\displaystyle{ X_t }$ are used to refer to the random variable with the index $\displaystyle{ t }$, 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.

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]

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]

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 $\displaystyle{ n }$-dimensional Euclidean space, then the stochastic process is called a $\displaystyle{ n }$-dimensional vector process or $\displaystyle{ n }$-vector process.[63][64]

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.

### 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, $\displaystyle{ p }$, or decreases by one with probability $\displaystyle{ 1-p }$, so the index set of this random walk is the natural numbers, while its state space is the integers. If the $\displaystyle{ p=0.5 }$, this random walk is called a 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]

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]

### 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.

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.

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 $\displaystyle{ n }$-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 $\displaystyle{ \mu }$, which is a real number, then the resulting stochastic process is said to have drift $\displaystyle{ \mu }$.

### Notation符号

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

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.

## 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.

### 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 $\displaystyle{ p }$ and zero with probability $\displaystyle{ 1-p }$. This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is $\displaystyle{ p }$ 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]

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 $\displaystyle{ t }$, 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.

### 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 $\displaystyle{ n }$-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.

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, $\displaystyle{ p }$, or decreases by one with probability $\displaystyle{ 1-p }$, so the index set of this random walk is the natural numbers, while its state space is the integers. If the $\displaystyle{ p=0.5 }$, this random walk is called a symmetric random walk.[127][128]

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

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

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

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

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 $\displaystyle{ S^T }$-valued random variable, where $\displaystyle{ S^T }$ is the space of all the possible $\displaystyle{ S }$-valued functions of $\displaystyle{ t\in T }$ that map from the set $\displaystyle{ T }$ into the space $\displaystyle{ S }$. 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 $\displaystyle{ T }$ the interpretation of time. such as the Cartesian plane $\displaystyle{ R^2 }$ or $\displaystyle{ n }$-dimensional Euclidean space, where an element $\displaystyle{ t\in T }$ can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.

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 $\displaystyle{ n }$-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 $\displaystyle{ \mu }$, which is a real number, then the resulting stochastic process is said to have drift $\displaystyle{ \mu }$.[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$\displaystyle{ \mu }$，即实数，由此产生的随机过程被称为漂移$\displaystyle{ \mu }$[150][151][152]

The mathematical space $\displaystyle{ S }$ of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, $\displaystyle{ n }$-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.

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]

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 $\displaystyle{ \{X(t,\omega):t\in T \} }$ is a stochastic process, then for any point $\displaystyle{ \omega\in\Omega }$, the mapping

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

[数学中心]

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

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]

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 $\displaystyle{ \{X(t):t\in T \} }$ is a stochastic process with state space $\displaystyle{ S }$ and index set $\displaystyle{ T=[0,\infty) }$, then for any two non-negative numbers $\displaystyle{ t_1\in [0,\infty) }$ and $\displaystyle{ t_2\in [0,\infty) }$ such that $\displaystyle{ t_1\leq t_2 }$, the difference $\displaystyle{ X_{t_2}-X_{t_1} }$ is a $\displaystyle{ S }$-valued random variable known as an increment.

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 $\displaystyle{ t }$, 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]

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

$\displaystyle{ \lt 中心 \gt \lt 数学 \gt Defined on the real line, the Poisson process can be interpreted as a stochastic process,\lt ref name="Applebaum2004page1337"/\gt \lt ref name="Rosenblatt1962page94"\gt {{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press|page=[https://archive.org/details/randomprocesses00rose_0/page/94 94]}}\lt /ref\gt among other random objects.\lt ref name="Haenggi2013page10and18"\gt {{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5|pages=10, 18}}\lt /ref\gt \lt ref name="ChiuStoyan2013page41and108"\gt {{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41, 108}}\lt /ref\gt But then it can be defined on the \lt math\gt n }$-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]

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

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

[/itex]

[数学中心]

## Definitions定义

so the law of a $\displaystyle{ X }$ can be written as:

### Stochastic process随机过程

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

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

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

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

[/itex]

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

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

For any measurable subset $\displaystyle{ C }$ of the $\displaystyle{ n }$-fold Cartesian power $\displaystyle{ S^n=S\times\dots \times S }$, the finite-dimensional distributions of a stochastic process $\displaystyle{ X }$ 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.

