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第2行： − + − [[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"/>]] + − 在[[概率论]]及相关领域中，'''随机过程 stochastic process'''（或random process）是一个数学对象，通常被定义为随机变量的集合。给出对一个随机过程的解释，该过程表示某个系统随机的数值随时间的变化，例如细菌种群的增长，电流由于热噪声而波动，或者一个气体分子的运动。<ref name="doob1953stochasticP46to47">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=7Bu8jgEACAAJ|year=1990|publisher=Wiley|pages=46, 47}}</ref><ref name="Parzen1999">{{cite book|author=Emanuel Parzen|title=Stochastic Processes|url=https://books.google.com/books?id=0mB2CQAAQBAJ|year= 2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|pages=7, 8}}</ref><ref name="GikhmanSkorokhod1969page1">{{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=q0lo91imeD0C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=1}}</ref><ref name=":0">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|location= NJ|pages=1–235}}</ref>随机过程被广泛用作以随机方式变化的系统和现象的数学模型。它们在许多学科都有应用，比如生物学<ref name="Bressloff2014">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ|year=2014|publisher=Springer|isbn=978-3-319-08488-6}}</ref>，[[化学]] <ref name="Kampen2011">{{cite book|author=N.G. Van Kampen|title=Stochastic Processes in Physics and Chemistry|url=https://books.google.com/books?id=N6II-6HlPxEC|year=2011|publisher=Elsevier|isbn=978-0-08-047536-3}}</ref> 生态学,<ref name="LandeEngen2003">{{cite book|author1=Russell Lande|author2=Steinar Engen|author3=Bernt-Erik Sæther|title=Stochastic Population Dynamics in Ecology and Conservation|url=https://books.google.com/books?id=6KClauq8OekC|year=2003|publisher=Oxford University Press|isbn=978-0-19-852525-7}}</ref> 神经科学<ref name="LaingLord2010">{{cite book|author1=Carlo Laing|author2=Gabriel J Lord|title=Stochastic Methods in Neuroscience|url=https://books.google.com/books?id=RaYSDAAAQBAJ|year=2010|publisher=OUP Oxford|isbn=978-0-19-923507-0}}</ref>, 物理学<ref name="PaulBaschnagel2013">{{cite book|author1=Wolfgang Paul|author2=Jörg Baschnagel|title=Stochastic Processes: From Physics to Finance|url=https://books.google.com/books?id=OWANAAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-319-00327-6}}</ref>, 图像处理, 信号处理,<ref name="Dougherty1999">{{cite book|author=Edward R. Dougherty|title=Random processes for image and signal processing|url=https://books.google.com/books?id=ePxDAQAAIAAJ|year=1999|publisher=SPIE Optical Engineering Press|isbn=978-0-8194-2513-3}}</ref> 控制理论, <ref name="Bertsekas1996">{{cite book|author=Dimitri P. Bertsekas|title=Stochastic Optimal Control: The Discrete-Time Case|url=http://www.athenasc.com/socbook.html|year=1996|publisher=Athena Scientific]|isbn=1-886529-03-5}}</ref> [[信息论]],<ref name="CoverThomas2012page71">{{cite book|author1=Thomas M. Cover|author2=Joy A. Thomas|title=Elements of Information Theory|url=https://books.google.com/books?id=VWq5GG6ycxMC=PT16|year=2012|publisher=John Wiley & Sons|isbn=978-1-118-58577-1|page=71}}</ref> 计算机科学,<ref name="Baron2015">{{cite book|author=Michael Baron|title=Probability and Statistics for Computer Scientists, Second Edition|url=https://books.google.com/books?id=CwQZCwAAQBAJ|year=2015|publisher=CRC Press|isbn=978-1-4987-6060-7|page=131}}</ref> 密码学<ref>{{cite book|author1=Jonathan Katz|author2=Yehuda Lindell|title=Introduction to Modern Cryptography: Principles and Protocols|url=https://archive.org/details/Introduction_to_Modern_Cryptography|year=2007|publisher=CRC Press|isbn=978-1-58488-586-3|page=[https://archive.org/details/Introduction_to_Modern_Cryptography/page/n44 26]}}</ref> 和 电信学.<ref name="BaccelliBlaszczyszyn2009">{{cite book|author1=François Baccelli|author2=Bartlomiej Blaszczyszyn|title=Stochastic Geometry and Wireless Networks|url=https://books.google.com/books?id=H3ZkTN2pYS4C|year=2009|publisher=Now Publishers Inc|isbn=978-1-60198-264-3}}</ref> 此外，金融市场中看似随机的变化促进了随机过程在金融领域的广泛应用。<ref name="Steele2001">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=H06xzeRQgV4C|year=2001|publisher=Springer Science & Business Media|isbn=978-0-387-95016-7}}</ref><ref name="MusielaRutkowski2006">{{cite book|author1=Marek Musiela|author2=Marek Rutkowski|title=Martingale Methods in Financial Modelling|url=https://books.google.com/books?id=iojEts9YAxIC|year= 2006|publisher=Springer Science & Business Media|isbn=978-3-540-26653-2}}</ref><ref name="Shreve2004">{{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}}</ref> + − 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" />+ − − − − + − − − − + − − − − − 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"/> − + − + − 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》，英语中“random”（随机）一词的最早出现时间可追溯到16世纪，而早期有记载的用法则始于14世纪，意思是“急躁、速度快、力量大或暴力（骑马、跑步、击打等）”。这个词本身来自法语中间的一个词，意思是“速度，匆忙”，它可能是从法语动词“奔跑”或“飞奔”衍生而来。术语“随机过程”的首次书面出现是在“随机过程”之前出现的，牛津英语词典也将其作为同义词出现，并被Francis Edgeworth于1888年发表的一篇文章中使用。<ref name="OxfordRandom">{Cite OED | random}</ref>+ − − <br> − − 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"/> − − <br> + − + − − <br> − − + − − <br> + − + − 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> − − − − − − <br> + − 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>）。]] − − ''' Wiener process维纳过程'''在概率论中起着中心作用，通常被认为是最重要和研究的随机过程，并与其他随机过程联系在一起<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29">{{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=29}}</ref><ref name="Florescu2014page471">{{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=471}}</ref><ref name="KarlinTaylor2012page21">{{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=21, 22}}</ref><ref name="KaratzasShreve2014pageVIII">{{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=VIII}}</ref><ref name="RevuzYor2013pageIX">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=IX}}</ref>其索引集和状态空间分别是非负数和实数，因此它既有连续索引集又有状态空间<ref name="Rosenthal2006page186">{{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|page=186}}</ref>但是过程可以定义得更广泛，这样它的状态空间可以是维欧几里德空间。<ref name="Klebaner2005page81"/><ref name="KarlinTaylor2012page21"/><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=33}}</ref>如果任何增量的[[平均值]]为零，则所得到的维纳或布朗运动过程称为零漂移。如果任意两个时间点的增量的平均值等于时间差乘以某个常数<math>\mu</math>，即实数，由此产生的随机过程被称为'''漂移'''。<ref name="Steele2012page118">{{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=118}}</ref><ref name="MörtersPeres2010page1"/><ref name="KaratzasShreve2014page78">{{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=78}}</ref> − − − 几乎可以肯定，维纳过程的样本路径处处连续，但无处可微。它可以看作是简单随机游走的一个连续版本。<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>当其他随机过程（如某些随机游动重新缩放）的数学极限时，该过程出现，<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>这是[[Donsker定理]]或不变性原理的主题，也被称为'''函数中心极限定理'''。<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> − − − 维纳过程是一些重要的随机过程家族的成员，包括马尔可夫过程，Lévy过程和高斯过程。<ref name="RogersWilliams2000page1"/><ref name="Applebaum2004page1337"/>该过程也有许多应用，是随机微积分中使用的主要随机过程。<ref name="Klebaner2005">{{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}}</ref><ref name="KaratzasShreve2014page">{{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}}</ref>它在数量金融中起着核心作用，<ref name="Applebaum2004page1341">{{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=1341}}</ref><ref name="KarlinTaylor2012page340">{{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=340}}</ref>在Black-Scholes-Merton模型中使用它。<ref name="Klebaner2005page124">{{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=124}}</ref>该过程也被用于不同的领域，包括大多数自然科学以及社会科学的一些分支，作为各种随机现象的数学模型。<ref name="Steele2012page29"/><ref name="KaratzasShreve2014page47">{{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=47}}</ref><ref name="Wiersema2008page2">{{cite book|author=Ubbo F. Wiersema|title=Brownian Motion Calculus|url=https://books.google.com/books?id=0h-n0WWuD9cC|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-02171-2|page=2}}</ref> − + − ===泊松过程 Poisson process=== − 泊松过程是一个随机过程，有不同的形式和定义。<ref name="Tijms2003page1">{{cite book|author=Henk C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=eBeNngEACAAJ|year=2003|publisher=Wiley|isbn=978-0-471-49881-0|pages=1, 2}}</ref><ref name="DaleyVere-Jones2006chap2">{{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|pages=19–36}}</ref>它可以定义为一个计数过程，它是一个随机过程，表示某个时间点或事件的随机数量。在从零到某个给定时间区间内的过程点的数目是一个泊松随机变量，它取决于该时间和某个参数。该过程以自然数为状态空间，非负数为索引集。此过程也称为泊松计数过程，因为它可以被解释为计数过程的一个示例。<ref name="tijms2303page1">Henk C. Tijms (2003). A First Course in Stochastic Models. Wiley. pp. 1, 2. ISBN 978-0-471-49881-0.</ref>+ + + − 如果一个泊松过程是用一个正常数定义的，那么这个过程称为齐次泊松过程。<ref name="Tijms2003page1"/><ref name="PinskyKarlin2011">{{cite book|author1=Mark A. Pinsky|author2=Samuel Karlin|title=An Introduction to Stochastic Modeling|url=https://books.google.com/books?id=PqUmjp7k1kEC|year=2011|publisher=Academic Press|isbn=978-0-12-381416-6|page=241}}</ref>齐次泊松过程是随机过程的一个重要类，如马尔可夫过程和Lévy过程。<ref name="Applebaum2004page1337"/> + − + − + + − 随机过程被定义为在一个公共概率空间<math>(\Omega, \mathcal{F}, P)</math>上定义的随机变量集合，其中<math>\Omega</math> 是[[样本空间]]，<math>\mathcal{F}</math>是一个<math>\sigma</math>-代数，<math>P</math>是概率测度；而随机变量，由某个集合<math>T</math>索引，所有值都取同一个数学空间<math>S</math>，对于某些<math>\sigma</math>-代数<math>\sigma</math><ref name="Lamperti1977page1"/> − − − 换言之，对于给定的概率空间<math>（\Omega，\mathcal{F}，P）</math>和可测空间<math>（S，Sigma）</math>，随机过程是一个值为<math>S</math>的随机变量的集合，可以写成：<ref name="Florescu2014page293">{{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=293}}</ref> − + 第125行：
