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第166行： − 随机游动的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，其值随概率增加1，例如，<math>p</math>，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<ref name=“Gut2012page88”>{cite book | author=Allan Gut | title=Probability:a Graduate Course=https://books.google.com/books？id=XDFA-n|M5hMC | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4614-4708-5 | page=88}</ref><ref name=“grimmetttstirzaker2001page71”>{引用图书| author1=Geoffrey Grimmett | author2=David Stirzaker | title=概率和随机过程| url=https://books.google.com/books？id=G3ig-0M4wSIC |年份=2001 | publisher=OUP Oxford | isbn=978-0-19-857222-0 | page=71}</ref>+ 第198行：
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→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.
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.
'''<font color="#ff8000"> 维纳过程Wiener process</font>'''是马尔可夫过程、 Lévy 过程和 Gaussian 过程等重要随机过程的一个成员。它在定量金融学中扮演着核心角色，例如，在'''<font color="#ff8000"> 布莱克-斯科尔斯-默顿模型Black–Scholes–Merton model</font>'''中就使用了它。这个过程也用于不同的领域，包括大多数自然科学和一些社会科学分支，作为各种随机现象的数学模型。
'''<font color="#ff8000"> 维纳过程Wiener process</font>'''是马尔可夫过程、 列维Lévy 过程和 高斯Gaussian 过程等重要随机过程的一个成员。它在定量金融学中扮演着核心角色，例如，在'''<font color="#ff8000"> 布莱克-斯科尔斯-默顿模型Black–Scholes–Merton model</font>'''中就使用了它。这个过程也用于不同的领域，包括大多数自然科学和一些社会科学分支，作为各种随机现象的数学模型。
===Random walk随机游动===
==='''<font color="#ff8000"> 随机游走Random walk</font>'''===
Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.
Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.
{{Main|Random walk}}
{{Main|Random walk}}
{{Main |随机漫步}}
{{Main |随机游走}}
[[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.<ref name="Klenke2013page347">{{cite book|author=Achim Klenke|title=Probability Theory: A Comprehensive Course|url=https://books.google.com/books?id=aqURswEACAAJ|year=2013|publisher=Springer|isbn=978-1-4471-5362-7|pages=347}}</ref><ref name="LawlerLimic2010page1">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1|page=1}}</ref><ref name="Kallenberg2002page136">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date= 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=136}}</ref><ref name="Florescu2014page383">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=383}}</ref><ref name="Durrett2010page277">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=277}}</ref> But some also use the term to refer to processes that change in continuous time,<ref name="Weiss2006page1">{{cite book|last1=Weiss|first1=George H.|title=Encyclopedia of Statistical Sciences|chapter=Random Walks|year=2006|doi=10.1002/0471667196.ess2180.pub2|page=1|isbn=978-0471667193}}</ref> particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism.<ref name="Spanos1999page454">{{cite book|author=Aris Spanos|title=Probability Theory and Statistical Inference: Econometric Modeling with Observational Data|url=https://books.google.com/books?id=G0_HxBubGAwC|year=1999|publisher=Cambridge University Press|isbn=978-0-521-42408-0|page=454}}</ref> 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.<ref name="Weiss2006page1"/><ref name="Klebaner2005page81">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=81}}</ref>
[[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.<ref name="Klenke2013page347">{{cite book|author=Achim Klenke|title=Probability Theory: A Comprehensive Course|url=https://books.google.com/books?id=aqURswEACAAJ|year=2013|publisher=Springer|isbn=978-1-4471-5362-7|pages=347}}</ref><ref name="LawlerLimic2010page1">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1|page=1}}</ref><ref name="Kallenberg2002page136">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date= 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=136}}</ref><ref name="Florescu2014page383">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=383}}</ref><ref name="Durrett2010page277">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=277}}</ref> But some also use the term to refer to processes that change in continuous time,<ref name="Weiss2006page1">{{cite book|last1=Weiss|first1=George H.|title=Encyclopedia of Statistical Sciences|chapter=Random Walks|year=2006|doi=10.1002/0471667196.ess2180.pub2|page=1|isbn=978-0471667193}}</ref> particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism.<ref name="Spanos1999page454">{{cite book|author=Aris Spanos|title=Probability Theory and Statistical Inference: Econometric Modeling with Observational Data|url=https://books.google.com/books?id=G0_HxBubGAwC|year=1999|publisher=Cambridge University Press|isbn=978-0-521-42408-0|page=454}}</ref> 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.<ref name="Weiss2006page1"/><ref name="Klebaner2005page81">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=81}}</ref>