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 $\displaystyle{ S^T }$-valued random variable, where $\displaystyle{ S^T }$ is the space of all the possible $\displaystyle{ S }$-valued functions of $\displaystyle{ t\in T }$ that map from the set $\displaystyle{ T }$ into the space $\displaystyle{ S }$.[31][87]

When the index set $\displaystyle{ T }$ 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.

### Index set索引集

The set $\displaystyle{ T }$ 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 $\displaystyle{ T }$ the interpretation of time.[1] In addition to these sets, the index set $\displaystyle{ T }$ can be other linearly ordered sets or more general mathematical sets,[1][67] such as the Cartesian plane $\displaystyle{ R^2 }$ or $\displaystyle{ n }$-dimensional Euclidean space, where an element $\displaystyle{ t\in T }$ 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]

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 $\displaystyle{ \{\mathcal{F}_t\}_{t\in T} }$, on a probability space $\displaystyle{ (\Omega, \mathcal{F}, P) }$ is a family of sigma-algebras such that $\displaystyle{ \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} }$ for all $\displaystyle{ s \leq t }$, where $\displaystyle{ t, s\in T }$ and $\displaystyle{ \leq }$ denotes the total order of the index set $\displaystyle{ T }$.

### State space 状态空间

The mathematical space $\displaystyle{ S }$ of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, $\displaystyle{ n }$-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]

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

“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$\displaystyle{ \{X（t，omega）：t\in t\} }$是一个随机过程，那么对于任何点$\displaystyle{ \omega\in\omega }$，则 mapping

[/itex]

[数学中心]

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

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 $\displaystyle{ n }$-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$\displaystyle{ n }$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。

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

### Increment增量

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

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 $\displaystyle{ \{X(t):t\in T \} }$ is a stochastic process with state space $\displaystyle{ S }$ and index set $\displaystyle{ T=[0,\infty) }$, then for any two non-negative numbers $\displaystyle{ t_1\in [0,\infty) }$ and $\displaystyle{ t_2\in [0,\infty) }$ such that $\displaystyle{ t_1\leq t_2 }$, the difference $\displaystyle{ X_{t_2}-X_{t_1} }$ is a $\displaystyle{ S }$-valued random variable known as an increment.[55][56] When interested in the increments, often the state space $\displaystyle{ S }$ is the real line or the natural numbers, but it can be $\displaystyle{ n }$-dimensional Euclidean space or more abstract spaces such as Banach spaces.[56]

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

[数学中心]

holds.

#### Law定律

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

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

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

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

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 $\displaystyle{ n }$-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.

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

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

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

[/itex]

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

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

$\displaystyle{ \mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}). If two stochastic processes \lt math\gt X }$ and $\displaystyle{ Y }$ 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, $\displaystyle{ D[0,1] }$ denotes the space of càdlàg functions defined on the unit interval $\displaystyle{ [0,1] }$.

[/itex]

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]

#### 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.

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

$\displaystyle{ \mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1}, }$

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

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.

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

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.

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

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.

#### 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 $\displaystyle{ n }$-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.

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 $\displaystyle{ X }$ is a stationary stochastic process, then for any $\displaystyle{ t\in T }$ the random variable $\displaystyle{ X_t }$ has the same distribution, which means that for any set of $\displaystyle{ n }$ index set values $\displaystyle{ t_1,\dots, t_n }$, the corresponding $\displaystyle{ n }$ 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$\displaystyle{ X }$是一个平稳随机过程，那么对于任何$\displaystyle{ t\in T }$，随机变量$\displaystyle{ X_t }$具有相同的分布，这意味着对于任何一组$\displaystyle{ n }$索引集值$\displaystyle{ t_1,\dots, t_n }$而言，对应的$\displaystyle{ n }$随机变量

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

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]

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.

When the index set $\displaystyle{ T }$ 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]

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.

