更改

[[File:BMonSphere.jpg|thumb|球体表面维纳或布朗运动过程的计算机模拟实现。维纳过程被广泛认为是概率论中研究最多和中心的随机过程。<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>]]

[[File:BMonSphere.jpg|thumb|球体表面维纳或布朗运动过程的计算机模拟实现。维纳过程被广泛认为是概率论中研究最多和中心的随机过程。<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>]]

 

'''随机函数 Random function'''这个术语也用来指随机或随机过程，<ref name="GusakKukush2010page21">{{cite book|first1=Dmytro|last1=Gusak|first2=Alexander|last2=Kukush|first3=Alexey|last3=Kulik|first4=Yuliya|last4=Mishura|first5=Andrey|last5=Pilipenko|title=Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory|url=https://books.google.com/books?id=8Nzn51YTbX4C|year=2010|publisher=Springer Science & Business Media|isbn=978-0-387-87862-1|page=21|ref=harv}}</ref><ref name="Skorokhod2005page42">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year= 2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=42}}</ref> 因为随机过程也可以被解释为函数空间中的随机元素。<ref name="Kallenberg2002page24" /><ref name="Lamperti1977page1">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=1–2}}</ref>stochastic和random process可以互换使用，通常没有专门的数学空间用于索引随机变量。<ref name="Kallenberg2002page24">{{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=24–25}}</ref><ref name="ChaumontYor2012">{{cite book|author1=Loïc Chaumont|author2=Marc Yor|title=Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning|url=https://books.google.com/books?id=1dcqV9mtQloC&pg=PR4|year= 2012|publisher=Cambridge University Press|isbn=978-1-107-60655-5|page=175}}</ref>但是，当随机变量被整数或实数的一个区间索引时，通常使用这两个术语。<ref name="GikhmanSkorokhod1969page1" /><ref name="ChaumontYor2012" />如果随机变量被笛卡尔平面或某些高维欧几里得空间索引，那么随机变量的集合通常被称为'''随机场 random field'''。<ref name="GikhmanSkorokhod1969page1" /><ref name="AdlerTaylor2009page7">{{cite book|author1=Robert J. Adler|author2=Jonathan E. Taylor|title=Random Fields and Geometry|url=https://books.google.com/books?id=R5BGvQ3ejloC|year=2009|publisher=Springer Science & Business Media|isbn=978-0-387-48116-6|pages=7–8}}</ref>随机过程的值并不总是数字，可以是向量或其他数学对象。<ref name="GikhmanSkorokhod1969page1" /><ref name="Lamperti1977page1" />

根据随机过程的数学性质，随机过程可以分为不同的类别，包括随机游走，<ref name="LawlerLimic2010">{{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}}</ref> 鞅（概率论）,<ref name="Williams1991">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5}}</ref> 马尔可夫过程,<ref name="RogersWilliams2000">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year= 2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7}}</ref> Lévy过程,<ref name="ApplebaumBook2004">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2}}</ref> 高斯过程,<ref>{{cite book|author=Mikhail Lifshits|title=Lectures on Gaussian Processes|url=https://books.google.com/books?id=03m2UxI-UYMC|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-24939-6}}</ref> 随机场，<ref name="Adler2010">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA1|year= 2010|publisher=SIAM|isbn=978-0-89871-693-1}}</ref> 更新过程, 和分支过程.<ref name="KarlinTaylor2012">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year= 2012|publisher=Academic Press|isbn=978-0-08-057041-9}}</ref>。随机过程的研究使用了概率、微积分、线性代数、集合论的数学知识和技术，和[[拓扑学]]<ref name="Hajek2015">{{cite book|author=Bruce Hajek|title=Random Processes for Engineers|url=https://books.google.com/books?id=Owy0BgAAQBAJ|year=2015|publisher=Cambridge University Press|isbn=978-1-316-24124-0}}</ref><ref name="LatoucheRamaswami1999">{{cite book|author1=G. Latouche|author2=V. Ramaswami|title=Introduction to Matrix Analytic Methods in Stochastic Modeling|url=https://books.google.com/books?id=Kan2ki8jqzgC|year=1999|publisher=SIAM|isbn=978-0-89871-425-8}}</ref><ref name="DaleyVere-Jones2007">{{cite book|author1=D.J. Daley|author2=David Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure|url=https://books.google.com/books?id=nPENXKw5kwcC|year= 2007|publisher=Springer Science & Business Media|isbn=978-0-387-21337-8}}</ref>以及数学分析的分支，如实分析，测量理论，傅立叶分析，和泛函分析。随机过程理论被认为是对数学的重要贡献<ref name="Applebaum2004">{{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|pages=1336–1347}}</ref>，不论由于理论还是应用，它都是一个活跃的研究课题。<ref name="BlathImkeller2011">{{cite book|author1=Jochen Blath|author2=Peter Imkeller|author3=Sylvie Rœlly|title=Surveys in Stochastic Processes|url=https://books.google.com/books?id=CyK6KAjwdYkC|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-072-2}}</ref><ref name="Talagrand2014">{{cite book|author=Michel Talagrand|title=Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems|url=https://books.google.com/books?id=tfa5BAAAQBAJ&pg=PR4|year=2014|publisher=Springer Science & Business Media|isbn=978-3-642-54075-2|pages=4–}}</ref><ref name="Bressloff2014VII">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-08488-6|pages=vii–ix}}</ref>

