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第10行： − 应用和现象研究反过来又启发了新随机过程的提出。这种随机过程的例子包括维纳过程（Wiener process）或布朗运动过程（Brownian motion process，“布朗运动”可以指物理过程，也被称为“布朗运动”，以及随机过程，一个数学对象，但为了避免歧义，本文使用“布朗运动过程”或“维纳过程”来表示后者，其风格类似于，例如，吉赫曼和斯科罗霍德 <ref name="GikhmanSkorokhod1969">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3}}</ref> 或罗森布拉特<ref name="Rosenblatt1962">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press}}</ref>）使用人路易斯·巴切勒为了研究巴黎证券交易所的价格变化，<ref name="JarrowProtter2004">{{cite book|last1=Jarrow|first1=Robert|title=A Festschrift for Herman Rubin|last2=Protter|first2=Philip|chapter=A short history of stochastic integration and mathematical finance: the early years, 1880–1970|year=2004|pages=75–80|issn=0749-2170|doi=10.1214/lnms/1196285381|citeseerx=10.1.1.114.632|series=Institute of Mathematical Statistics Lecture Notes - Monograph Series|isbn=978-0-940600-61-4}}</ref> 以及爱尔朗使用的泊松过程来研究某段时间内拨出的电话号码。<ref name="Stirzaker2000">{{cite journal|last1=Stirzaker|first1=David|title=Advice to Hedgehogs, or, Constants Can Vary|journal=The Mathematical Gazette|volume=84|issue=500|year=2000|pages=197–210|issn=0025-5572|doi=10.2307/3621649|jstor=3621649}}</ref>这两个随机过程被认为是随机过程理论中最重要和最核心的，<ref name="doob1953stochasticP46to47" /><ref name="Parzen1999" /><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=32}}</ref> 并且被巴切勒和爱尔朗先后于不同的环境和国家被多次独立地发现<ref name="JarrowProtter2004" /><ref name="GuttorpThorarinsdottir2012">{{cite journal|last1=Guttorp|first1=Peter|last2=Thorarinsdottir|first2=Thordis L.|title=What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes|journal=International Statistical Review|volume=80|issue=2|year=2012|pages=253–268|issn=0306-7734|doi=10.1111/j.1751-5823.2012.00181.x}}</ref>。 + − '''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。 第30行：
第28行： − 当解释为时间时，如果随机过程的指标集有有限个或可数个元素，例如有限的一组数、一组整数或自然数，那么随机过程被称为离散时间<ref name="Billingsley2008page482"/><ref name="Borovkov2013page527">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=527}}</ref> 。如果索引集是实数轴上的某个区间，则时间被称为连续时间。这两类随机过程分别被称为'''离散时间随机过程'''和'''连续时间随机过程'''<ref name="KarlinTaylor2012page27"/><ref name="Brémaud2014page120"/><ref name="Rosenthal2006page177">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|pages=177–178}}</ref>。离散时间随机过程被认为更容易研究，因为连续时间过程需要更先进的数学技术和知识，特别当索引集不可数时。<ref name="KloedenPlaten2013page63">{{cite book|author1=Peter E. Kloeden|author2=Eckhard Platen|title=Numerical Solution of Stochastic Differential Equations|url=https://books.google.com/books?id=r9r6CAAAQBAJ=PA1|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-12616-5|page=63}}</ref><ref name="Khoshnevisan2006page153">{{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|pages=153–155}}</ref> 如果索引集是整数或整数的子集，则随机过程也可以称为'''随机序列 random sequence'''。<ref name="Borovkov2013page527"/> + − 如果状态空间是整数或自然数，则随机过程称为“离散随机过程”或“整值随机过程”。如果状态空间是实数轴，则随机过程被称为“实值随机过程”或“具有连续状态空间的过程”。如果状态空间是<math>n</math>-维欧几里德空间，则随机过程称为<math>n</math>-“维向量过程”或<math>n</math>—“向量过程”。<ref name="florescu214page294">Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 294, 295. ISBN 978-1-118-59320-2.</ref><ref name="KarlinTaylor2012page26"/>+ + + 第39行：
第39行： − + + + 第47行：