第82行： − + − − − 还有其他方法可以考虑随机过程，上面的定义被认为是传统的。<ref name="RogersWilliams2000page121">{{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|pages=121, 122}}</ref><ref name="Asmussen2003page408">{{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=408}}</ref>例如，一个随机过程可以解释或定义为一个<math>S^T</math>值的随机变量，其中<math>S^T</math>是所有可能的<math>S</math>-值函数的空间T</math>从集合<math>T</math>到空间<math>S</math>。<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/> − − − ===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> − ===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>+ − + + − + + − + − + − 第153行：
第100行： − + − − <br> − − ===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"/> − + + − − + − − − 第176行：
第114行： − + − + − − 第188行：
第124行： − + − − 第197行：
第131行： − + − − <br> − + − + − − − 第212行：
第141行： − + − − 对于<math>n</math>级[[笛卡尔幂]]<math>S^n=S\times\dots \times S</math>的任何可测子集<math>C</math>，<math>X</math>的有限维分布可以写成：<ref name="Lamperti1977page1"/> − + 第224行：
第151行： − + − − <br> − − ====稳定性 Stationarity==== − − “稳定性”是当随机过程的所有随机变量都是相同分布时随机过程所具有的数学性质。换言之，如果<math>X</math>是一个平稳随机过程，那么对于任何<math>t\in T</math>，随机变量<math>X_t</math>具有相同的分布，这意味着对于任何一组<math>n</math>索引集值<math>t_1,\dots, t_n</math>而言，对应的<math>n</math>随机变量 − + + 第239行：
第161行： − + − + − + − + + − ====过滤 Filtration====+ − 过滤是定义在某个概率空间中的sigma代数的递增序列和具有某种[[总阶]]关系的索引集，例如在索引集是实数的某个子集的情况下。更为正式的是，如果随机过程有一个指数集总排序的随机过程，则如果随机过程有一个指数集的总序为总序，那么在概率空间<math>(\Omega, \mathcal{F}, P)</math>上的过滤<math>\{\mathcal{F}_t\}_{t\in T} </math> 是一个sigma代数族，使得<math> \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} </math>对所有<math>s \leq t</math>，其中<math>t, s\in T</math>和<math>\leq</math>表示指标集<math>T</math>的总阶<ref name="Florescu2014page294"/>通过过滤的概念，可以研究<math>t\in T</math>中随机过程<math>X_t</math>所包含的信息量，这可以解释为时间<math>t</math><ref name="Florescu2014page294"/><ref name="Williams1991page93"/>过滤背后的直觉是，随着时间的流逝，关于<math>t</math>的更多信息是已知的或可用的，这些信息可以在<math>\mathcal{F}t</math>中获得，使<math>\Omega</math>的分区越来越细。<ref name="Klebaner2005page22">{{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|pages=22–23}}</ref><ref name="MörtersPeres2010page37">{{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=37}}</ref> − − <br> − − − 第265行：
第182行： − + − − − 除了修改，还使用了“版本”一词，<ref name="Adler2010page14"/><ref name="Klebaner2005page48">{{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=48}}</ref><ref name="Øksendal2003page14">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=14}}</ref><ref name="Florescu2014page472">{{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=472}}</ref>然而，当两个随机过程具有相同的有限维分布，但它们可能定义在不同的概率空间上，因此两个过程是相互修改的，在后一种意义上，它们也是彼此的版本，但不是相反。<ref name="RevuzYor2013page18">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|pages=18–19}}</ref><ref name="FrizVictoir2010page571"/> − . − − 如果一个连续时间的实值随机过程在其增量上满足一定的矩条件，则[[Kolmogorov连续性定理]]指出，该过程存在一个修正，其具有概率为1的连续样本路径，因此随机过程有一个连续的修改或版本<ref name="Øksendal2003page14"/><ref name="Florescu2014page472"/><ref name="ApplebaumBook2004page20">{{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=20}}</ref>该定理也可以推广到随机域，因此索引集是<math>n</math>-维欧几里德空间<ref name="Kunita1997page31">{{cite book|author=Hiroshi Kunita|title=Stochastic Flows and Stochastic Differential Equations|url=https://books.google.com/books?id=_S1RiCosqbMC|year=1997|publisher=Cambridge University Press|isbn=978-0-521-59925-2|page=31}}</ref>以及以[[度量空间]]为状态空间的随机过程。<ref name="Kallenberg2002page">{{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|page=35}}</ref> − <br> − + − 两个随机过程<math>X</math>和<math>Y</math>定义在同一概率空间<math>（\Omega，\mathcal{F}，P）</math>上，具有相同的索引集<math>T</math>和集空间<math>S</math>上的两个随机过程如果+ − + + 第286行：
第197行： − + − − − ====可分性 Separability==== − − − “可分性”是随机过程的一种性质，它基于与概率测度有关的指标集。假设随机过程或具有不可数指标集的随机场的泛函可以形成随机变量。对于一个随机过程是可分离的，除了其他条件外，它的指标集必须是一个[[可分离空间]]，{efn |术语“可分离”在这里出现了两次，有两种不同的含义，第一种含义来自概率，第二种含义来自拓扑和分析。对于一个随机过程是可分的（概率意义上），它的指标集必须是一个可分空间（在拓扑或分析意义上），除了其他条件。<ref name="Skorokhod2005page93"/>}}，这意味着索引集有一个稠密的可数子集。<ref name="Adler2010page14"/><ref name="Ito2006page32">{{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|pages=32–33}}</ref> − − 更精确地说，具有概率空间<math>(\Omega,{\cal F},P)</math>的实值连续时间随机过程<math>X</math>是可分离的，如果它的指数集<math>T</math>有一个稠密的可数子集<math>\Omega_0 \subset \Omega</math>，因此<<math>P(\Omega_0)=0</math>，这样对于每个开集<math>G\subset T</math>和每个闭集<math>F\subset \textstyle R =(-\infty,\infty) </math>，<math>\{ X_t \in F \text{ for all } t \in G\cap U\}</math>和<math>\{ X_t \in F \text{ for all } t \in G\}</math>这两个事件最多在<math>\Omega_0</math>的一个子集上不同。<ref name="GikhmanSkorokhod1969page150">{{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=150}}</ref><ref name="Todorovic2012page19">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|pages=19–20}}</ref><ref name="Molchanov2005page340">{{cite book|author=Ilya Molchanov|title=Theory of Random Sets|url=https://books.google.com/books?id=kWEwk1UL42AC|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-892-3|page=340}}</ref>+ − 可分离性的定义（连续时间实值随机过程的可分性定义可以用其他方式表述。<ref name="Billingsley2008page526">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=526–527}}</ref><ref name="Borovkov2013page535">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=535}}</ref>）也可以为其他索引集和状态空间而声明，<ref name="GusakKukush2010page22">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.</ref>例如在随机场的情况下，索引集和状态空间可以是<math>n</math>维欧几里德空间。<ref name="AdlerTaylor2009page7"/><ref name="Adler2010page14"/>+ − 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> − + − 两个在相同的概率空间<math>(\Omega,\mathcal{F},P)</math>上定义，具有相同索引集<math>T</math>的随机过程<math>X</math>和<math>Y</math>被称为“相互独立”，如果对于所有<math>n \in \mathbb{N}</math>，以及每个特定的<math>t_1,\ldots,t_n \in T</math>，随机向量<math>\left( X(t_1),\ldots,X(t_n) \right)</math> 和<math>\left( Y(t_1),\ldots,Y(t_n) \right)</math>是独立的。<ref name=Lapidoth>Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.</ref> + − + − 两个随机过程<math>\left\{X_t\right\}</math>和<math>\left\{Y_t\right\}</math> 称为“不相关的”的，如果它们的互协方差<math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right]</math>始终为零。<ref name=KunIlPark>Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3</ref>最后：+ + + − + − − − − <br>+ − + − − − + + − ====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> − − − − <br>+ − + − − − − − <br> − + − − − 布朗运动过程和泊松过程（一维）都是马尔可夫过程的例子<ref name="Ross1996page235and358">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|pages=235, 358}}</ref>，整数上的[[随机游走]]和[[赌徒破产]]问题是离散时间中马尔可夫过程的例子。<ref name="Florescu2014page373">{{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=373, 374}}</ref><ref name="KarlinTaylor2012page49">{{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=49}}</ref> − − − 马尔可夫链是一种具有离散状态空间或离散索引集（通常表示时间）的马尔可夫过程，但马尔可夫链的精确定义各不相同<ref name="Asmussen2003page7">{{cite book|url=https://books.google.com/books?id=BeYaTxesKy0C|title=Applied Probability and Queues|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=7|author=Søren Asmussen}}</ref>例如，通常将马尔可夫链定义为具有可数状态空间的[[连续变量|离散或连续时间]]中的马尔可夫过程（因此不管时间的性质），<ref name="Parzen1999page188">{{cite book|url=https://books.google.com/books?id=0mB2CQAAQBAJ|title=Stochastic Processes|year=2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|page=188|author=Emanuel Parzen}}</ref><ref name="KarlinTaylor2012page29">{{cite book|url=https://books.google.com/books?id=dSDxjX9nmmMC|title=A First Course in Stochastic Processes|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=29, 30|author1=Samuel Karlin|author2=Howard E. Taylor}}</ref><ref name="Lamperti1977chap6">{{cite book|url=https://books.google.com/books?id=Pd4cvgAACAAJ|title=Stochastic processes: a survey of the mathematical theory|publisher=Springer-Verlag|year=1977|isbn=978-3-540-90275-1|pages=106–121|author=John Lamperti}}</ref><ref name="Ross1996page174and231">{{cite book|url=https://books.google.com/books?id=ImUPAQAAMAAJ|title=Stochastic processes|publisher=Wiley|year=1996|isbn=978-0-471-12062-9|pages=174, 231|author=Sheldon M. Ross}}</ref>但通常将马尔可夫链定义为在可数状态空间或连续状态空间中具有离散时间（因此与状态空间无关），<ref name="Asmussen2003page7" />有人认为，现在倾向于使用具有离散时间的马尔可夫链的第一个定义，尽管Joseph Doob和Kai Lai Chung等研究人员使用了第二个定义。