 

 

==简介==

==简介==

随机过程可以定义为随机变量的集合，这些随机变量由一些数学集合构成索引，这意味着随机过程中的每个随机变量都与集合中的一个元素唯一关联。<ref name="Parzen1999"/><ref name="GikhmanSkorokhod1969page1"/>用于索引随机变量的集合称为“索引集”。从历史上看，索引集是实线的一些子集，例如自然数，为索引集提供了对时间的解释。<ref name="doob1953stochasticP46to47"/> 集合中的每个随机变量都从相同的数学空间中获取值，称为“状态空间 state space”。例如，这个状态空间可以是整数、实线或维欧几里德空间。<ref name="doob1953stochasticP46to47"/> '''增量 increment'''是随机过程在两个索引值之间变化的量，通常被解释为两个时间点。<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/>由于随机性，随机过程可以有许多结果，随机过程的单个结果称为其他名称中的一个，“示例函数”或“实现”。<ref name="Lamperti1977page1"/><ref name="RogersWilliams2000page121b"/>

随机过程可以被定义为随机变量的集合，这些随机变量由一些数学集合构成索引，这意味着随机过程中的每个随机变量都与集合中的一个元素唯一关联。<ref name="Parzen1999"/><ref name="GikhmanSkorokhod1969page1"/>用于索引随机变量的集合称为“索引集”。从历史上看，索引集是实数的一些子集，例如自然数，为索引集提供了对时间的解释。<ref name="doob1953stochasticP46to47"/> 集合中的每个随机变量都取值于相同的数学空间中，称为“状态空间（state space）”。例如，这个状态空间可以是整数、实数或维欧几里德空间。<ref name="doob1953stochasticP46to47"/> '''增量'''是随机过程在两个索引值之间变化的量，通常被解释为两个时间点。<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/>由于随机性，随机过程可以有许多结果，随机过程的单个结果称为其他名称中的一个，“抽样函数”或“实现”。<ref name="Lamperti1977page1"/><ref name="RogersWilliams2000page121b"/>

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

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

[[File:Wiener_process_3d.png|thumb|right|单个计算机模拟时间0≤t≤2的三维维纳或布朗运动过程的“样本函数”或“实现”。这个随机过程的指标集是非负数，而其状态空间是三维欧几里德空间]]

[[File:Wiener_process_3d.png|thumb|right|单个计算机模拟时间0≤t≤2的三维维纳或布朗运动过程的“样本函数”或“实现”。这个随机过程的指标集是非负数，而其状态空间是三维欧几里德空间]]

===分类===

===分类===

术语“随机函数”也用于指随机或随机过程，<ref name="GikhmanSkorokhod1969page1"/><ref name="Loeve1978">{{cite book|author=M. Loève|title=Probability Theory II|url=https://books.google.com/books?id=1y229yBbULIC|year=1978|publisher=Springer Science & Business Media|isbn=978-0-387-90262-3|page=163}}</ref><ref name="Brémaud2014page133">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=133}}</ref>尽管有时它只在随机过程取实值时使用。<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/>当索引集是数学空间而不是实线时，也使用这个术语，<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"> p. 1</ref>，而术语“随机过程”和“随机过程”通常在指数集被解释为时间时使用，<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"/><ref name="Bass2011page1">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=1}}</ref>和其他术语，例如当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或流形。<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>