A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If the <math>p=0.5</math>, this random walk is called a symmetric random walk.<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>
A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If the <math>p=0.5</math>, this random walk is called a symmetric random walk.<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>
'''<font color="#ff8000"> 随机游走Random walk</font>'''的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，例如其值随概率<math>p</math>增加1，，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<ref name=“Gut2012page88”>{cite book | author=Allan Gut | title=Probability:a Graduate Course=https://books.google.com/books？id=XDFA-n|M5hMC | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4614-4708-5 | page=88}</ref><ref name=“grimmetttstirzaker2001page71”>{引用图书| author1=Geoffrey Grimmett | author2=David Stirzaker | title=概率和随机过程| url=https://books.google.com/books？id=G3ig-0M4wSIC |年份=2001 | publisher=OUP Oxford | isbn=978-0-19-857222-0 | page=71}</ref>
A stochastic process is defined as a collection of random variables defined on a common probability space <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a sample space, <math>\mathcal{F}</math> is a <math>\sigma</math>-algebra, and <math>P</math> is a probability measure; and the random variables, indexed by some set <math>T</math>, all take values in the same mathematical space <math>S</math>, which must be measurable with respect to some <math>\sigma</math>-algebra <math>\Sigma</math>.
A stochastic process is defined as a collection of random variables defined on a common probability space <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a sample space, <math>\mathcal{F}</math> is a <math>\sigma</math>-algebra, and <math>P</math> is a probability measure; and the random variables, indexed by some set <math>T</math>, all take values in the same mathematical space <math>S</math>, which must be measurable with respect to some <math>\sigma</math>-algebra <math>\Sigma</math>.
Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>. A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.
Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>. A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.
从历史上看，在自然科学的许多问题中，t 中的一个点“ math”代表时间，所以“ math” x (t)是一个随机变量，代表时间观察到的值。一个随机过程也可以写成{ math > { x (t，Omega) : t in t } </math > 来反映它实际上是一个双变量的函数，t </math > 中的 < math > 和 ω </math > 中的 < math > Omega。
从历史上看，在自然科学的许多问题中，t 中的一个点<math>t\in T</math>代表时间，所以<math>X(t)</math>是一个随机变量，代表时间<math>t</math>观察到的值。一个随机过程也可以写成<math> \{X(t,\omega):t\in T \}</math> 来反映它实际上是一个双变量的函数，<math>t\in T</math> 且<math>\omega\in \Omega</math>。
[[File:DriftedWienerProcess1D.svg|thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).]]
[[File:DriftedWienerProcess1D.svg|thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).]]
[[文件：floadWienerProcess1d.svg|拇指|左|实现Wiener过程（或布朗运动过程），具有漂移（{color |蓝色}且不漂移（{color |红色}红色}）。]]
[[文件：floadWienerProcess1d.svg|拇指|左|实现维纳Wiener过程（或布朗运动过程），具有漂移（{color |蓝色}且不漂移（{color |红色}红色}）。]]
There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued functions of <math>t\in T</math> that map from the set <math>T</math> into the space <math>S</math>. of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set <math>T</math> the interpretation of time. such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.
There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued functions of <math>t\in T</math> that map from the set <math>T</math> into the space <math>S</math>. of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set <math>T</math> the interpretation of time. such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.