第49行： − + + − 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"/>+ − + − + − + − + − + + + + − 随机游走是随机过程，通常定义为欧几里德空间中独立同分布的随机变量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。<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>但是有些人也使用这个术语来指代连续时间变化的过程，<ref name=“Weiss2006page1”>{cite book | last1=Weiss | first1=George H.| title=Statistical Sciences | chapter=Random Walks | year=2006 | doi=10.1002/0471667196.ess2180.pub2 | page=1 | isbn=978-0471667193}}</ref>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。<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>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。<ref name="Weiss2006page1">Weiss, George H. (2006). "Random Walks". Encyclopedia of Statistical Sciences. p. 1. doi:10.1002/0471667196.ess2180.pub2. ISBN 978-0471667193.</ref><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> − 随机游走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|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>+ − + − 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.<ref name="RogersWilliams2000page1">{{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|page=1}}</ref><ref name="Klebaner2005page56">{{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=56}}</ref>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。<ref name="Brush1968page1">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=1–2|issn=0003-9519|doi=10.1007/BF00328110}}</ref><ref name="Applebaum2004page1338">{{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=1338}}</ref><ref name="Applebaum2004page1338"/><ref name="GikhmanSkorokhod1969page21">{{cite book|author1=Iosif Ilyich 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=21}}</ref>+
编辑到维纳过程
应用和现象研究反过来又启发了新随机过程的提出。这种随机过程的例子包括维纳过程（Wiener process）或布朗运动过程（Brownian motion process，“布朗运动”可以指物理过程，也被称为“布朗运动”，以及随机过程，一个数学对象，但为了避免歧义，本文使用“布朗运动过程”或“维纳过程”来表示后者，其风格类似于：例如吉赫曼和斯科罗霍德 <ref name="GikhmanSkorokhod1969">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3}}</ref> 或罗森布拉特<ref name="Rosenblatt1962">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press}}</ref>）使用路易斯·巴切勒来研究巴黎证券交易所的价格变化，<ref name="JarrowProtter2004">{{cite book|last1=Jarrow|first1=Robert|title=A Festschrift for Herman Rubin|last2=Protter|first2=Philip|chapter=A short history of stochastic integration and mathematical finance: the early years, 1880–1970|year=2004|pages=75–80|issn=0749-2170|doi=10.1214/lnms/1196285381|citeseerx=10.1.1.114.632|series=Institute of Mathematical Statistics Lecture Notes - Monograph Series|isbn=978-0-940600-61-4}}</ref> 以及爱尔朗使用的泊松过程来研究某段时间内拨出的电话号码。<ref name="Stirzaker2000">{{cite journal|last1=Stirzaker|first1=David|title=Advice to Hedgehogs, or, Constants Can Vary|journal=The Mathematical Gazette|volume=84|issue=500|year=2000|pages=197–210|issn=0025-5572|doi=10.2307/3621649|jstor=3621649}}</ref>这两个随机过程被认为是随机过程理论中最重要和最核心的，<ref name="doob1953stochasticP46to47" /><ref name="Parzen1999" /><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=32}}</ref> 并且被巴切勒和爱尔朗先后于不同的环境和国家多次被独立地发现<ref name="JarrowProtter2004" /><ref name="GuttorpThorarinsdottir2012">{{cite journal|last1=Guttorp|first1=Peter|last2=Thorarinsdottir|first2=Thordis L.|title=What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes|journal=International Statistical Review|volume=80|issue=2|year=2012|pages=253–268|issn=0306-7734|doi=10.1111/j.1751-5823.2012.00181.x}}</ref>。
'''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，也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，以及拓扑学<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>和数学分析的分支，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>，不论关于理论还是应用，它都是一个活跃的研究主题。<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>