术语“随机函数”也用于指随机或随机过程，<ref name="GikhmanSkorokhod1969page1"/><ref name="Loeve1978">{{cite book|author=M. Loève|title=Probability Theory II|url=https://books.google.com/books?id=1y229yBbULIC|year=1978|publisher=Springer Science & Business Media|isbn=978-0-387-90262-3|page=163}}</ref><ref name="Brémaud2014page133">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=133}}</ref>尽管有时它只在随机过程取实值时使用。<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/>当索引集是数学空间而不是实数时，也使用这个术语，<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"> p. 1</ref>，而术语“随机过程”和“随机过程”通常在指数集被解释为时间时使用，<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"/><ref name="Bass2011page1">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=1}}</ref>和其他术语，例如当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或流形。<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>

<br>

<br>

==示例==

==示例==

===伯努利过程 Bernoulli process===

===伯努利过程 Bernoulli process===

最简单的随机过程之一是伯努利过程，<ref name="Florescu2014page293"/>它是独立且相同分布随机变量的序列，其中每个随机变量取1或0，比如概率<math>p</math>的值为1，概率<math>1-p</math>为零。这个过程可以与反复翻动硬币有关，其中获得头部的概率为<math>p</math>，其值为1，而尾部的值为零。<ref name= "Florescu2014page301">{{cite book| first= Ionut |last= 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=301}}</ref>换句话说，伯努利过程是一系列独立且同分布的伯努利随机变量，<ref name="BertsekasTsitsiklis2002page273">{{cite book|author1=Dimitri P. Bertsekas|author2=John N. Tsitsiklis|title=Introduction to Probability|url=https://books.google.com/books?id=bcHaAAAAMAAJ|year=2002|publisher=Athena Scientific|isbn=978-1-886529-40-3|page=273}}</ref>每一次抛硬币都是[[伯努利试验]]的一个例子。<ref name="Ibe2013page11">{{cite book|author=Oliver C. Ibe|title=Elements of Random Walk and Diffusion Processes|url=https://books.google.com/books?id=DUqaAAAAQBAJ&pg=PT10|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-61793-9|page=11}}</ref>

最简单的随机过程之一是伯努利过程，<ref name="Florescu2014page293"/>它是独立且相同分布随机变量的序列，其中每个随机变量取1或0，比如概率<math>p</math>的值为1，概率<math>1-p</math>为零。这个过程可以与反复翻动硬币有关，其中获得头部的概率为<math>p</math>，其值为1，而尾部的值为零。<ref name="Florescu2014page301">{{cite book| first= Ionut |last= 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=301}}</ref>换句话说，伯努利过程是一系列独立且同分布的伯努利随机变量，<ref name="BertsekasTsitsiklis2002page273">{{cite book|author1=Dimitri P. Bertsekas|author2=John N. Tsitsiklis|title=Introduction to Probability|url=https://books.google.com/books?id=bcHaAAAAMAAJ|year=2002|publisher=Athena Scientific|isbn=978-1-886529-40-3|page=273}}</ref>每一次抛硬币都是[[伯努利试验]]的一个例子。<ref name="Ibe2013page11">{{cite book|author=Oliver C. Ibe|title=Elements of Random Walk and Diffusion Processes|url=https://books.google.com/books?id=DUqaAAAAQBAJ&pg=PT10|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-61793-9|page=11}}</ref>

<br>

<br>

齐次泊松过程可以用不同的方法定义和推广。它的指标集可以定义为实线，这个随机过程也被称为平稳泊松过程<ref name="Kingman1992page38">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=38}}</ref><ref name="DaleyVere-Jones2006page19">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=19}}</ref>如果泊松过程的参数常数被某个非负可积函数的<math>t</math>代替，则得到的过程称为非齐次或非齐次Poisson过程，其中过程点的平均密度不再是常数。<ref name="Kingman1992page22">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=22}}</ref>作为排队论中的一个基本过程，泊松过程是数学模型的一个重要过程，在这里，它找到了在特定时间窗口中随机发生的事件模型的应用程序。<ref name="KarlinTaylor2012page118">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=118, 119}}</ref><ref name="Kleinrock1976page61">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems00klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1|page=[https://archive.org/details/queueingsystems00klei/page/61 61]}}</ref>

 