<ref name="MeynTweedie2009">{{cite book|author1=Sean Meyn|author2=Richard L. Tweedie|title=Markov Chains and Stochastic Stability|url=https://books.google.com/books?id=Md7RnYEPkJwC|year=2009|publisher=Cambridge University Press|isbn=978-0-521-73182-9|page=19}}</ref> − 马尔可夫过程是一类重要的随机过程，在许多领域有着广泛的应用。<ref name="LatoucheRamaswami1999"/><ref name="KarlinTaylor2012page47">{{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=47}}</ref>例如，它们是一种称为[[Markov chain Monte Carlo]]的一般随机模拟方法的基础，该方法用于模拟具有特定概率分布的随机对象，并在贝叶斯统计中得到应用。<ref name="RubinsteinKroese2011page225">{{cite book|author1=Reuven Y. Rubinstein|author2=Dirk P. Kroese|title=Simulation and the Monte Carlo Method|url=https://books.google.com/books?id=yWcvT80gQK4C|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-21052-9|page=225}}</ref><ref name="GamermanLopes2006">{{cite book|author1=Dani Gamerman|author2=Hedibert F. Lopes|title=Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition|url=https://books.google.com/books?id=yPvECi_L3bwC|year=2006|publisher=CRC Press|isbn=978-1-58488-587-0}}</ref>+ − 马尔可夫特性的概念最初是针对连续和离散时间的随机过程，但它也适用于其它指标集，如<math>n</math>维欧氏空间，这导致随机变量的集合被称为马尔可夫随机场。<ref name="Rozanov2012page61">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=61}}</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=27}}</ref><ref name="Bremaud2013page253">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=253}}</ref>+ − + − '''鞅 Martingale'''是一个离散时间或连续时间的随机过程，其性质是在给定过程的当前值和所有过去值的情况下，每个未来值的条件期望值等于当前值。在离散时间中，如果此属性适用于下一个值，则它适用于所有未来值。鞅的精确数学定义需要另外两个条件与过滤的数学概念相结合，这与随时间推移增加可用信息的直觉有关。鞅通常被定义为实值，<ref name="Klebaner2005page65">{{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=65}}</ref><ref name="KaratzasShreve2014page11">{{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=11}}</ref><ref name="Williams1991page93">{{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|pages=93, 94}}</ref> but they can also be complex-valued<ref name="Doob1990page292">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref>或更一般的。<ref name="Pisier2016">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref> + − 对称随机游动和Wiener过程（具有零漂移）分别是离散时间和连续时间的鞅的例子。<ref name="Klebaner2005page65"/><ref name="KaratzasShreve2014page11"/>对于一个[[独立且同分布]]随机变量的[[序列]]<math>X_1, X_2, X_3, \dots</math>且平均值为零，由连续部分和<math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> 构成的随机过程是一个离散时间鞅<ref name="Steele2012page12">{{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|pages=12, 13}}</ref>，离散时间鞅推广了独立随机变量的部分和的概念。<ref name="HallHeyde2014page2">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</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> + − 数学上的鞅形式化了公平博弈的概念，<ref name="Ross1996page295">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=295}}</ref>它们最初的开发目的是表明不可能赢得一场公平的比赛。<ref name="Steele2012page11"/>但现在它们被用于许多概率领域，这是研究它们的主要原因之一。<ref name="Williams1991page93"/><ref name="Steele2012page11">{{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=11}}</ref><ref name="Kallenberg2002page96">{{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=96}}</ref>许多概率问题已经通过在问题中找到鞅并加以研究而得到解决。<ref name="Steele2012page371">{{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=371}}</ref>在给定鞅矩的条件下，鞅会收敛，因此经常使用鞅得到收敛结果，这主要是由于[[鞅收敛定理]]s。<ref name="HallHeyde2014page2"/><ref name="Steele2012page22">{{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=22}}</ref><ref name="GrimmettStirzaker2001page336">{{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=336}}</ref> + − 鞅在统计学中有许多应用，但有人指出，它的使用和应用并不像它在统计学领域那样广泛，尤其是统计推断。<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref>他们在排队论和棕榈微积分等概率论领域找到了应用<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref>以及其他领域，如经济学。<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/> + − + − + − − 第388行：
第273行： − + − + − + − + − + − − − − − − − − − − − 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> − + + − 概率论起源于机会博弈，它有着悠久的历史，有些游戏在几千年前就已经开始了，<ref name=":1">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|location=US|pages=1–2}}</ref><ref name="David1955">{{cite journal|last1=David|first1=F. N.|title=Studies in the History of Probability and Statistics I. Dicing and Gaming (A Note on the History of Probability)|journal=Biometrika|volume=42|issue=1/2|pages=1–15|year=1955|issn=0006-3444|doi=10.2307/2333419|jstor=2333419}}</ref>但很少从概率的角度对其进行分析。<ref name=":1" /><ref name="Maistrov2014page1">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=1}}</ref>在概率论上有过书面通信时，1654年通常被认为是概率论的诞生，受[[点数问题|赌博问题].<ref name=":1" /><ref name="Seneta2006page1">{{cite book|last1=Seneta|first1=E.|title=Encyclopedia of Statistical Sciences|chapter=Probability, History of|year=2006|doi=10.1002/0471667196.ess2065.pub2|page=1|isbn=978-0471667193}}</ref><ref name="Tabak2014page24to26">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|pages=24–26}}</ref>但是早期有关于赌博游戏概率的数学研究，比如[[Gerolamo Cardano]]的“Liber de Ludo Aleae”，16世纪写于16世纪，死后于1663年发表。<ref name=":1" /><ref name="Bellhouse2005">{{cite journal|last1=Bellhouse|first1=David|title=Decoding Cardano's Liber de Ludo Aleae|journal=Historia Mathematica|volume=32|issue=2|year=2005|pages=180–202|issn=0315-0860|doi=10.1016/j.hm.2004.04.001|doi-access=free}}</ref> + + + − 继Cardano之后，[[雅各布·伯努利]] Jakob Bernoulli（也被称为James Bernoulli或Jacques Bernoulli<ref name="Hald2005page221">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=221}}</ref>）写了[[魔术师Ars conjuctandi]]，在概率论史上被认为是重大事件。<ref name=":1" />伯努利的书出版于1713年，也是在他死后出版的，激发了许多数学家研究概率。<ref name=":1" /><ref name="Maistrov2014page56">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=56}}</ref><ref name="Tabak2014page37">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|page=37}}</ref> − 但尽管一些著名的数学家对概率论做出了贡献，比如[[皮埃尔-西蒙-拉普拉斯]、[[亚伯拉罕-德-莫伊夫]]、[[卡尔-高斯]]、[[西蒙-泊阿松]] Siméon Poisson和[[帕夫努蒂·切比雪夫]] Pafnuty Chebyshev，<ref name="Chung1998">{{cite journal|last1=Chung|first1=Kai Lai|title=Probability and Doob|journal=The American Mathematical Monthly|volume=105|issue=1|pages=28–35|year=1998|issn=0002-9890|doi=10.2307/2589523|jstor=2589523}}</ref><ref name="Bingham2000">{{cite journal|last1=Bingham|first1=N.|title=Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov|journal=Biometrika|volume=87|issue=1|year=2000|pages=145–156|issn=0006-3444|doi=10.1093/biomet/87.1.145}}</ref>指导20世纪，大多数数学界人士（一个显著的例外是俄罗斯的圣彼得堡学派，在那里，以切比雪夫为首的数学家研究概率论<ref name="BenziBenzi2007">{{cite journal|last1=Benzi|first1=Margherita|last2=Benzi|first2=Michele|last3=Seneta|first3=Eugene|title=Francesco Paolo Cantelli. b. 20 December 1875 d. 21 July 1966|journal=International Statistical Review|volume=75|issue=2|year=2007|page=128|issn=0306-7734|doi=10.1111/j.1751-5823.2007.00009.x}}</ref>）才认为概率论是数学的一部分。<ref name="Chung1998"/><ref name="BenziBenzi2007"/><ref name="Doob1996">{{cite journal|last1=Doob|first1=Joseph L.|title=The Development of Rigor in Mathematical Probability (1900-1950)|journal=The American Mathematical Monthly|volume=103|issue=7|pages=586–595|year=1996|issn=0002-9890|doi=10.2307/2974673|jstor=2974673}}</ref><ref name="Cramer1976">{{cite journal|last1=Cramer|first1=Harald|title=Half a Century with Probability Theory: Some Personal Recollections|journal=The Annals of Probability|volume=4|issue=4|year=1976|pages=509–546|issn=0091-1798|doi=10.1214/aop/1176996025|doi-access=free}}</ref> − + + + − ===统计力学 Statistical mechanics=== − 在物理科学中，科学家们在19世纪发展了[[统计力学]]这门学科，在这个学科中，物理系统，例如装满气体的容器，可以从数学上看作或处理为许多运动粒子的集合。尽管有些科学家试图将随机性纳入统计物理学，比如[[鲁道夫·克劳修斯]]，大部分工作没有或几乎没有随机性。<ref name="Truesdell1975page22">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=22–23|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1967page150">{{cite journal|last1=Brush|first1=Stephen G.|title=Foundations of statistical mechanics 1845?1915|journal=Archive for History of Exact Sciences|volume=4|issue=3|year=1967|pages=150–151|issn=0003-9519|doi=10.1007/BF00412958}}</ref>+ + + − 这种情况在1859年发生了变化，当时[[詹姆斯·克拉克·麦克斯韦]] James Clerk Maxwell对该领域做出了重大贡献，更具体地说，是对气体动力学理论的贡献，通过介绍他的工作，他假设气体粒子以随机速度随机方向移动<ref name="Truesdell1975page31">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=31–32|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1958">{{cite journal|last1=Brush|first1=S.G.|title=The development of the kinetic theory of gases IV. Maxwell|journal=Annals of Science|volume=14|issue=4|year=1958|pages=243–255|issn=0003-3790|doi=10.1080/00033795800200147}}</ref>气体动力学理论和统计物理在19世纪后半叶继续发展，主要由克劳修斯，[[路德维希·玻尔兹曼]]和[[约西亚·威拉德·吉布斯]]完成，这项工作后来对[[阿尔伯特·爱因斯坦]]关于[[布朗运动]]的数学模型产生了影响。<ref name="Brush1968page15">{{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=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>+ − + + + − 1900年在巴黎举行的国际数学家大会上，David Hilbert提出了一份[[Hilbert问题|数学问题]]的清单，其中他的第六个问题要求对涉及公理的物理和概率进行数学处理。<ref name="Bingham2000"/>大约在20世纪初，数学家发展了测量理论，这是研究数学函数积分的数学分支，其中两位创始人是法国数学家Henri Lebesgue和Émile Borel。1925年，另一位法国数学家Paul Lévy出版了第一本使用测度论思想的概率论书籍。