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> 莱维过程,<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，以及拓扑学<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>和数学分析的分支，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>，不论关于理论还是应用，它都是活跃的研究主题。<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"/>
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被称为“抽样函数”或“实现”。<ref name="Lamperti1977page1"/><ref name="RogersWilliams2000page121b"/>
[[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]]
当解释为时间时，如果随机过程的指标集有有限个或可数个元素，例如有限的一组数、一组整数或自然数，那么随机过程被认为在离散时间域上<ref name="Billingsley2008page482"/><ref name="Borovkov2013page527">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=527}}</ref> 。如果索引集是实数上的某个区间，则时间被称为连续时间。这两类随机过程分别被称为'''离散时间随机过程'''和'''连续时间随机过程'''<ref name="KarlinTaylor2012page27"/><ref name="Brémaud2014page120"/><ref name="Rosenthal2006page177">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|pages=177–178}}</ref>。离散时间随机过程被认为更容易研究，因为连续时间过程需要更先进的数学技术和知识，特别当索引集不可数时。<ref name="KloedenPlaten2013page63">{{cite book|author1=Peter E. Kloeden|author2=Eckhard Platen|title=Numerical Solution of Stochastic Differential Equations|url=https://books.google.com/books?id=r9r6CAAAQBAJ=PA1|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-12616-5|page=63}}</ref><ref name="Khoshnevisan2006page153">{{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|pages=153–155}}</ref> 如果索引集是整数或整数的子集，则随机过程也可以称为'''随机序列 random sequence'''。<ref name="Borovkov2013page527"/>
如果状态空间是整数或自然数，则随机过程称为“离散随机过程”或“整值随机过程”。如果状态空间是实数，则随机过程被称为“实值随机过程”或“具有连续状态空间的过程”。如果状态空间是<math>n</math>-维欧几里德空间，则随机过程称为<math>n</math>-“维向量过程”或<math>n</math>—“向量过程”。<ref name="florescu214page294">Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 294, 295. ISBN 978-1-118-59320-2.</ref><ref name="KarlinTaylor2012page26"/>
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===词源学===
===词源学===
在英语中，“随机”一词最初用作形容词，其定义是“与推测有关”，源于一个希腊语词，意思是“瞄准一个标记，猜测”，而牛津英语词典将1662年作为最早出现的年份。<ref name="OxfordStochastic">{Cite OED | random}</ref>在他关于概率“Ars conquectandi”的著作中，最初于1713年以拉丁文出版，[[雅各布·伯努利 Jakob Bernoulli]]使用了“Ars conquectandi istice”这个短语，这本书已经被翻译成“猜想或随机的艺术”。<ref name="Sheĭnin2006page5">{{cite book|author=O. B. Sheĭnin|title=Theory of probability and statistics as exemplified in short dictums|url=https://books.google.com/books?id=XqMZAQAAIAAJ|year=2006|publisher=NG Verlag|isbn=978-3-938417-40-9|page=5}}</ref>这一短语是[adislaus Bortkiewicz在关于伯努利问题中使用，<ref name="SheyninStrecker2011page136">{{cite book|author1=Oscar Sheynin|author2=Heinrich Strecker|title=Alexandr A. Chuprov: Life, Work, Correspondence|url=https://books.google.com/books?id=1EJZqFIGxBIC&pg=PA9|year=2011|publisher=V&R unipress GmbH|isbn=978-3-89971-812-6|page=136}}</ref>他在1917年用德语写下了“随机”一词。术语“随机过程”最早出现在1934年Joseph Doob的一篇论文中。<ref name="OxfordStochastic"/> 对于这个术语和一个具体的数学定义，Doob引用了另一篇1934年的论文，其中Aleksandr Khinchin在德语中使用了术语“随机过程”，<ref name="Doob1934"/><ref name="Khintchine1934">{{cite journal|last1=Khintchine|first1=A.|title=Korrelationstheorie der stationeren stochastischen Prozesse|journal=Mathematische Annalen|volume=109|issue=1|year=1934|pages=604–615|issn=0025-5831|doi=10.1007/BF01449156}}</ref>尽管德语这个词在早些时候就被使用过，例如，Andrei Kolmogorov在1931年就使用过。<ref name="Kolmogoroff1931page1">{{cite journal|last1=Kolmogoroff|first1=A.|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung|journal=Mathematische Annalen|volume=104|issue=1|year=1931|page=1|issn=0025-5831|doi=10.1007/BF01457949}}</ref>