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 $\displaystyle{ X }$ is said to be stationary in the wide sense, then the process $\displaystyle{ X }$ has a finite second moment for all $\displaystyle{ t\in T }$ and the covariance of the two random variables $\displaystyle{ X_t }$ and $\displaystyle{ X_{t+h} }$ depends only on the number $\displaystyle{ h }$ for all $\displaystyle{ t\in T }$.[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]

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.

#### 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 $\displaystyle{ X }$ is a Lévy process if for $\displaystyle{ n }$ non-negatives numbers, $\displaystyle{ 0\leq t_1\leq \dots \leq t_n }$, the corresponding $\displaystyle{ n-1 }$ increments

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 $\displaystyle{ \{\mathcal{F}_t\}_{t\in T} }$, on a probability space $\displaystyle{ (\Omega, \mathcal{F}, P) }$ is a family of sigma-algebras such that $\displaystyle{ \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} }$ for all $\displaystyle{ s \leq t }$, where $\displaystyle{ t, s\in T }$ and $\displaystyle{ \leq }$ denotes the total order of the index set $\displaystyle{ T }$.[63] With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process $\displaystyle{ X_t }$ at $\displaystyle{ t\in T }$, which can be interpreted as time $\displaystyle{ t }$.[63][233] The intuition behind a filtration $\displaystyle{ \mathcal{F}_t }$ is that as time $\displaystyle{ t }$ passes, more and more information on $\displaystyle{ X_t }$ is known or available, which is captured in $\displaystyle{ \mathcal{F}_t }$, resulting in finer and finer partitions of $\displaystyle{ \Omega }$.[234][235]

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

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

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.

$\displaystyle{ P(X_t=Y_t)=1 , }$

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]

A point process is a collection of points randomly located on some mathematical space such as the real line, $\displaystyle{ n }$-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 $\displaystyle{ n }$-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.

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]

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.

#### Indistinguishable无法识别

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

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.

$\displaystyle{ P(X_t=Y_t \text{ for all } t\in T )=1 , }$

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.

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

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.

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

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.

#### 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.

Two stochastic processes $\displaystyle{ X }$ and $\displaystyle{ Y }$ defined on the same probability space $\displaystyle{ (\Omega,\mathcal{F},P) }$ with the same index set $\displaystyle{ T }$ are said be independent if for all $\displaystyle{ n \in \mathbb{N} }$ and for every choice of epochs $\displaystyle{ t_1,\ldots,t_n \in T }$, the random vectors $\displaystyle{ \left( X(t_1),\ldots,X(t_n) \right) }$ and $\displaystyle{ \left( Y(t_1),\ldots,Y(t_n) \right) }$ are independent.[266]: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 $\displaystyle{ \left\{X_t\right\} }$ and $\displaystyle{ \left\{Y_t\right\} }$ are called uncorrelated if their cross-covariance $\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right] }$ is zero for all times.[267]:p. 142 Formally:

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

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

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.

#### Independence implies uncorrelatedness独立意味着不相关

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

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

For random walks in $\displaystyle{ n }$-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.

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

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.

#### 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

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 $\displaystyle{ [0,1] }$ or $\displaystyle{ [0,\infty) }$, 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 $\displaystyle{ D }$,[268][269][270][271] so the function space is also referred to as space $\displaystyle{ D }$.[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, $\displaystyle{ D[0,1] }$ denotes the space of càdlàg functions defined on the unit interval $\displaystyle{ [0,1] }$.[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$\displaystyle{ [0,1] }$$\displaystyle{ [0，\infty） }$，取实线或度量空间上的值=图书https://books.com/？id=LkQOBwAAQBAJ&pg=PR5 | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21748-2 |页=78–79}</ref>[275][276]这些函数被称为cádLag或cadlag函数，这是基于法语表达式“continue a droite，limiteégauche”的首字母缩略词，因为这些函数是右连续的，具有左极限。[277]<refname=“Klebaner2005page4”>{cite book | author=Fima C.Klebaner | title=随机微积分及其应用简介|网址=图书https://books.com/？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=4}</ref>由Anatoliy Skorokod引入的Skorokod函数空间，[276]通常用字母$\displaystyle{ D }$表示，[277][278][276][279]因此函数空间也被称为空间$\displaystyle{ D }$[277][280][281]此函数空间的表示法还可以包括定义所有cádlág函数的间隔，因此，例如，$\displaystyle{ D[0,1] }$表示在单位间隔$\displaystyle{ [0上定义的c|dla g函数的空间，1] }$[279][281][282]

process.