<ref name="Bingham2000"/> + − 20世纪20年代，苏联的数学家们对概率论做出了重大贡献，比如Sergei Bernstein，Aleksandr Khinchin，（Khinchin这个名字也用英语写成（或音译成）Khintchine。<ref name="Doob1934">{{cite journal|last1=Doob|first1=Joseph|title=Stochastic Processes and Statistics|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=20|issue=6|year=1934|pages=376–379|doi=10.1073/pnas.20.6.376|pmid=16587907|pmc=1076423|bibcode=1934PNAS...20..376D}}</ref>）和Andrei Kolmogorov。<ref name="Cramer1976"/> Kolmogorov于1984年发表了基于测量理论的数学基础的首次尝试。概率论的概率论。<ref name="KendallBatchelor1990page33">{{cite journal|last1=Kendall|first1=D. G.|last2=Batchelor|first2=G. K.|last3=Bingham|first3=N. H.|last4=Hayman|first4=W. K.|last5=Hyland|first5=J. M. E.|last6=Lorentz|first6=G. G.|last7=Moffatt|first7=H. K.|last8=Parry|first8=W.|last9=Razborov|first9=A. A.|last10=Robinson|first10=C. A.|last11=Whittle|first11=P.|title=Andrei Nikolaevich Kolmogorov (1903–1987)|journal=Bulletin of the London Mathematical Society|volume=22|issue=1|year=1990|page=33|issn=0024-6093|doi=10.1112/blms/22.1.31}}</ref>在20世纪30年代初，Khinchin和Kolmogorov在20世纪30年代初建立了概率研讨会，这些研讨会由研究者参加，如Eugene Slutsky等和Nikolai Smirnov，<ref name="Vere-Jones2006page1">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=1|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>还有Khinchin给出了第一个随机变量的数学定义，把随机过程作为以实数线索引的一组随机变量。<ref name="Doob1934"/><ref name="Vere-Jones2006page4">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=4|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>（Doob在引用Khinchin时，使用了“机会变量”这个词，它曾经是“随机变量”的替代词。<ref name="Snell2005">{{cite journal|last1=Snell|first1=J. Laurie|title=Obituary: Joseph Leonard Doob|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|page=251|issn=0021-9002|doi=10.1239/jap/1110381384|doi-access=free}}</ref>） − − − ===现代概率论的诞生 Birth of modern probability theory=== − 1933年，Andrei Kolmogorov在德国出版了一本关于概率论基础的书，名为“概率计算的基本概念”，后来翻译成英文，1950年出版，作为概率论的基础。这本书的出版现在被广泛认为是现代概率论的诞生，当时概率论和随机过程理论成为数学的一部分。<ref name="Bingham2000"/><ref name="Cramer1976"/>+ − 在Kolmogorov的书出版后，钦钦和科尔莫戈洛夫以及其他数学家如Joseph Doob、William Feller、Maurice Fréchet、Paul Lévy、Wolfgang Doeblin等对概率论和随机过程进行了进一步的基础性工作，和Harald Cramér。<ref name="Bingham2000"/><ref name="Cramer1976"/>+ − 几十年后，Cramér把20世纪30年代称为“数学概率论的英雄时期”。<ref name="Cramer1976"/>第二次世界大战极大地中断了概率论的发展，例如，Feller从瑞典迁移到美国。<ref name="Lindvall1991">{{cite journal|last1=Lindvall|first1=Torgny|title=W. Doeblin, 1915-1940|journal=The Annals of Probability|volume=19|issue=3|year=1991|pages=929–934|issn=0091-1798|doi=10.1214/aop/1176990329|doi-access=free}}</ref>+ + + − [[File:Joseph_Doob.jpg|thumb|right|数学家[[Joseph Doob]]在随机过程理论方面做了早期的工作，做出了基本贡献，尤其是在鞅理论方面。<ref name="Getoor2009">{{cite journal|last1=Getoor|first1=Ronald|title=J. L. Doob: Foundations of stochastic processes and probabilistic potential theory|journal=The Annals of Probability|volume=37|issue=5|year=2009|page=1655|issn=0091-1798|doi=10.1214/09-AOP465|arxiv=0909.4213|bibcode=2009arXiv0909.4213G|s2cid=17288507}}</ref><ref name="Snell2005"/>他的书《随机过程》被认为在概率论领域具有很高的影响力。<ref name="Bingham2005"/>]]+ + − + + − ===二战后的随机过程 Stochastic processes after World War II===+ − 第二次世界大战后，概率论和随机过程的研究得到了数学家的更多关注，在概率论和数学的许多领域做出了重大贡献，并开创了新的领域统计学<ref name="Cramer1976"/><ref name="Meyer2009">{{cite journal|last1=Meyer|first1=Paul-André|title=Stochastic Processes from 1950 to the Present|journal=Electronic Journal for History of Probability and Statistics|volume=5|issue=1|year=2009|pages=1–42}}</ref>从20世纪40年代开始，[[Kiyosi Itô]]发表了发展[[随机微积分]]领域的论文，它包括基于维纳或布朗运动过程的随机积分和随机[[微分方程]]。<ref name="Ito1998Prize">{{cite journal|title=Kiyosi Itô receives Kyoto Prize|journal=Notices of the AMS|volume=45|issue=8|year=1998|pages=981–982}}</ref>+ + + − 同样从20世纪40年代开始，随机过程（尤其是鞅）与[[势理论]]的数学领域之间建立了联系，Shizuo Kakutani的早期思想和Joseph Doob后来的工作。<ref name="Meyer2009"/>在1950年代[[Gilbert Hunt]]完成了被认为是开创性的进一步工作，把马尔可夫过程和势理论联系起来，这对Lévy过程理论产生了重大影响，并使人们对用Itô开发的方法研究马尔可夫过程产生了更多的兴趣<ref name="JarrowProtter2004"/><ref name="Bertoin1998pageVIIIandIX">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii and ix}}</ref><ref name="Steele2012page176">{{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=176}}</ref>+ + − 1953年，Joseph Doob出版了《随机过程 Stochastic processes》一书，这本书对随机过程理论产生了重大影响，并强调了测度理论在概率论中的重要性。<ref name="Meyer2009"/>+ + + − <ref name="Bingham2005">{{cite journal|last1=Bingham|first1=N. H.|title=Doob: a half-century on|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|pages=257–266|issn=0021-9002|doi=10.1239/jap/1110381385|doi-access=free}}</ref>Doob还主要发展了鞅理论，后来Paul-André Meyer也作出了重大贡献。早期的研究是由Sergei Bernstein、Paul Lévy和Jean Ville进行的，后者采用了随机过程的鞅项。<ref name="HallHeyde2014page1">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|pages=1, 2}}</ref><ref name="Dynkin1989">{{cite journal|last1=Dynkin|first1=E. B.|title=Kolmogorov and the Theory of Markov Processes|journal=The Annals of Probability|volume=17|issue=3|year=1989|pages=822–832|issn=0091-1798|doi=10.1214/aop/1176991248|doi-access=free}}</ref>鞅理论中的方法已成为解决各种概率问题的常用方法。研究马尔可夫过程的技术和理论发展到鞅上。相反地，从鞅理论中也建立了处理Markov过程的方法。<ref name="Meyer2009"/>+ + − 概率的其他领域也被发展和用于研究随机过程，其中一个主要方法是大偏差理论。<ref name="Meyer2009"/> 该理论在统计物理等领域有许多应用，其核心思想至少可以追溯到20世纪30年代。20世纪60年代和70年代后期，苏联的亚历山大·温策尔和美利坚合众国的[[Monroe D.Donsker]]和[[Srinivasa Varadhan]]完成了基础工作，<ref name="Ellis1995page98">{{cite journal|last1=Ellis|first1=Richard S.|title=An overview of the theory of large deviations and applications to statistical mechanics|journal=Scandinavian Actuarial Journal|volume=1995|issue=1|year=1995|page=98|issn=0346-1238|doi=10.1080/03461238.1995.10413952}}</ref>这将使瓦拉丹获得2007年阿贝尔奖。<ref name="RaussenSkau2008">{{cite journal|last1=Raussen|first1=Martin|last2=Skau|first2=Christian|title=Interview with Srinivasa Varadhan|journal=Notices of the AMS|volume=55|issue=2|year=2008|pages=238–246}}</ref>上世纪90年代和2000年代的理论[[施拉姆–Loewner演化]]]<ref name="HenkelKarevski2012page113">{{cite book|author1=Malte Henkel|author2=Dragi Karevski|title=Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution|url=https://books.google.com/books?id=fnCQWd0GEZ8C&pg=PA113|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-27933-1|page=113}}</ref>和[[粗略路径]]<ref name="FrizVictoir2010page571">{{cite book|author1=Peter K. Friz|author2=Nicolas B. Victoir|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|url=https://books.google.com/books?id=CVgwLatxfGsC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48721-4|page=571}}</ref>被引入和发展来研究概率论中的随机过程和其他数学对象，分别在2008年和2014年分别授予Wendelin Werner<ref name="Werner2004Fields">{{cite journal|title=2006 Fields Medals Awarded|journal=Notices of the AMS|volume=53|issue=9|year=2015|pages=1041–1044}}</ref>和Martin Hairer菲尔兹奖。<ref name="Hairer2004Fields">{{cite journal|last1=Quastel|first1=Jeremy|title=The Work of the 2014 Fields Medalists|journal=Notices of the AMS|volume=62|issue=11|year=2015|pages=1341–1344}}</ref>+ + + − 随机过程理论仍然是研究的焦点，每年都有关于随机过程的国际会议。<ref name="BlathImkeller2011"/><ref name="Applebaum2004page1336"/>+ + + − + − − − 第483行：
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|description=被广泛用作以随机方式变化的系统和现象的数学模型
|description=被广泛用作以随机方式变化的系统和现象的数学模型
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}}[[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="doob1953stochasticP46to472">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=7Bu8jgEACAAJ|year=1990|publisher=Wiley|pages=46, 47}}</ref><ref name="RogersWilliams2000page12">{{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="Steele2012page292">{{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=29}}</ref>|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:BMonSphere.jpg]]