在英语中，“随机”一词最初用作形容词，其定义是“与推测有关”，源于一个希腊语词，意思是“瞄准一个标记，猜测”，而牛津英语词典将1662年作为最早出现的年份。<ref name="OxfordStochastic">{Cite OED | random}</ref>[[雅各布·伯努利 Jakob Bernoulli]]在他关于概率“Ars conquectandi”的著作中使用了“Ars conquectandi istice”这个短语，最初于1713年以拉丁文出版，目前在译文中已经被翻译成“猜想或随机的艺术”。<ref name="Sheĭnin2006page5">{{cite book|author=O. B. Sheĭnin|title=Theory of probability and statistics as exemplified in short dictums|url=https://books.google.com/books?id=XqMZAQAAIAAJ|year=2006|publisher=NG Verlag|isbn=978-3-938417-40-9|page=5}}</ref>这一短语根据伯努利的引用，是由拉迪斯劳斯-博特凯维茨在1917是用德语stochastic表示同样的意思“随机”时使用的。<ref name="SheyninStrecker2011page136">{{cite book|author1=Oscar Sheynin|author2=Heinrich Strecker|title=Alexandr A. Chuprov: Life, Work, Correspondence|url=https://books.google.com/books?id=1EJZqFIGxBIC&pg=PA9|year=2011|publisher=V&R unipress GmbH|isbn=978-3-89971-812-6|page=136}}</ref>。术语“随机过程”最早出现在1934年约瑟夫-杜布的一篇论文中。<ref name="OxfordStochastic"/> 对于这个术语和明确的数学定义，杜布引用了另一篇1934年的论文，其中亚历山大-金钦在德语中使用了术语"''stochastischer Prozeß'' ”，<ref name="Doob1934"/><ref name="Khintchine1934">{{cite journal|last1=Khintchine|first1=A.|title=Korrelationstheorie der stationeren stochastischen Prozesse|journal=Mathematische Annalen|volume=109|issue=1|year=1934|pages=604–615|issn=0025-5831|doi=10.1007/BF01449156}}</ref>尽管德语这个词在早些时候就被使用过，例如，安德烈-科尔莫戈罗夫在1931年就使用过。<ref name="Kolmogoroff1931page1">{{cite journal|last1=Kolmogoroff|first1=A.|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung|journal=Mathematische Annalen|volume=104|issue=1|year=1931|page=1|issn=0025-5831|doi=10.1007/BF01457949}}</ref>
根据牛津英语词典的研究，英语中和随机含义相同的这个词的早期出现，可以追溯到16世纪，而早期记录的类似用法开始于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。
===术语===
===术语===
随机过程的定义是不同的，<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> 但是随机过程传统上被定义为一组随机变量的集合<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/>。术语“随机（random）过程”和“随机（stochastic）过程”被视为同义词，可以互换使用，而无需精确指定索引集。<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{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/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref>。两个“集合”<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/>，或“家庭”使用<ref name="Parzen1999"/><ref name="Ito2006page13">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|page=13}}</ref>术语“参数集”<ref name="Lamperti1977page1"/> 或“参数空间”<ref name="AdlerTaylor2009page7"/> ，而不是“索引集”。
随机过程的定义是不同的，<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> 但是随机过程传统上被定义为由一些集合索引的随机变量的集族<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/>。术语“随机（random）过程”和“随机（stochastic）过程”被视为同义词，可以互换使用，而无需精确指定索引集。<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{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/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref>。两个“集族”<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/>，或“族”都在使用着<ref name="Parzen1999"/><ref name="Ito2006page13">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|page=13}}</ref>，而除了“索引集”的叫法，还会叫做“参数集”<ref name="Lamperti1977page1"/> 或“参数空间”<ref name="AdlerTaylor2009page7"/> 。
术语“随机函数”也用于指代随机或随机过程，<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”通常在索引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>其他的叫法，例如当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或流形时，会称为随机场。<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>
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===符号===
===记号===