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]

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.

#### 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.

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]

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.

## 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.

### 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 $\displaystyle{ n }$-dimensional Euclidean space.

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]

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]

{{columns-list|colwidth=30em|

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]

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]

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 $\displaystyle{ n }$-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.[316][317][318]

### 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]

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 $\displaystyle{ X_1, X_2, X_3, \dots }$ with zero mean, the stochastic process formed from the successive partial sums $\displaystyle{ X_1,X_1+ X_2, X_1+ X_2+X_3, \dots }$ is a discrete-time martingale.[330] In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.[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]

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]

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]

### 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 $\displaystyle{ X }$ is a Lévy process if for $\displaystyle{ n }$ non-negatives numbers, $\displaystyle{ 0\leq t_1\leq \dots \leq t_n }$, the corresponding $\displaystyle{ n-1 }$ increments

$\displaystyle{ X_{t_2}-X_{t_1}, \dots , X_{t_{n-1}}-X_{t_n}, }$

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]

-->

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]

### 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]

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]

### 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]

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]

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]

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]

[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]

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

Category:Statistical data types

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

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149. {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=33}
150. {cite book | author=J.Michael Steele | title=随机微积分和金融应用程序| url=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=118}
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153. Ioannis Karatzas; Steven Shreve (1991). Brownian Motion and Stochastic Calculus. Springer. p. 61. ISBN 978-1-4612-0949-2.
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155. Olav Kallenberg (2002). Foundations of Modern Probability. Springer Science & Business Media. pp. 225, 260. ISBN 978-0-387-95313-7.
156. Ioannis Karatzas; Steven Shreve (1991). Brownian Motion and Stochastic Calculus. Springer. p. 70. ISBN 978-1-4612-0949-2.
157. Peter Mörters; Yuval Peres (2010). Brownian Motion. Cambridge University Press. p. 131. ISBN 978-1-139-48657-6.
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159. {cite book |作者=Steven E.Shreve | title=金融随机微积分II：连续时间模型| url=https://books.google.com/books？id=O8kD1NwQBsQC | year=2004 | publisher=Springer Science&Business Media | isbn=978-0-387-40101-0 | page=93}
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162. Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS. 51 (11): 1341.
163. Samuel Karlin; Howard E. Taylor (2012). A First Course in Stochastic Processes. Academic Press. p. 340. ISBN 978-0-08-057041-9.
164. Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 124. ISBN 978-1-86094-555-7.
165. Ioannis Karatzas; Steven Shreve (1991). Brownian Motion and Stochastic Calculus. Springer. p. 47. ISBN 978-1-4612-0949-2.
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168. {引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2}
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170. Henk C. Tijms (2003). A First Course in Stochastic Models. Wiley. pp. 1, 2. ISBN 978-0-471-49881-0.