在[https://wiki.swarma.org/index.php/%E6%A6%82%E7%8E%87%E8%AE%BA 概率论]及相关领域中，'''随机过程 stochastic process'''（或random process）是一个数学对象，通常被定义为随机变量的集合。给出对一个随机过程的解释，该过程表示某个系统随机的数值随时间的变化，例如细菌种群的增长，电流由于热噪声而波动，或者一个气体分子的运动。<ref name="doob1953stochasticP46to472" /><ref name="Parzen19992">{{cite book|author=Emanuel Parzen|title=Stochastic Processes|url=https://books.google.com/books?id=0mB2CQAAQBAJ|year= 2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|pages=7, 8}}</ref><ref name="GikhmanSkorokhod1969page12">{{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=q0lo91imeD0C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=1}}</ref><ref name=":02">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|location= NJ|pages=1–235}}</ref>随机过程被广泛用作以随机方式变化的系统和现象的数学模型。它们在许多学科都有应用，比如生物学<ref name="Bressloff20142">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ|year=2014|publisher=Springer|isbn=978-3-319-08488-6}}</ref>，化学 <ref name="Kampen20112">{{cite book|author=N.G. Van Kampen|title=Stochastic Processes in Physics and Chemistry|url=https://books.google.com/books?id=N6II-6HlPxEC|year=2011|publisher=Elsevier|isbn=978-0-08-047536-3}}</ref> 生态学,<ref name="LandeEngen20032">{{cite book|author1=Russell Lande|author2=Steinar Engen|author3=Bernt-Erik Sæther|title=Stochastic Population Dynamics in Ecology and Conservation|url=https://books.google.com/books?id=6KClauq8OekC|year=2003|publisher=Oxford University Press|isbn=978-0-19-852525-7}}</ref> 神经科学<ref name="LaingLord20102">{{cite book|author1=Carlo Laing|author2=Gabriel J Lord|title=Stochastic Methods in Neuroscience|url=https://books.google.com/books?id=RaYSDAAAQBAJ|year=2010|publisher=OUP Oxford|isbn=978-0-19-923507-0}}</ref>, 物理学<ref name="PaulBaschnagel20132">{{cite book|author1=Wolfgang Paul|author2=Jörg Baschnagel|title=Stochastic Processes: From Physics to Finance|url=https://books.google.com/books?id=OWANAAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-319-00327-6}}</ref>, 图像处理, 信号处理,<ref name="Dougherty19992">{{cite book|author=Edward R. Dougherty|title=Random processes for image and signal processing|url=https://books.google.com/books?id=ePxDAQAAIAAJ|year=1999|publisher=SPIE Optical Engineering Press|isbn=978-0-8194-2513-3}}</ref> 控制理论, <ref name="Bertsekas19962">{{cite book|author=Dimitri P. Bertsekas|title=Stochastic Optimal Control: The Discrete-Time Case|url=http://www.athenasc.com/socbook.html|year=1996|publisher=Athena Scientific]|isbn=1-886529-03-5}}</ref> [https://wiki.swarma.org/index.php/%E4%BF%A1%E6%81%AF%E8%AE%BA 信息论],<ref name="CoverThomas2012page712">{{cite book|author1=Thomas M. Cover|author2=Joy A. Thomas|title=Elements of Information Theory|url=https://books.google.com/books?id=VWq5GG6ycxMC=PT16|year=2012|publisher=John Wiley & Sons|isbn=978-1-118-58577-1|page=71}}</ref> 计算机科学,<ref name="Baron20152">{{cite book|author=Michael Baron|title=Probability and Statistics for Computer Scientists, Second Edition|url=https://books.google.com/books?id=CwQZCwAAQBAJ|year=2015|publisher=CRC Press|isbn=978-1-4987-6060-7|page=131}}</ref> 密码学<ref>{{cite book|author1=Jonathan Katz|author2=Yehuda Lindell|title=Introduction to Modern Cryptography: Principles and Protocols|url=https://archive.org/details/Introduction_to_Modern_Cryptography|year=2007|publisher=CRC Press|isbn=978-1-58488-586-3|page=[https://archive.org/details/Introduction_to_Modern_Cryptography/page/n44 26]}}</ref> 和 电信学.<ref name="BaccelliBlaszczyszyn20092">{{cite book|author1=François Baccelli|author2=Bartlomiej Blaszczyszyn|title=Stochastic Geometry and Wireless Networks|url=https://books.google.com/books?id=H3ZkTN2pYS4C|year=2009|publisher=Now Publishers Inc|isbn=978-1-60198-264-3}}</ref> 此外，金融市场中看似随机的变化促进了随机过程在金融领域的广泛应用。<ref name="Steele20012">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=H06xzeRQgV4C|year=2001|publisher=Springer Science & Business Media|isbn=978-0-387-95016-7}}</ref><ref name="MusielaRutkowski20062">{{cite book|author1=Marek Musiela|author2=Marek Rutkowski|title=Martingale Methods in Financial Modelling|url=https://books.google.com/books?id=iojEts9YAxIC|year= 2006|publisher=Springer Science & Business Media|isbn=978-3-540-26653-2}}</ref><ref name="Shreve20042">{{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}}</ref>
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="GikhmanSkorokhod19692">{{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="Rosenblatt19622">{{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="JarrowProtter20042">{{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="Stirzaker20002">{{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="doob1953stochasticP46to472" /><ref name="Parzen19992" /><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="JarrowProtter20042" /><ref name="GuttorpThorarinsdottir20122">{{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="GusakKukush2010page212">{{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="Skorokhod2005page422">{{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="Kallenberg2002page242">{{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="Lamperti1977page12">{{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="Kallenberg2002page242" /><ref name="ChaumontYor20122">{{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="GikhmanSkorokhod1969page12" /><ref name="ChaumontYor20122" />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="GikhmanSkorokhod1969page12" /><ref name="AdlerTaylor2009page72">{{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="GikhmanSkorokhod1969page12" /><ref name="Lamperti1977page12" />
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="LawlerLimic20102">{{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="Williams19912">{{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="RogersWilliams20002">{{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="ApplebaumBook20042">{{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="Adler20102">{{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="KarlinTaylor20122">{{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="Hajek20152">{{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="LatoucheRamaswami19992">{{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-Jones20072">{{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="Applebaum20042">{{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="BlathImkeller20112">{{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="Talagrand20142">{{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="Bressloff2014VII2">{{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="Parzen19992" /><ref name="GikhmanSkorokhod1969page12" />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="doob1953stochasticP46to472" /> 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="doob1953stochasticP46to472" /> '''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="KarlinTaylor2012page272">{{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 name="Applebaum2004page13372">{{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>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="Lamperti1977page12" /><ref name="RogersWilliams2000page121b2">{{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|pages=121–124}}</ref>[[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|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:Wiener_process_3d.png]]
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]]
===分类===
===分类===
随机过程可以用不同的方法进行分类，例如，根据其状态空间、索引集或随机变量之间的相关性。一种常见的分类方法是通过索引集和状态空间的基数进行分类。<ref name="Florescu2014page2942">{{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="KarlinTaylor2012page262">{{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=26}}</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|pages=24, 25}}</ref>
随机过程可以用不同的方法进行分类，例如，根据其状态空间、索引集或随机变量之间的相关性。一种常见的分类方法是通过索引集和状态空间的基数进行分类。<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26">{{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=26}}</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|pages=24, 25}}</ref>
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="Billingsley2008page4822">{{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><ref name="Borovkov2013page5272">{{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="KarlinTaylor2012page272" /><ref name="Brémaud2014page1202">{{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><ref name="Rosenthal2006page1772">{{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="KloedenPlaten2013page632">{{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="Khoshnevisan2006page1532">{{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="Borovkov2013page5272" />
<br>
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="florescu214page2942">Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 294, 295. ISBN 978-1-118-59320-2.</ref><ref name="KarlinTaylor2012page262" />
===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“瞄准一个目标，猜测”，而牛津英语词典将1662年作为其最早出现的年份。<ref name="OxfordStochastic2">{Cite OED | random}</ref>雅各布·伯努利 Jakob Bernoulli在他关于概率的著作《猜想的艺术》（Ars conquectandi）中使用了“猜想的艺术或随机的艺术”（“Ars Conjectandi sive Stochastice”）这个短语，该著作最初于1713年以拉丁文出版。<ref name="Sheĭnin2006page52">{{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>在提到伯努利时，这一短语被拉迪斯劳斯-博特凯维茨 Ladislaus Bortkiewicz 使用，他在1917用德语stochastic表示“随机”的意思。<ref name="SheyninStrecker2011page1362">{{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 Joseph Doob 1934年的一篇论文中。<ref name="OxfordStochastic2" /> 对于该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 Aleksandr Khinchin用德语使用了"''stochastischer Prozeß'' ”一词，<ref name="Doob19342">{{cite journal|last1=Doob|first1=Joseph|title=Stochastic Processes and Statistics|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=20|issue=6|year=1934|pages=376–379|doi=10.1073/pnas.20.6.376|pmid=16587907|pmc=1076423|bibcode=1934PNAS...20..376D}}</ref><ref name="Khintchine19342">{{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>尽管这个德语术语在早些时候就被使用过，例如，安德烈-科尔莫戈罗夫 Andrei Kolmogorov在1931年就使用了它。<ref name="Kolmogoroff1931page12">{{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。