齐次泊松过程可以用不同的方法定义和推广。它的指标集可以定义为实数，这个随机过程也被称为平稳泊松过程<ref name="Kingman1992page38">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=38}}</ref><ref name="DaleyVere-Jones2006page19">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=19}}</ref>如果泊松过程的参数常数被某个非负可积函数的<math>t</math>代替，则得到的过程称为非齐次或非齐次Poisson过程，其中过程点的平均密度不再是常数。<ref name="Kingman1992page22">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=22}}</ref>作为排队论中的一个基本过程，泊松过程是数学模型的一个重要过程，在这里，它找到了在特定时间窗口中随机发生的事件模型的应用程序。<ref name="KarlinTaylor2012page118">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=118, 119}}</ref><ref name="Kleinrock1976page61">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems00klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1|page=[https://archive.org/details/queueingsystems00klei/page/61 61]}}</ref>

在实线上定义的泊松过程可以解释为一个随机过程，<ref name="Applebaum2004page1337"/><ref name="Rosenblatt1962page94">{{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]}}</ref>等随机变量对象。<ref name="Haenggi2013page10and18">{{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}}</ref><ref name="ChiuStoyan2013page41and108">{{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}}</ref>但是它可以定义在<math>n</math>维欧几里德空间或其他数学空间上，<ref name="Kingman1992page11">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=11}}</ref>其中它通常被解释为随机集或随机计数度量，而不是随机过程。<ref name="Haenggi2013page10and18"/><ref name="ChiuStoyan2013page41and108"/>在此设置中，是泊松过程，也称为泊松点过程，是概率论中最重要的研究对象之一，无论是应用还是理论原因。<ref name="Stirzaker2000"/><ref name="Streit2010page1">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC&pg=PA11|year=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|page=1}}</ref>但有人指出，Poisson过程并没有得到应有的重视，部分原因是它经常被认为只是在实线上，而不是在其他数学空间中。<ref name="Streit2010page1"/><ref name="Kingman1992pagev">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=v}}</ref>

在实数上定义的泊松过程可以解释为一个随机过程，<ref name="Applebaum2004page1337"/><ref name="Rosenblatt1962page94">{{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]}}</ref>等随机变量对象。<ref name="Haenggi2013page10and18">{{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}}</ref><ref name="ChiuStoyan2013page41and108">{{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}}</ref>但是它可以定义在<math>n</math>维欧几里德空间或其他数学空间上，<ref name="Kingman1992page11">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=11}}</ref>其中它通常被解释为随机集或随机计数度量，而不是随机过程。<ref name="Haenggi2013page10and18"/><ref name="ChiuStoyan2013page41and108"/>在此设置中，是泊松过程，也称为泊松点过程，是概率论中最重要的研究对象之一，无论是应用还是理论原因。<ref name="Stirzaker2000"/><ref name="Streit2010page1">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC&pg=PA11|year=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|page=1}}</ref>但有人指出，Poisson过程并没有得到应有的重视，部分原因是它经常被认为只是在实数上，而不是在其他数学空间中。<ref name="Streit2010page1"/><ref name="Kingman1992pagev">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=v}}</ref>

==定义==

==定义==

===索引集 Index set===

===索引集 Index set===

集合<math>T</math>称为“索引集”<ref name="Parzen1999"/><ref name="Florescu2014page294"/>或“参数集”<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|pages=93, 94}}</ref>。通常，这个集合是实线的一个子集，例如自然数或一个区间，使集合<math>T</math>能够解释时间。<ref name="doob1953stochasticP46to47"/>除了这些集合，索引集<math>T</math>可以是其他线性有序集或更一般的数学集，<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=482}}</ref>例如笛卡尔平面<math>R^2</math>或<math>n</math>维欧几里得空间，其中t中的元素可以表示空间中的一个点。<ref name="KarlinTaylor2012page27">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=27}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=25}}</ref>但一般情况下，当索引集有序时，随机过程可以得到更多的结果和定理。<ref name="Skorokhod2005page104">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=104}}</ref>