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176. D.J. Daley; D. Vere-Jones (2006). An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Springer Science & Business Media. p. 19. ISBN 978-0-387-21564-8.
177. J. F. C. Kingman (1992). Poisson Processes. Clarendon Press. p. 22. ISBN 978-0-19-159124-2.
178. Samuel Karlin; Howard E. Taylor (2012). A First Course in Stochastic Processes. Academic Press. pp. 118, 119. ISBN 978-0-08-057041-9.
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181. {cite book | author=Leonard Kleinrock | title=排队系统：理论|网址=https://archive.org/details/queueingsystems00klei%7Curl access=registration |年份=1976 | publisher=Wiley | isbn=978-0-471-49110-1 |页=[1]}}
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185. Roy L. Streit (2010). Poisson Point Processes: Imaging, Tracking, and Sensing. Springer Science & Business Media. p. 1. ISBN 978-1-4419-6923-1.
186. J. F. C. Kingman (1992). Poisson Processes. Clarendon Press. p. v. ISBN 978-0-19-159124-2.
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188. {cite book | author=Martin Haenggi | title=无线网络随机几何| url=https://books.google.com/books？id=CLtDhblwWEgC | year=2013 | publisher=Cambridge University Press | isbn=978-1-107-01469-5 | pages=10，18}
189. {cite book | author1=Sung Nok Chiu | author2=Dietrich Stoyan | author3=Wilfrid S.Kendall | author4=Joseph Mecke | title=随机几何及其应用| url=https://books.google.com/books？id=825NfM6Nc EC | year=2013 | publisher=John Wiley&Sons | isbn=978-1-118-65825-3 | pages=41108}
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199. Valeriy Skorokhod (2005). Basic Principles and Applications of Probability Theory. Springer Science & Business Media. pp. 93, 94. ISBN 978-3-540-26312-8.
200. Donald L. Snyder; Michael I. Miller (2012). Random Point Processes in Time and Space. Springer Science & Business Media. p. 25. ISBN 978-1-4612-3166-0.
201. Valeriy Skorokhod (2005). Basic Principles and Applications of Probability Theory. Springer Science & Business Media. p. 104. ISBN 978-3-540-26312-8.
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204. {cite book | author=Patrick Billingsley | title=Probability and Measure |网址=https://books.google.com/books？id=qyxqoxyeic | year=2008 | publisher=Wiley India Pvt.Limited | isbn=978-81-265-1771-8 | page=482}}
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208. Bernt Øksendal (2003). Stochastic Differential Equations: An Introduction with Applications. Springer Science & Business Media. p. 10. ISBN 978-3-540-04758-2.
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210. {cite book | author=BerntØksendal | title=随机微分方程：应用简介| url=https://books.google.com/books？id=VgQDWyihxKYC |年=2003 | publisher=Springer Science&Business Media | isbn=978-3-540-04758-2 | page=10}
211. Peter K. Friz; Nicolas B. Victoir (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge University Press. p. 571. ISBN 978-1-139-48721-4.
212. Sidney I. Resnick (2013). Adventures in Stochastic Processes. Springer Science & Business Media. pp. 40–41. ISBN 978-1-4612-0387-2.
213. Ward Whitt (2006). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Science & Business Media. p. 23. ISBN 978-0-387-21748-2.
214. David Applebaum (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press. p. 4. ISBN 978-0-521-83263-2.
215. Daniel Revuz; Marc Yor (2013). Continuous Martingales and Brownian Motion. Springer Science & Business Media. p. 10. ISBN 978-3-662-06400-9.
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220. L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. pp. 123. ISBN 978-1-107-71749-7.
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222. John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. pp. 6 and 7. ISBN 978-3-540-90275-1.
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228. {cite book | author=Joseph L.Doob | title=randours | url=图书https://books.com/？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=94–96}}
229. Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 298, 299. ISBN 978-1-118-59320-2.
230. Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. p. 8. ISBN 978-0-486-69387-3.
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232. {cite book | author1=Iosif Ilyich Gikhman | author2=Anatoly Vladimirovich skorokod | title=随机过程理论导论| url=图书https://books.com/？id=yJyLzG7N7r8C&pg=PR2 |年份=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=8}