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。
根据《牛津英语词典》，“随机”一词在英语中的早期出现及其目前的含义——即与机会或运气有关，可以追溯到16世纪，而更早的使用记录始于14世纪，是一个名词，意思是 "急躁、巨大的速度、力量或暴力（在骑马、跑步、击球等方面）"。这个词本身来自一个中古法语单词，意思是 "速度、匆忙"，它可能是从法语动词“奔跑”或“飞奔”衍生而来。随机过程（random process）一词的首次书面出现早于随机过程（stochastic process），《牛津英语词典》也将其作为同义词，并被弗朗西斯·艾其沃斯 Francis Edgeworth 于1888年发表的一篇文章中使用。<ref name="OxfordRandom2">{Cite OED | random}</ref>
===术语===
===术语===
随机过程的定义各不相同<ref name="FristedtGray2013page5802">{{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> ，但是传统上随机过程被定义为由某个集合索引的随机变量的集族<ref name="RogersWilliams2000page1212">{{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|pages=121, 122}}</ref><ref name="Asmussen2003page4082">{{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=408}}</ref>。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="Kallenberg2002page242" /><ref name="ChaumontYor20122" /><ref name="AdlerTaylor2009page72" /><ref name="Stirzaker2005page452">{{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="Rosenblatt1962page912">{{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="Gubner2006page3832">{{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>。 "集合"<ref name="Lamperti1977page12" /><ref name="Stirzaker2005page452" /> 和"族"<ref name="Parzen19992" /><ref name="Ito2006page132">{{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>都会使用，而有时则用 "参数集"<ref name="Lamperti1977page12" /> 或 "参数空间"<ref name="AdlerTaylor2009page72" />来代替 "索引集"。
术语“随机函数”也用于指代随机过程（stochastic process）或随机过程（random process），<ref name="GikhmanSkorokhod1969page12" /><ref name="Loeve19782">{{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="Brémaud2014page1332">{{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="Lamperti1977page12" /><ref name="Ito2006page132" />当索引集是实数以外的数学空间时，也使用这个术语<ref name="GikhmanSkorokhod1969page12" /><ref name="GusakKukush2010page12">p. 1</ref>，而当索引集被解释为时间时，通常使用“随机过程”（stochastic process）和“随机过程”（random process），<ref name="GikhmanSkorokhod1969page12" /><ref name="GusakKukush2010page12" /><ref name="Bass2011page12">{{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>另外当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或流形时，会用随机场这一术语。<ref name="GikhmanSkorokhod1969page12" /><ref name="Lamperti1977page12" /><ref name="AdlerTaylor2009page72" />
===记号===
===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>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"/>
随机过程可以用<math>\{X(t)\}_{t\in T} </math>，<ref name="Brémaud2014page1202" /> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page4082" /> <math>\{X_t\}</math><ref name="Lamperti1977page32">,{{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>被视为函数表示法的滥用。<ref name="Klebaner2005page552">{{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> 比如， <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="Lamperti1977page32" />如果索引集是<math>T=[0,\infty)</math>，那么我们可以这样写，例如用<math>(X_t , t \geq 0)</math>来表示随机过程。<ref name="ChaumontYor20122" />
==示例==
==示例==
===伯努利过程===
===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>
最简单的随机过程之一是伯努利过程，<ref name="Florescu2014page2932">{{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=293}}</ref>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="Florescu2014page3012">{{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="BertsekasTsitsiklis2002page2732">{{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[https://wiki.swarma.org/index.php/%E4%BC%AF%E5%8A%AA%E5%88%A9%E8%AF%95%E9%AA%8C 伯努利试验]的一个例子。<ref name="Ibe2013page112">{{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>
===随机游走===
===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="Klenke2013page3472">{{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="LawlerLimic2010page12">{{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="Kallenberg2002page1362">{{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="Florescu2014page3832">{{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="Durrett2010page2772">{{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”2"><nowiki>{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}}</nowiki></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="Spanos1999page4542">{{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="Weiss2006page12">Weiss, George H. (2006). "Random Walks". Encyclopedia of Statistical Sciences. p. 1. doi:10.1002/0471667196.ess2180.pub2. ISBN 978-0471667193.</ref><ref name="Klebaner2005page812">{{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="Gut2012page882">{{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="GrimmettStirzaker2001page712">{{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>
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>
===维纳过程===
===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="RogersWilliams2000page12" /><ref name="Klebaner2005page562">{{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="Brush1968page12">{{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="Applebaum2004page13382">{{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="Applebaum2004page13382" /><ref name="GikhmanSkorokhod1969page212">{{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>）。|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:%E7%BB%B4%E7%BA%B3%E8%BF%87%E7%A8%8B.png]]''' 维纳过程'''在概率论中起着中心作用，通常被认为是最重要和研究的随机过程，并与其他随机过程联系在一起<ref name="doob1953stochasticP46to472" /><ref name="RogersWilliams2000page12" /><ref name="Steele2012page292" /><ref name="Florescu2014page4712">{{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=471}}</ref><ref name="KarlinTaylor2012page212">{{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=21, 22}}</ref><ref name="KaratzasShreve2014pageVIII2">{{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=VIII}}</ref><ref name="RevuzYor2013pageIX2">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=IX}}</ref>。其索引集和状态空间分别是非负数和实数，因此它既有连续索引集又有状态空间<ref name="Rosenthal2006page1862">{{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|page=186}}</ref>但是过程可以定义得更广泛，这样它的状态空间可以是维欧几里德空间。<ref name="Klebaner2005page812" /><ref name="KarlinTaylor2012page212" /><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=33}}</ref>如果任何增量的平均值为零，则所得到的维纳或布朗运动过程称为零漂移。如果任意两个时间点的增量的平均值等于时间差乘以某个常数<math>\mu</math>，即实数，由此产生的随机过程被称为'''漂移'''。<ref name="Steele2012page1182">{{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=118}}</ref><ref name="MörtersPeres2010page12">{{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><ref name="KaratzasShreve2014page782">{{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=78}}</ref>
<br>
几乎可以肯定，维纳过程的样本路径处处连续，但无处可微。它可以看作是简单随机游走的一个连续版本。<ref name="Applebaum2004page13372" /><ref name="MörtersPeres2010page12" />当其他随机过程（如某些随机游动重新缩放）的数学极限时，该过程出现，<ref name="KaratzasShreve2014page612">{{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="Shreve2004page932">{{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>这是唐斯克定理或不变性原理的主题，也被称为'''函数中心极限定理'''。<ref name="Kallenberg2002page225and2602">{{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="KaratzasShreve2014page702">{{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="MörtersPeres2010page1312">{{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>
维纳过程是一些重要的随机过程家族的成员，包括马尔可夫过程，莱维过程和高斯过程。<ref name="RogersWilliams2000page12" /><ref name="Applebaum2004page13372" />该过程也有许多应用，是随机微积分中使用的主要随机过程。<ref name="Klebaner20052">{{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}}</ref><ref name="KaratzasShreve2014page2">{{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}}</ref>它在数量金融中起着核心作用，<ref name="Applebaum2004page13412">{{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=1341}}</ref><ref name="KarlinTaylor2012page3402">{{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=340}}</ref>在Black-Scholes-Merton模型中使用它。<ref name="Klebaner2005page1242">{{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=124}}</ref>该过程也被用于不同的领域，包括大多数自然科学以及社会科学的一些分支，作为各种随机现象的数学模型。<ref name="Steele2012page292" /><ref name="KaratzasShreve2014page472">{{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=47}}</ref><ref name="Wiersema2008page22">{{cite book|author=Ubbo F. Wiersema|title=Brownian Motion Calculus|url=https://books.google.com/books?id=0h-n0WWuD9cC|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-02171-2|page=2}}</ref>
===泊松过程===