集合<math>T</math>称为“索引集”<ref name="Parzen1999"/><ref name="Florescu2014page294"/>或“参数集”<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|pages=93, 94}}</ref>。通常，这个集合是实数的一个子集，例如自然数或一个区间，使集合<math>T</math>能够解释时间。<ref name="doob1953stochasticP46to47"/>除了这些集合，索引集<math>T</math>可以是其他线性有序集或更一般的数学集，<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=482}}</ref>例如笛卡尔平面<math>R^2</math>或<math>n</math>维欧几里得空间，其中t中的元素可以表示空间中的一个点。<ref name="KarlinTaylor2012page27">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=27}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=25}}</ref>但一般情况下，当索引集有序时，随机过程可以得到更多的结果和定理。<ref name="Skorokhod2005page104">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=104}}</ref>

<br>

<br>

===状态空间 State space ===

===状态空间 State space ===

随机过程的数学空间<math>S</math>称为其“状态空间”。这个数学空间可以用整数、实线、<math>n</math>维欧几里得空间、复杂平面或更抽象的数学空间来定义。状态空间是用反映随机过程可以采用的不同值的元素来定义的进程。<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="Florescu2014page294">{{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|pages=294, 295}}</ref><ref name="Brémaud2014page120">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=120}}</ref>

随机过程的数学空间<math>S</math>称为其“状态空间”。这个数学空间可以用整数、实数、<math>n</math>维欧几里得空间、复杂平面或更抽象的数学空间来定义。状态空间是用反映随机过程可以采用的不同值的元素来定义的进程。<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="Florescu2014page294">{{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|pages=294, 295}}</ref><ref name="Brémaud2014page120">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=120}}</ref>

<br>

<br>

===增量 Increment===

===增量 Increment===

随机过程的增量是同一随机过程的两个随机变量之间的差值。对于一个指数集可以解释为时间的随机过程，增量是随机过程在某个时间段内的变化量。例如，如果<math>\{X（t）：t\in t\}</math> 是具有状态空间的随机过程<math>S</math>且索引集<math>T=[0,\infty)</math>中的任意两个非负数<math>t_1\in [0,\infty)</math>和<math>t_2\in [0,\infty)</math>且<math>t_1\leq t_2</math>，差异<math>X{tu 2}-X{t_1}</math>是一个称为增量的<math>S</math>值随机变量。<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/>当对增量感兴趣时，通常状态空间<math>S</math>是实线或自然数，但它可以是<math>n</math>维欧几里德空间或更抽象的空间，如[[巴拿赫空间]] Banach spaces。<ref name="Applebaum2004page1337"/>

随机过程的增量是同一随机过程的两个随机变量之间的差值。对于一个指数集可以解释为时间的随机过程，增量是随机过程在某个时间段内的变化量。例如，如果<math>\{X（t）：t\in t\}</math> 是具有状态空间的随机过程<math>S</math>且索引集<math>T=[0,\infty)</math>中的任意两个非负数<math>t_1\in [0,\infty)</math>和<math>t_2\in [0,\infty)</math>且<math>t_1\leq t_2</math>，差异<math>X{tu 2}-X{t_1}</math>是一个称为增量的<math>S</math>值随机变量。<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/>当对增量感兴趣时，通常状态空间<math>S</math>是实数或自然数，但它可以是<math>n</math>维欧几里德空间或更抽象的空间，如[[巴拿赫空间]] Banach spaces。<ref name="Applebaum2004page1337"/>

<br>

<br>

它们都有相同的[[概率分布]]。平稳随机过程的指标集通常被解释为时间，因此可以是整数或实线。<ref name="Lamperti1977page6">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=6 and 7}}</ref><ref name="GikhmanSkorokhod1969page4">{{cite book|author1=Iosif I. Gikhman |author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=4}}</ref> 但对于点过程和随机场也存在平稳性的概念，其中指标集不被解释为时间。<ref name="Lamperti1977page6"/><ref name="Adler2010page14">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA263|year=2010|publisher=SIAM|isbn=978-0-89871-693-1|pages=14, 15}}</ref><ref name="ChiuStoyan2013page112">{{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|page=112}}</ref>

它们都有相同的[[概率分布]]。平稳随机过程的指标集通常被解释为时间，因此可以是整数或实数。<ref name="Lamperti1977page6">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=6 and 7}}</ref><ref name="GikhmanSkorokhod1969page4">{{cite book|author1=Iosif I. Gikhman |author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=4}}</ref> 但对于点过程和随机场也存在平稳性的概念，其中指标集不被解释为时间。<ref name="Lamperti1977page6"/><ref name="Adler2010page14">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA263|year=2010|publisher=SIAM|isbn=978-0-89871-693-1|pages=14, 15}}</ref><ref name="ChiuStoyan2013page112">{{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|page=112}}</ref>