还有其他的方法来考虑随机过程，上面的定义被认为是传统的定义。例如，随机过程可以被解释或定义为一个 < math > s ^ t </math > 值随机变量，其中 < math > s ^ t </math > 是 t </math > 中所有可能的 < math > s </math > 值函数的空间，这些函数从集合 < math > t </math > 映射到空间 < math > s </math > 。随机过程。这个集合通常是实数线的一些子集，比如自然数或者区间，赋予集合 < math > t </math > 时间的解释。比如笛卡尔平面 < math > r ^ 2 </math > 或 < math > n </math > 维欧氏空间，其中 t </math > 中的一个元素 < t 可以表示空间中的一个点。但是一般来说，当指标集是有序的时候，对于随机过程可能有更多的结果和定理。
还有其他的方法来考虑随机过程，上面的定义被认为是传统的定义。例如，随机过程可以被解释或定义为一个 <math>S^T</math> 值随机变量，其中 <math>S^T</math> 是 <math>t\in T</math> 中所有可能的 <math>S</math>值函数的空间，这些函数从集合 <math>T</math> 映射到空间 <math>S</math> 。随机过程。这个集合通常是实数线的一些子集，比如使集合 <math>T</math> 时间有意义的自然数集或者区间。比如笛卡尔平面 <math>R^2</math> 或 <math>n</math> 维欧氏空间，其中的一个元素 <math>t\in T</math>可以表示空间中的一个点。但是一般来说，当指标集是有序的时候，对于随机过程可能有更多的结果和定理。
The mathematical space <math>S</math> of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, <math>n</math>-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.
The mathematical space <math>S</math> of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, <math>n</math>-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.
随机过程的数学空间 s 称为状态空间。这个数学空间可以使用整数、实数线、维欧氏空间、复平面或更抽象的数学空间来定义。状态空间使用元素定义，这些元素反映了随机过程可以采用的不同值。
随机过程的数学空间 <math>S</math>称为状态空间。这个数学空间可以使用整数、实数线、维欧氏空间、复平面或更抽象的数学空间来定义。状态空间使用元素定义，这些元素反映了随机过程可以采用的不同值。
[[Almost surely]], a sample path of a Wiener process is continuous everywhere but [[nowhere differentiable function|nowhere differentiable]]. It can be considered as a continuous version of the simple random walk.<ref name="Applebaum2004page1337">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|page=1337}}</ref><ref name="MörtersPeres2010page1">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|pages=1, 3}}</ref> The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,<ref name="KaratzasShreve2014page61">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=61}}</ref><ref name="Shreve2004page93">{{cite book|author=Steven E. Shreve|title=Stochastic Calculus for Finance II: Continuous-Time Models|url=https://books.google.com/books?id=O8kD1NwQBsQC|year=2004|publisher=Springer Science & Business Media|isbn=978-0-387-40101-0|page=93}}</ref> which is the subject of [[Donsker's theorem]] or invariance principle, also known as the functional central limit theorem.<ref name="Kallenberg2002page225and260">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=225, 260}}</ref><ref name="KaratzasShreve2014page70">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=70}}</ref><ref name="MörtersPeres2010page131">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=131}}</ref>
[[Almost surely]], a sample path of a Wiener process is continuous everywhere but [[nowhere differentiable function|nowhere differentiable]]. It can be considered as a continuous version of the simple random walk.<ref name="Applebaum2004page1337">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|page=1337}}</ref><ref name="MörtersPeres2010page1">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|pages=1, 3}}</ref> The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,<ref name="KaratzasShreve2014page61">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=61}}</ref><ref name="Shreve2004page93">{{cite book|author=Steven E. Shreve|title=Stochastic Calculus for Finance II: Continuous-Time Models|url=https://books.google.com/books?id=O8kD1NwQBsQC|year=2004|publisher=Springer Science & Business Media|isbn=978-0-387-40101-0|page=93}}</ref> which is the subject of [[Donsker's theorem]] or invariance principle, also known as the functional central limit theorem.<ref name="Kallenberg2002page225and260">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=225, 260}}</ref><ref name="KaratzasShreve2014page70">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=70}}</ref><ref name="MörtersPeres2010page131">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=131}}</ref>