泊松过程是一个随机过程，有不同的形式和定义。<ref name="Tijms2003page12">{{cite book|author=Henk C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=eBeNngEACAAJ|year=2003|publisher=Wiley|isbn=978-0-471-49881-0|pages=1, 2}}</ref><ref name="DaleyVere-Jones2006chap22">{{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|pages=19–36}}</ref>它可以定义为一个计数过程，它是一个随机过程，表示某个时间点或事件的随机数量。在从零到某个给定时间区间内的过程点的数目是一个泊松随机变量，它取决于该时间和某个参数。该过程以自然数为状态空间，非负数为索引集。此过程也称为泊松计数过程，因为它可以被解释为计数过程的一个示例。<ref name="tijms2303page12">Henk C. Tijms (2003). A First Course in Stochastic Models. Wiley. pp. 1, 2. ISBN 978-0-471-49881-0.</ref>
如果一个泊松过程是用一个正常数定义的，那么这个过程称为齐次泊松过程。<ref name="Tijms2003page12" /><ref name="PinskyKarlin20112">{{cite book|author1=Mark A. Pinsky|author2=Samuel Karlin|title=An Introduction to Stochastic Modeling|url=https://books.google.com/books?id=PqUmjp7k1kEC|year=2011|publisher=Academic Press|isbn=978-0-12-381416-6|page=241}}</ref>齐次泊松过程是随机过程的一个重要类，如马尔可夫过程和Lévy过程。<ref name="Applebaum2004page13372" />
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>
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="Kingman1992page382">{{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-Jones2006page192">{{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="Kingman1992page222">{{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="KarlinTaylor2012page1182">{{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="Kleinrock1976page612">{{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>
在实数上定义的泊松过程可以解释为一个随机过程，<ref name="Applebaum2004page13372" /><ref name="Rosenblatt1962page942">{{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="Haenggi2013page10and182">{{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="ChiuStoyan2013page41and1082">{{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="Kingman1992page112">{{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="Haenggi2013page10and182" /><ref name="ChiuStoyan2013page41and1082" />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="Stirzaker20002" /><ref name="Streit2010page12">{{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="Streit2010page12" /><ref name="Kingman1992pagev2">{{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==
===随机过程 Stochastic process===
===随机过程 Stochastic process===
随机过程被定义为在一个公共概率空间<math>(\Omega, \mathcal{F}, P)</math>上定义的随机变量集合，其中<math>\Omega</math> 是[https://wiki.swarma.org/index.php/%E6%A0%B7%E6%9C%AC%E7%A9%BA%E9%97%B4 样本空间]，<math>\mathcal{F}</math>是一个<math>\sigma</math>-代数，<math>P</math>是概率测度；而随机变量，由某个集合<math>T</math>索引，所有值都取同一个数学空间<math>S</math>，对于某些<math>\sigma</math>-代数<math>\sigma</math><ref name="Lamperti1977page12" />
<center><math>
换言之，对于给定的概率空间<math>（\Omega，\mathcal{F}，P）</math>和可测空间<math>（S，Sigma）</math>，随机过程是一个值为<math>S</math>的随机变量的集合，可以写成：<ref name="Florescu2014page2932" /><center><math>
\{X(t):t\in T \}.
\{X(t):t\in T \}.
历史上，在许多自然科学问题中，一个点<math>t\in T</math> 具有时间的意义，因此，<math>X(t)</math>表示是一个在时间<math>t</math>的随机变量。<ref name="Borovkov2013page528">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=528}}</ref>随机过程也可以写成<math>\{X（t，omega）：t\ in t\}</math>来反映它实际上是两个变量的函数，<math>t\in t</math>和<math>\omega\in\omega</math><ref name="Lamperti1977page1"/><ref name="LindgrenRootzen2013page11">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>
历史上，在许多自然科学问题中，一个点<math>t\in T</math> 具有时间的意义，因此，<math>X(t)</math>表示是一个在时间<math>t</math>的随机变量。<ref name="Borovkov2013page5282">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=528}}</ref>随机过程也可以写成<math>\{X（t，omega）：t\ in t\}</math>来反映它实际上是两个变量的函数，<math>t\in t</math>和<math>\omega\in\omega</math><ref name="Lamperti1977page12" /><ref name="LindgrenRootzen2013page112">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>
还有其他方法可以考虑随机过程，上面的定义被认为是传统的。<ref name="RogersWilliams2000page1212" /><ref name="Asmussen2003page4082" />例如，一个随机过程可以解释或定义为一个<math>S^T</math>值的随机变量，其中<math>S^T</math>是所有可能的<math>S</math>-值函数的空间T<nowiki></math></nowiki>从集合<math>T</math>到空间<math>S</math>。<ref name="Kallenberg2002page242" /><ref name="RogersWilliams2000page1212" />
<br>
===索引集===
集合<math>T</math>称为“索引集”<ref name="Parzen19992" /><ref name="Florescu2014page2942" />或“参数集”<ref name="Lamperti1977page12" /><ref name="Skorokhod2005page932">{{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="doob1953stochasticP46to472" />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="doob1953stochasticP46to472" /><ref name="Billingsley2008page4822" />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="KarlinTaylor2012page272" /><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="Skorokhod2005page1042">{{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>
===样本函数 Sample function===
===状态空间 ===
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="doob1953stochasticP46to472" /><ref name="GikhmanSkorokhod1969page12" /><ref name="Lamperti1977page12" /><ref name="Florescu2014page2942" /><ref name="Brémaud2014page1202" />
'''样本函数'''是随机过程的单个结果，因此，它是由随机过程中每个随机变量的一个可能值构成的。<ref name="Lamperti1977page1"/><ref name="Florescu2014page296">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=296}}</ref>更准确地说，如果<math>\{X（t，omega）：t\in t\}</math>是一个随机过程，那么对于任何点<math>\omega\in\omega</math>，映射
===样本函数===
'''样本函数'''是随机过程的单个结果，因此，它是由随机过程中每个随机变量的一个可能值构成的。<ref name="Lamperti1977page12" /><ref name="Florescu2014page2962">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=296}}</ref>更准确地说，如果<math>\{X（t，omega）：t\in t\}</math>是一个随机过程，那么对于任何点<math>\omega\in\omega</math>，映射<center><math>
<center><math>
X(\cdot,\omega): T \rightarrow S,
X(\cdot,\omega): T \rightarrow S,
称为样本函数，称为“实现”，或者，特别是当<math>T</math>被解释为时间时，随机过程的“样本路径”<math>\{X（T，omega）：T\in T\}</math>。<ref name="RogersWilliams2000page121b">{{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|pages=121–124}}</ref>这意味着对于一个固定的<math>\omega\in\omega</math>，存在一个将索引集<math>T</math>映射到状态空间<math>S</math><ref name="Lamperti1977page1"/> 的示例函数的其他名称随机过程包括“轨迹”、“路径函数”<ref name="Billingsley2008page493">{{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=493}}</ref>或“路径”.<ref name="Øksendal2003page10">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=10}}</ref>
称为样本函数，称为“实现”，或者，特别是当<math>T</math>被解释为时间时，随机过程的“样本路径”<math>\{X（T，omega）：T\in T\}</math>。<ref name="RogersWilliams2000page121b2" />这意味着对于一个固定的<math>\omega\in\omega</math>，存在一个将索引集<math>T</math>映射到状态空间<math>S</math><ref name="Lamperti1977page12" /> 的示例函数的其他名称随机过程包括“轨迹”、“路径函数”<ref name="Billingsley2008page4932">{{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=493}}</ref>或“路径”.<ref name="Øksendal2003page102">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=10}}</ref>
<br>
===增量===
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="KarlinTaylor2012page272" /><ref name="Applebaum2004page13372" />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 。<ref name="Applebaum2004page13372" />
===进一步定义===
===进一步定义===
====定律====
====定律====
对于定义在概率空间<math>（\Omega，\mathcal{F}，P）</math>上的随机过程<math>X\colon\Omega\rightarrow S^T</math>，随机过程X</math>的定律被定义为'''前推度量 Pushforward measure'''：<center><math>
对于定义在概率空间<math>（\Omega，\mathcal{F}，P）</math>上的随机过程<math>X\colon\Omega\rightarrow S^T</math>，随机过程X</math>的定律被定义为'''前推度量 Pushforward measure'''：
<center><math>
\mu=P\circ X^{-1},
\mu=P\circ X^{-1},
其中<math>P</math>是一个概率度量，符号<math>\circ</math>表示函数组合，<math>X^{-1}</math>是可测量函数的前映像，或者等价地，<math>S^T</math>值随机变量<math>X</math>，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，所以随机过程的规律就是一个概率测度。<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/><ref name="FrizVictoir2010page571"/><ref name="Resnick2013page40">{{cite book|author=Sidney I. Resnick|title=Adventures in Stochastic Processes|url=https://books.google.com/books?id=VQrpBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4612-0387-2|pages=40–41}}</ref>
其中<math>P</math>是一个概率度量，符号<math>\circ</math>表示函数组合，<math>X^{-1}</math>是可测量函数的前映像，或者等价地，<math>S^T</math>值随机变量<math>X</math>，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，所以随机过程的规律就是一个概率测度。<ref name="Kallenberg2002page242" /><ref name="RogersWilliams2000page1212" /><ref name="FrizVictoir2010page5712">{{cite book|author1=Peter K. Friz|author2=Nicolas B. Victoir|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|url=https://books.google.com/books?id=CVgwLatxfGsC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48721-4|page=571}}</ref><ref name="Resnick2013page402">{{cite book|author=Sidney I. Resnick|title=Adventures in Stochastic Processes|url=https://books.google.com/books?id=VQrpBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4612-0387-2|pages=40–41}}</ref>
对于<math>S^T</math>的可测子集<math>B</math>，预图像<math>X</math>给出
对于<math>S^T</math>的可测子集<math>B</math>，预图像<math>X</math>给出<center><math>
<center><math>
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
所以a<math>X</math>定律可以写成：<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>
所以a<math>X</math>定律可以写成：<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><center><math>
<center><math>
\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).