随机过程可分性的概念是由[[Joseph Doob]]，<ref name="Ito2006page32"/>提出的。可分性的基本思想是使指标集的可数点集决定随机过程的性质，<ref name="Billingsley2008page526"/>因此离散时间随机过程总是可分离的。<ref name="Doob1990page56">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=56}}</ref>Doob的一个定理，有时被称为Doob的可分性定理，表示任何实值连续时间随机过程都有一个可分离的修改。<ref name="Ito2006page32"/><ref name="Todorovic2012page19"/><ref name="Khoshnevisan2006page155">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|page=155}}</ref>对于具有索引集和状态空间而不是实线的更一般的随机过程，也存在该定理的版本。<ref name="Skorokhod2005page93"/>

随机过程可分性的概念是由[[Joseph Doob]]，<ref name="Ito2006page32"/>提出的。可分性的基本思想是使指标集的可数点集决定随机过程的性质，<ref name="Billingsley2008page526"/>因此离散时间随机过程总是可分离的。<ref name="Doob1990page56">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=56}}</ref>Doob的一个定理，有时被称为Doob的可分性定理，表示任何实值连续时间随机过程都有一个可分离的修改。<ref name="Ito2006page32"/><ref name="Todorovic2012page19"/><ref name="Khoshnevisan2006page155">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|page=155}}</ref>对于具有索引集和状态空间而不是实数的更一般的随机过程，也存在该定理的版本。<ref name="Skorokhod2005page93"/>

<br>

<br>

====斯科罗霍德空间 Skorokhod space====

====斯科罗霍德空间 Skorokhod space====

''skorokod space''也写为''Skorohod space''，是所有右连续左极限的函数的数学空间，定义在实线的某个区间上，例如<math>[0,1]</math>或<math>[0，\infty）</math>，取实线或度量空间上的值。<ref name="Whitt2006page78">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|pages=78–79}}</ref><ref name="GusakKukush2010page24">Gusak, Dmytro; Kukush, Alexander; Kulik, Alexey; Mishura, Yuliya; Pilipenko, Andrey (2010). Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory. Springer Science & Business Media. p. 21. ISBN 978-0-387-87862-1., p. 24</ref><ref name="Bogachev2007Vol2page53">{{cite book|author=Vladimir I. Bogachev|title=Measure Theory (Volume 2)|url=https://books.google.com/books?id=CoSIe7h5mTsC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-34514-5|page=53}}</ref>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。<ref name="Whitt2006page78"/><ref name="Klebaner2005page4">{{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=4}}</ref>由[[Anatoliy Skorokod]]引入的Skorokod函数空间，<ref name="Bogachev2007Vol2page53"/>通常用字母<math>D</math>表示，<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/>因此函数空间也被称为空间<math>D</math><ref name="Whitt2006page78"/><ref name="Asmussen2003page420">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=420}}</ref><ref name="Billingsley2013page121">{{cite book|author=Patrick Billingsley|title=Convergence of Probability Measures|url=https://books.google.com/books?id=6ItqtwaWZZQC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-62596-5|page=121}}</ref>此函数空间的表示法还可以包括定义所有cádlág函数的间隔，因此，例如，<math>D[0,1]</math>表示在单位间隔<math>[0，1] </math>。<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>

''skorokod space''也写为''Skorohod space''，是所有右连续左极限的函数的数学空间，定义在实数的某个区间上，例如<math>[0,1]</math>或<math>[0，\infty）</math>，取实数或度量空间上的值。<ref name="Whitt2006page78">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|pages=78–79}}</ref><ref name="GusakKukush2010page24">Gusak, Dmytro; Kukush, Alexander; Kulik, Alexey; Mishura, Yuliya; Pilipenko, Andrey (2010). Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory. Springer Science & Business Media. p. 21. ISBN 978-0-387-87862-1., p. 24</ref><ref name="Bogachev2007Vol2page53">{{cite book|author=Vladimir I. Bogachev|title=Measure Theory (Volume 2)|url=https://books.google.com/books?id=CoSIe7h5mTsC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-34514-5|page=53}}</ref>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。<ref name="Whitt2006page78"/><ref name="Klebaner2005page4">{{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=4}}</ref>由[[Anatoliy Skorokod]]引入的Skorokod函数空间，<ref name="Bogachev2007Vol2page53"/>通常用字母<math>D</math>表示，<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/>因此函数空间也被称为空间<math>D</math><ref name="Whitt2006page78"/><ref name="Asmussen2003page420">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=420}}</ref><ref name="Billingsley2013page121">{{cite book|author=Patrick Billingsley|title=Convergence of Probability Measures|url=https://books.google.com/books?id=6ItqtwaWZZQC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-62596-5|page=121}}</ref>此函数空间的表示法还可以包括定义所有cádlág函数的间隔，因此，例如，<math>D[0,1]</math>表示在单位间隔<math>[0，1] </math>。<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>