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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.
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236. {引用图书|作者=David Williams | title=Probability with鞅| url=https://books.google.com/books？id=e9saZ0YSi AC | year=1991 | publisher=Cambridge University Press | isbn=978-0-521-40605-5 | pages=93，94}
237. {cite book | author1=Peter Mörters | author2=Yuval Peres | title=布朗运动| url=图书https://books.com/？id=e-TbA-dSrzYC | year=2010 | publisher=剑桥大学出版社| isbn=978-1-139-48657-6 | page=37}
238. L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. p. 130. ISBN 978-1-107-71749-7.
239. Alexander A. Borovkov (2013). Probability Theory. Springer Science & Business Media. p. 530. ISBN 978-1-4471-5201-9.
240. Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 48. ISBN 978-1-86094-555-7.
241. Bernt Øksendal (2003). Stochastic Differential Equations: An Introduction with Applications. Springer Science & Business Media. p. 14. ISBN 978-3-540-04758-2.
242. Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 472. ISBN 978-1-118-59320-2.
243. Daniel Revuz; Marc Yor (2013). Continuous Martingales and Brownian Motion. Springer Science & Business Media. pp. 18–19. ISBN 978-3-662-06400-9.
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248. David Applebaum (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press. p. 20. ISBN 978-0-521-83263-2.
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250. Olav Kallenberg (2002). Foundations of Modern Probability. Springer Science & Business Media. p. 35. ISBN 978-0-387-95313-7.
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252. Monique Jeanblanc; Marc Yor; Marc Chesney (2009). Mathematical Methods for Financial Markets. Springer Science & Business Media. p. 11. ISBN 978-1-85233-376-8.
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254. {cite book | author1=Monique Jeanblanc | author2=Marc Yor | author2 link=Marc Yor | author3=Marc Chesney | title=金融市场数学方法| url=图书https://books.com/？id=ZhbROxoQ ZMC |年=2009 | publisher=Springer Science&Business Media | isbn=978-1-85233-376-8 | page=11}
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258. Petar Todorovic (2012). An Introduction to Stochastic Processes and Their Applications. Springer Science & Business Media. pp. 19–20. ISBN 978-1-4613-9742-7.
259. Ilya Molchanov (2005). Theory of Random Sets. Springer Science & Business Media. p. 340. ISBN 978-1-85233-892-3.
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262. Davar Khoshnevisan (2006). Multiparameter Processes: An Introduction to Random Fields. Springer Science & Business Media. p. 155. ISBN 978-0-387-21631-7.
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266. 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
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271. Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 4. ISBN 978-1-86094-555-7.
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273. Patrick Billingsley (2013). Convergence of Probability Measures. John Wiley & Sons. p. 121. ISBN 978-1-118-62596-5.
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278. 引用错误：无效<ref>标签；未给name属性为“GusakKukush2010page24”的引用提供文字
279. 引用错误：无效<ref>标签；未给name属性为“Klebaner2005page4”的引用提供文字
280. {cite book | author=S|ren Asmussen | title=应用概率和队列| url=图书https://books.com/？id=BeYaTxesKy0C | year=2003 | publisher=Springer Science&Business Media | isbn=978-0-387-00211-8 | page=420}
281. {cite book |作者=Patrick Billingsley | title=Convergence of Probability Measures|网址=图书https://books.com/？id=6ItqtwaWZZQC | year=2013 | publisher=John Wiley&Sons | isbn=978-1-118-62596-5 | page=121}
282. {cite book | author=Richard F.Bass | title=random Processes |网址=图书https://books.com/？id=Ll0T7PIkcKMC |年=2011 | publisher=Cambridge University Press | isbn=978-1-139-50147-7 | page=34}
283. Nicholas H. Bingham; Rüdiger Kiesel (2013). Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. Springer Science & Business Media. p. 154. ISBN 978-1-4471-3856-3.
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}
285. Alexander A. Borovkov (2013). Probability Theory. Springer Science & Business Media. p. 532. ISBN 978-1-4471-5201-9.
286. Davar Khoshnevisan (2006). Multiparameter Processes: An Introduction to Random Fields. Springer Science & Business Media. pp. 148–165. ISBN 978-0-387-21631-7.