随机过程可以用<math>\{X(t)\}_{t\in T} </math>，<ref name="Brémaud2014page120"/> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page408"/> <math>\{X_t\}</math><ref name="Lamperti1977page3">,{{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|page=3}}</ref>或简单地称为<math>X</math>或<math>X（t）</math>，尽管<math>X（t）</math>被视为函数表示法滥用。<ref name="Klebaner2005page55">{{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=55}}</ref> 例如， <math>X(t)</math> 或 <math>X_t</math>引用具有索引<math>t</math>的随机变量，而不是整个随机过程。<ref name="Lamperti1977page3"/>如果索引集是<math>T=[0,\infty)</math>，然后，我们可以写，例如，<math>(X_t , t \geq 0)</math>来表示随机过程。<ref name="ChaumontYor2012"/>
随机过程可以用<math>\{X(t)\}_{t\in T} </math>，<ref name="Brémaud2014page120"/> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page408"/> <math>\{X_t\}</math><ref name="Lamperti1977page3">,{{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|page=3}}</ref>或简单地称为<math>X</math>或<math>X（t）</math>，尽管<math>X（t）</math>被视为函数表示法滥用。<ref name="Klebaner2005page55">{{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=55}}</ref> 例如， <math>X(t)</math> 或 <math>X_t</math>表示具有索引<math>t</math>的随机变量，而不是整个随机过程。<ref name="Lamperti1977page3"/>如果索引集是<math>T=[0,\infty)</math>，那么我们可以这样写，例如，<math>(X_t , t \geq 0)</math>来表示随机过程。<ref name="ChaumontYor2012"/>
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==示例==
==示例==
===伯努利过程 Bernoulli process===
===伯努利过程===
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"/>它是独立同分布随机变量的序列，其中每个随机变量取值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投一枚硬币关联，其中正面的概率为<math>p</math>，其值为1，而反面的值为零。<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>
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===随机游走 Random walk===
===随机游走===
随机游走这样一类随机过程，通常定义为欧几里德空间中独立同分布的随机变量或者或随机向量的和，因此它们是在离散时间上变化的过程。<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>但是有些人也使用这个术语来指代在连续时间上变化的过程，<ref name=“Weiss2006page1”>{cite book | last1=Weiss | first1=George H.| title=Statistical Sciences | chapter=Random Walks | year=2006 | doi=10.1002/0471667196.ess2180.pub2 | page=1 | isbn=978-0471667193}}</ref>尤其是金融中使用的维纳过程，这导致了一些误解，从而招来了一些批评。<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>还有其他各种类型的随机游走，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，它们都是被充分研究的，在不同的学科中有许多应用。<ref name="Weiss2006page1">Weiss, George H. (2006). "Random Walks". Encyclopedia of Statistical Sciences. p. 1. doi:10.1002/0471667196.ess2180.pub2. ISBN 978-0471667193.</ref><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>
随机游走的一个经典例子被称为“简单随机游动”，它是一个离散时间上的随机过程，以整数为状态空间，基于伯努利过程，其中每个伯努利变量取+1或-1。换言之，简单随机游走发生在整数上，其值要么随概率<math>p</math>增加1，要么随着概率<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走。<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>
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===维纳过程 Wiener process===
===维纳过程===
维纳过程是一个具有平稳独立增量并且基于增量大小呈正态分布的随机过程。<ref name="RogersWilliams2000page1">{{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|page=1}}</ref><ref name="Klebaner2005page56">{{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=56}}</ref>维纳过程是以诺伯特-维纳命名的，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布朗运动，因为它和液体中的[[布朗运动]]有历史渊源。<ref name="Brush1968page1">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=1–2|issn=0003-9519|doi=10.1007/BF00328110}}</ref><ref name="Applebaum2004page1338">{{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=1338}}</ref><ref name="Applebaum2004page1338"/><ref name="GikhmanSkorokhod1969page21">{{cite book|author1=Iosif Ilyich 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=21}}</ref>
[[File:维纳过程.png|thumb|left|实现维纳Wiener过程（或布朗运动过程），具有漂移（<font color=blue>蓝色</font>）且不漂移（<font color=red>红色</font>）。]]
[[File:维纳过程.png|thumb|left|实现维纳Wiener过程（或布朗运动过程），具有漂移（<font color=blue>蓝色</font>）且不漂移（<font color=red>红色</font>）。]]