在随机过程理论中，由于通常假定连续时间随机过程的样本函数属于一个Skorokod空间，<ref name="Bogachev2007Vol2page53"/><ref name="Asmussen2003page420"/>因此经常使用Skorokod函数空间，对应于Wiener过程的样本函数。但是空间也有间断函数，这意味着随机过程的样本函数具有跳跃性，例如泊松过程（在实线上），同时也是这一领域的成员。<ref name="Billingsley2013page121"/><ref name="BinghamKiesel2013page154">{{cite book|author1=Nicholas H. Bingham|author2=Rüdiger Kiesel|title=Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives|url=https://books.google.com/books?id=AOIlBQAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-3856-3|page=154}}</ref>

在随机过程理论中，由于通常假定连续时间随机过程的样本函数属于一个Skorokod空间，<ref name="Bogachev2007Vol2page53"/><ref name="Asmussen2003page420"/>因此经常使用Skorokod函数空间，对应于Wiener过程的样本函数。但是空间也有间断函数，这意味着随机过程的样本函数具有跳跃性，例如泊松过程（在实数上），同时也是这一领域的成员。<ref name="Billingsley2013page121"/><ref name="BinghamKiesel2013page154">{{cite book|author1=Nicholas H. Bingham|author2=Rüdiger Kiesel|title=Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives|url=https://books.google.com/books?id=AOIlBQAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-3856-3|page=154}}</ref>

<br>

<br>

通过应用适当的变换，也可以从随机过程中产生鞅'，这就是齐次泊松过程（在实线上）的情形，其结果是一个称为“补偿泊松过程”的鞅。<ref name="KaratzasShreve2014page11"/>也可以从其他鞅中构建鞅。<ref name="Steele2012page12"/>例如，有基于鞅的鞅Wiener过程，形成连续时间鞅。<ref name="Klebaner2005page65"/><ref name="Steele2012page115">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=115}}</ref>

 

通过应用适当的变换，也可以从随机过程中产生鞅'，这就是齐次泊松过程（在实数上）的情形，其结果是一个称为“补偿泊松过程”的鞅。<ref name="KaratzasShreve2014page11"/>也可以从其他鞅中构建鞅。<ref name="Steele2012page12"/>例如，有基于鞅的鞅Wiener过程，形成连续时间鞅。<ref name="Klebaner2005page65"/><ref name="Steele2012page115">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=115}}</ref>

===随机 Random field===

===随机 Random field===

随机场是由一个<math>n</math>维欧几里德空间或流形索引的随机变量的集合。一般来说，随机场可以看作是随机过程的一个例子，其中，索引集不一定是实行的子集。<ref name="AdlerTaylor2009page7"/>但是有一个约定，当索引具有两个或多个维度时，随机变量的索引集合称为随机字段。<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="KoralovSinai2007page171">{{cite book|author1=Leonid Koralov|author2=Yakov G. Sinai|title=Theory of Probability and Random Processes|url=https://books.google.com/books?id=tlWOphOFRgwC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-68829-7|page=171}}</ref>如果随机过程的具体定义要求索引集是实线的子集，那么随机场可以看作是随机过程的一个推广。<ref name="ApplebaumBook2004page19">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=19}}</ref>

随机场是由一个<math>n</math>维欧几里德空间或流形索引的随机变量的集合。一般来说，随机场可以看作是随机过程的一个例子，其中，索引集不一定是实行的子集。<ref name="AdlerTaylor2009page7"/>但是有一个约定，当索引具有两个或多个维度时，随机变量的索引集合称为随机字段。<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="KoralovSinai2007page171">{{cite book|author1=Leonid Koralov|author2=Yakov G. Sinai|title=Theory of Probability and Random Processes|url=https://books.google.com/books?id=tlWOphOFRgwC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-68829-7|page=171}}</ref>如果随机过程的具体定义要求索引集是实数的子集，那么随机场可以看作是随机过程的一个推广。<ref name="ApplebaumBook2004page19">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=19}}</ref>