287. Petar Todorovic (2012). An Introduction to Stochastic Processes and Their Applications. Springer Science & Business Media. p. 22. ISBN 978-1-4613-9742-7.
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.
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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.
293. Y.A. Rozanov (2012). Markov Random Fields. Springer Science & Business Media. p. 58. ISBN 978-1-4613-8190-7.
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296. Sheldon M. Ross (1996). Stochastic processes. Wiley. pp. 235, 358. ISBN 978-0-471-12062-9.  引用错误：无效<ref>标签；name属性“Ross1996page235and358”使用不同内容定义了多次
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298. Samuel Karlin; Howard E. Taylor (2012). A First Course in Stochastic Processes. Academic Press. p. 49. ISBN 978-0-08-057041-9.
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300. Søren Asmussen (2003). Applied Probability and Queues. Springer Science & Business Media. p. 7. ISBN 978-0-387-00211-8.
301. Emanuel Parzen (2015). Stochastic Processes. Courier Dover Publications. p. 188. ISBN 978-0-486-79688-8.
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303. John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. pp. 106–121. ISBN 978-3-540-90275-1.
304. Sheldon M. Ross (1996). Stochastic processes. Wiley. pp. 174, 231. ISBN 978-0-471-12062-9.
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311. Samuel Karlin; Howard E. Taylor (2012). A First Course in Stochastic Processes. Academic Press. p. 47. ISBN 978-0-08-057041-9.
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313. Dani Gamerman; Hedibert F. Lopes (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition. CRC Press. ISBN 978-1-58488-587-0.
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315. {引用图书| author1=Dani Gamerman | author2=Hedibert F.Lopes | title=Markov Chain montecarlo:贝叶斯推断随机模拟，第二版|网址=https://books.google.com/books？id=yPvECi%7CL3bwC | year=2006 | publisher=CRC Press | isbn=978-1-58488-587-0}
316. Y.A. Rozanov (2012). Markov Random Fields. Springer Science & Business Media. p. 61. ISBN 978-1-4613-8190-7.
317. Donald L. Snyder; Michael I. Miller (2012). Random Point Processes in Time and Space. Springer Science & Business Media. p. 27. ISBN 978-1-4612-3166-0.
318. Pierre Bremaud (2013). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer Science & Business Media. p. 253. ISBN 978-1-4757-3124-8.
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320. {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=27}
321. {cite book |作者=Pierre Bremaud | title=Markov Chains:Gibbs Fields，montecarlo Simulation，and Queues |网址=https://books.google.com/books？id=jrpvwwaaqbaj |年份=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4757-3124-8 | page=253}
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330. J. Michael Steele (2012). Stochastic Calculus and Financial Applications. Springer Science & Business Media. pp. 12, 13. ISBN 978-1-4684-9305-4.
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334. 引用错误：无效<ref>标签；未给name属性为“Steele2012page12”的引用提供文字
335. {cite book | author=J.Michael Steele | title=随机微积分与金融应用=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=115}
336. Sheldon M. Ross (1996). Stochastic processes. Wiley. p. 295. ISBN 978-0-471-12062-9.
337. J. Michael Steele (2012). Stochastic Calculus and Financial Applications. Springer Science & Business Media. p. 11. ISBN 978-1-4684-9305-4.
338. Olav Kallenberg (2002). Foundations of Modern Probability. Springer Science & Business Media. pp. 96. ISBN 978-0-387-95313-7.
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341. Geoffrey Grimmett; David Stirzaker (2001). Probability and Random Processes. OUP Oxford. p. 336. ISBN 978-0-19-857222-0.
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343. {cite book | author=J.Michael Steele | title=随机微积分和金融应用|网址=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=11}
344. {cite book | author=J.Michael Steele| title=随机微积分和金融应用程序| url=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=371}}
345. Glasserman, Paul; Kou, Steven (2006). "A Conversation with Chris Heyde". Statistical Science. 21 (2): 292, 293. arXiv:math/0609294. Bibcode:2006math......9294G. doi:10.1214/088342306000000088. ISSN 0883-4237.
346. Francois Baccelli; Pierre Bremaud (2013). Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences. Springer Science & Business Media. ISBN 978-3-662-11657-9.
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348. {cite book | author1=Francois Baccelli | author2=Pierre Bremaud | title=排队论的元素：Palm鞅演算和随机递归| url=https://books.google.com/books？id=dh3pcaaqbaj&pg=PR2 | year=2013 | publisher=Springer科学与商业媒体| isbn=978-3-662-11657-9}
349. {cite book | author1=P.Hall | author2=C.C.Heyde | title=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=x}
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351. Jean Bertoin (1998). Lévy Processes. Cambridge University Press. p. viii. ISBN 978-0-521-64632-1.
352. Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS. 51 (11): 1336.
353. David Applebaum (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press. p. 69. ISBN 978-0-521-83263-2.
354. {引用图书|作者=Jean Bertoin | title=莱维过程 |网址=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=Cambridge University Press | isbn=978-0-521-64632-1 | page=viii}}
355. 引用错误：无效<ref>标签；未给name属性为:1的引用提供文字
356. L. E. Maistrov (2014). Probability Theory: A Historical Sketch. Elsevier Science. p. 56. ISBN 978-1-4832-1863-2.
357. John Tabak (2014). Probability and Statistics: The Science of Uncertainty. Infobase Publishing. p. 37. ISBN 978-0-8160-6873-9.
358. Chung, Kai Lai (1998). "Probability and Doob". The American Mathematical Monthly. 105 (1): 28–35. doi:10.2307/2589523. ISSN 0002-9890. JSTOR 2589523.
359. Bingham, N. (2000). "Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov". Biometrika. 87 (1): 145–156. doi:10.1093/biomet/87.1.145. ISSN 0006-3444.
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361. Doob, Joseph L. (1996). "The Development of Rigor in Mathematical Probability (1900-1950)". The American Mathematical Monthly. 103 (7): 586–595. doi:10.2307/2974673. ISSN 0002-9890. JSTOR 2974673.
362. Cramer, Harald (1976). "Half a Century with Probability Theory: Some Personal Recollections". The Annals of Probability. 4 (4): 509–546. doi:10.1214/aop/1176996025. ISSN 0091-1798.
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364. {引用书| author=L.E.Maistrov | title=概率论：历史素描|网址=https://books.google.com/books？id=2ZbiBQAAQBAJ&pg=PR9 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-1863-2 | page=56}
365. {cite book | author=John Tabak | title=概率与统计学：不确定性科学|网址=https://books.google.com/books？id=h3WVqBPHboAC |年=2014 | publisher=Infobase Publishing | isbn=978-0-8160-6873-9 | page=37}
366. {引用期刊| last1=Chung | first1=Kai Lai | title=Probability and Doob | journal=The American Mathematic Monthly | volume=105 | isson=1 | pages=28-35 | year=1998 | issn=0002-9890 | doi=10.2307/2589523 | jstor=2589523}
367. {cite journal | last1=Bingham | first1=N.| title=概率统计史研究XLVI。概率度量：从Lebesgue到Kolmogorov | journal=Biometrika | volume=87 | Isse=1 | year=2000 | pages=145-156 | issn=0006-3444 | doi=10.1093/biomet/87.1.145}
368. 模板:引用期刊
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371. Doob, Joseph L.；title=数学概率中严谨性的发展（1900-1950年）. 美国数学月刊. doi:10.2307/2974673. ISSN 0002-9890. JSTOR 2974673. Unknown parameter |卷= ignored (help); Unknown parameter |页= ignored (help); Unknown parameter |问题= ignored (help); Unknown parameter |年= ignored (help); Missing or empty |title= (help)
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373. Truesdell, C. (1975). "Early kinetic theories of gases". Archive for History of Exact Sciences. 15 (1): 22–23. doi:10.1007/BF00327232. ISSN 0003-9519.
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376. Brush, S.G. (1958). "The development of the kinetic theory of gases IV. Maxwell". Annals of Science. 14 (4): 243–255. doi:10.1080/00033795800200147. ISSN 0003-3790.
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378. Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W.; Razborov, A. A.; Robinson, C. A.; Whittle, P. (1990). "Andrei Nikolaevich Kolmogorov (1903–1987)". Bulletin of the London Mathematical Society. 22 (1): 33. doi:10.1112/blms/22.1.31. ISSN 0024-6093.
379. Vere-Jones, David (2006). "Khinchin, Aleksandr Yakovlevich". Encyclopedia of Statistical Sciences. p. 1. doi:10.1002/0471667196.ess6027.pub2. ISBN 978-0471667193.
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