===点过程 Point process===

===点过程 Point process===

点过程是随机分布在某些数学空间（如实线、<math>n</math>维欧几里德空间或更抽象的空间）上的点的集合。有时“点过程”一词并不可取，因为历史上“过程”一词表示某个系统在时间上的演变，因此，点过程也被称为“随机点域”。<ref name="ChiuStoyan2013page109">{{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|page=109}}</ref>>点过程有不同的解释，这样一个随机计数度量或随机集。<ref name="ChiuStoyan2013page108">{{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|page=108}}</ref><ref name="Haenggi2013page10">{{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|page=10}}</ref>一些作者将点过程和随机过程视为两个不同的对象，因此点过程是随机过程产生或与随机过程相关联的随机对象，<ref name="DaleyVere-Jones2006page194">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=194}}</ref><ref name="CoxIsham1980page3">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8|page=3}}</ref>尽管已经注意到点过程和随机过程之间的区别并不清楚。<ref name="CoxIsham1980page3"/>

点过程是随机分布在某些数学空间（如实数、<math>n</math>维欧几里德空间或更抽象的空间）上的点的集合。有时“点过程”一词并不可取，因为历史上“过程”一词表示某个系统在时间上的演变，因此，点过程也被称为“随机点域”。<ref name="ChiuStoyan2013page109">{{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|page=109}}</ref>>点过程有不同的解释，这样一个随机计数度量或随机集。<ref name="ChiuStoyan2013page108">{{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|page=108}}</ref><ref name="Haenggi2013page10">{{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|page=10}}</ref>一些作者将点过程和随机过程视为两个不同的对象，因此点过程是随机过程产生或与随机过程相关联的随机对象，<ref name="DaleyVere-Jones2006page194">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=194}}</ref><ref name="CoxIsham1980page3">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8|page=3}}</ref>尽管已经注意到点过程和随机过程之间的区别并不清楚。<ref name="CoxIsham1980page3"/>

 

另一些作者认为点过程是一个随机过程，其中过程由一组底层空间（在点过程的上下文中，“状态空间”一词可以指定义点过程的空间，如实线，<ref name="Kingman1992page8">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=8}}</ref><ref name="MollerWaagepetersen2003page7">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|year=2003|publisher=CRC Press|isbn=978-0-203-49693-0|page=7}}</ref>其中与随机过程术语中的指标集相对应的指标集。）其上定义它的地方，如实线或<math>n</math>-维的欧几里得空间。<ref name="KarlinTaylor2012page31">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=31}}</ref><ref name="Schmidt2014page99">{{cite book|author=Volker Schmidt|title=Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms|url=https://books.google.com/books?id=brsUBQAAQBAJ&pg=PR5|date= 2014|publisher=Springer|isbn=978-3-319-10064-7|page=99}}</ref>其他随机过程，如更新和计数过程，在点过程理论中进行了研究一、基本理论与方法。<ref name="DaleyVere-Jones200">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8}}</ref><ref name="CoxIsham1980">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8}}</ref>

另一些作者认为点过程是一个随机过程，其中过程由一组底层空间（在点过程的上下文中，“状态空间”一词可以指定义点过程的空间，如实数，<ref name="Kingman1992page8">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=8}}</ref><ref name="MollerWaagepetersen2003page7">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|year=2003|publisher=CRC Press|isbn=978-0-203-49693-0|page=7}}</ref>其中与随机过程术语中的指标集相对应的指标集。）其上定义它的地方，如实数或<math>n</math>-维的欧几里得空间。<ref name="KarlinTaylor2012page31">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=31}}</ref><ref name="Schmidt2014page99">{{cite book|author=Volker Schmidt|title=Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms|url=https://books.google.com/books?id=brsUBQAAQBAJ&pg=PR5|date= 2014|publisher=Springer|isbn=978-3-319-10064-7|page=99}}</ref>其他随机过程，如更新和计数过程，在点过程理论中进行了研究一、基本理论与方法。<ref name="DaleyVere-Jones200">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8}}</ref><ref name="CoxIsham1980">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8}}</ref>

<br>

<br>