\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).
随机过程或随机变量的规律也被称为“概率定律 probability law”，“概率分布 probability distribution”，或“分布”。<ref name="Borovkov2013page528"/><ref name="FrizVictoir2010page571"/><ref name="Whitt2006page23">{{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|page=23}}</ref><ref name="ApplebaumBook2004page4">{{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=4}}</ref><ref name="RevuzYor2013page10">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=10}}</ref>
随机过程或随机变量的规律也被称为“概率定律 probability law”，“概率分布 probability distribution”，或“分布”。<ref name="Borovkov2013page5282" /><ref name="FrizVictoir2010page5712" /><ref name="Whitt2006page232">{{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|page=23}}</ref><ref name="ApplebaumBook2004page42">{{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=4}}</ref><ref name="RevuzYor2013page102">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=10}}</ref>
====有限维概率分布 Finite-dimensional probability distributions====
====有限维概率分布====
对于随机过程<math>X</math>，其“有限维分布”定义为：<center><math>
对于随机过程<math>X</math>，其“有限维分布”定义为：
<center><math>
\mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1},
\mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1},
这项措施<math>\mu_{t_1,..,t_n}</math>是随机向量的联合分布 <math>(X({t_1}),\dots, X({t_n}))</math>；它可以被视为法律的“投影”<math>\mu</math>到一个有限子集<math>T</math>。<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page123">{{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=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=123}}</ref>
这项措施<math>\mu_{t_1,..,t_n}</math>是随机向量的联合分布 <math>(X({t_1}),\dots, X({t_n}))</math>；它可以被视为法律的“投影”<math>\mu</math>到一个有限子集<math>T</math>。<ref name="Kallenberg2002page242" /><ref name="RogersWilliams2000page1232">{{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=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=123}}</ref>
<center><math>
对于<math>n</math>级笛卡尔幂<math>S^n=S\times\dots \times S</math>的任何可测子集<math>C</math>，<math>X</math>的有限维分布可以写成：<ref name="Lamperti1977page12" /><center><math>
\mu_{t_1,\dots,t_n}(C) =P \Big(\big\{\omega\in \Omega: \big( X_{t_1}(\omega), \dots, X_{t_n}(\omega) \big) \in C \big\} \Big).
\mu_{t_1,\dots,t_n}(C) =P \Big(\big\{\omega\in \Omega: \big( X_{t_1}(\omega), \dots, X_{t_n}(\omega) \big) \in C \big\} \Big).
随机过程的有限维分布满足两个称为一致性条件的数学条件。<ref name="Rosenthal2006page177"/>
随机过程的有限维分布满足两个称为一致性条件的数学条件。<ref name="Rosenthal2006page1772" />
<center><math>
====稳定性====
“稳定性”是当随机过程的所有随机变量都是相同分布时随机过程所具有的数学性质。换言之，如果<math>X</math>是一个平稳随机过程，那么对于任何<math>t\in T</math>，随机变量<math>X_t</math>具有相同的分布，这意味着对于任何一组<math>n</math>索引集值<math>t_1,\dots, t_n</math>而言，对应的<math>n</math>随机变量<center><math>
X_{t_1}, \dots X_{t_n},
X_{t_1}, \dots X_{t_n},
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>
它们都有相同的[https://wiki.swarma.org/index.php/%E6%A6%82%E7%8E%87%E5%88%86%E5%B8%83 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="Lamperti1977page62">{{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="GikhmanSkorokhod1969page42">{{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="Lamperti1977page62" /><ref name="Adler2010page142">{{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="ChiuStoyan2013page1122">{{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>
当指标集<math>T</math>可以解释为时间时，如果随机过程的有限维分布在时间平移下是不变的，则称其为平稳过程。这种随机过程可以用来描述处于稳态的物理系统，但是仍然会经历随机波动。<ref name="Lamperti1977page6"/>平稳性背后的直觉是，随着时间的推移，平稳随机过程的分布保持不变。<ref name="Doob1990page94">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=94–96}}</ref>只有当随机变量相同分布时，一系列随机变量才会形成平稳随机过程。<ref name="Lamperti1977page6"/>
当指标集<math>T</math>可以解释为时间时，如果随机过程的有限维分布在时间平移下是不变的，则称其为平稳过程。这种随机过程可以用来描述处于稳态的物理系统，但是仍然会经历随机波动。<ref name="Lamperti1977page62" />平稳性背后的直觉是，随着时间的推移，平稳随机过程的分布保持不变。<ref name="Doob1990page942">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=94–96}}</ref>只有当随机变量相同分布时，一系列随机变量才会形成平稳随机过程。<ref name="Lamperti1977page62" />
具有上述平稳性定义的随机过程有时被称为严格平稳的，但也有其他形式的平稳性。一个例子是当离散时间或连续时间随机过程<math>X</math>被称为广义平稳时，那么这个过程<math>X</math>，有一个有限的第二时刻对于所有<math>t\in T</math>和两个随机变量的协方差 <math>X_t</math> 和 <math>X_{t+h}</math> 只取决于在<math>t\in T</math>时的数值<math>h</math><ref name="Doob1990page94"/><ref name="Florescu2014page298">{{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=298, 299}}</ref> [[Aleksandr Khinchin | Khinchin]]介绍了“广义平稳性”的相关概念，其他名称包括“协方差平稳性”或“广义平稳性”。<ref name="Florescu2014page298"/><ref name="GikhmanSkorokhod1969page8">{{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=8}}</ref>
具有上述平稳性定义的随机过程有时被称为严格平稳的，但也有其他形式的平稳性。一个例子是当离散时间或连续时间随机过程<math>X</math>被称为广义平稳时，那么这个过程<math>X</math>，有一个有限的第二时刻对于所有<math>t\in T</math>和两个随机变量的协方差 <math>X_t</math> 和 <math>X_{t+h}</math> 只取决于在<math>t\in T</math>时的数值<math>h</math><ref name="Doob1990page942" /><ref name="Florescu2014page2982">{{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=298, 299}}</ref> 辛钦介绍了“广义平稳性”的相关概念，其他名称包括“协方差平稳性”或“广义平稳性”。<ref name="Florescu2014page2982" /><ref name="GikhmanSkorokhod1969page82">{{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=8}}</ref>
<br>
====过滤====
过滤是定义在某个概率空间中的sigma代数的递增序列和具有某种总阶关系的索引集，例如在索引集是实数的某个子集的情况下。更为正式的是，如果随机过程有一个指数集总排序的随机过程，则如果随机过程有一个指数集的总序为总序，那么在概率空间<math>(\Omega, \mathcal{F}, P)</math>上的过滤<math>\{\mathcal{F}_t\}_{t\in T} </math> 是一个sigma代数族，使得<math> \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} </math>对所有<math>s \leq t</math>，其中<math>t, s\in T</math>和<math>\leq</math>表示指标集<math>T</math>的总阶<ref name="Florescu2014page2942" />通过过滤的概念，可以研究<math>t\in T</math>中随机过程<math>X_t</math>所包含的信息量，这可以解释为时间<math>t</math><ref name="Florescu2014page2942" /><ref name="Williams1991page932">{{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|pages=93, 94}}</ref>过滤背后的直觉是，随着时间的流逝，关于<math>t</math>的更多信息是已知的或可用的，这些信息可以在<math>\mathcal{F}t</math>中获得，使<math>\Omega</math>的分区越来越细。<ref name="Klebaner2005page222">{{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|pages=22–23}}</ref><ref name="MörtersPeres2010page372">{{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=37}}</ref>
====修正====
====修正 Modification====
随机过程的“修正”是另一个随机过程，它与原始随机过程密切相关。更确切地说，一个随机过程<math>X</math>，与另一个随机过程<math>Y</math> 具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>（\Omega，{\cal F}，P）</math>具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>(\Omega,{\cal F},P)</math>，被称为<math>Y</math>的修改，如果对所有<math>t\in T</math>有
随机过程的“修正”是另一个随机过程，它与原始随机过程密切相关。更确切地说，一个随机过程<math>X</math>，与另一个随机过程<math>Y</math> 具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>（\Omega，{\cal F}，P）</math>具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>(\Omega,{\cal F},P)</math>，被称为<math>Y</math>的修改，如果对所有<math>t\in T</math>有
<center><math>
<center><math>
持有。两个相互修正的随机过程具有相同的有限维法则<ref name="RogersWilliams2000page130">{{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=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=130}}</ref>它们被称为“随机等价”或“等价物”<ref name="Borovkov2013page530">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>
持有。两个相互修正的随机过程具有相同的有限维法则<ref name="RogersWilliams2000page1302">{{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=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=130}}</ref>它们被称为“随机等价”或“等价物”<ref name="Borovkov2013page5302">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>
====难以区分 Indistinguishable====
除了修改，还使用了“版本”一词，<ref name="Adler2010page142" /><ref name="Klebaner2005page482">{{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=48}}</ref><ref name="Øksendal2003page142">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=14}}</ref><ref name="Florescu2014page4722">{{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=472}}</ref>然而，当两个随机过程具有相同的有限维分布，但它们可能定义在不同的概率空间上，因此两个过程是相互修改的，在后一种意义上，它们也是彼此的版本，但不是相反。<ref name="RevuzYor2013page182">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|pages=18–19}}</ref><ref name="FrizVictoir2010page5712" /> .
如果一个连续时间的实值随机过程在其增量上满足一定的矩条件，则[https://wiki.swarma.org/index.php/Kolmogorov%E8%BF%9E%E7%BB%AD%E6%80%A7%E5%AE%9A%E7%90%86 Kolmogorov连续性定理]指出，该过程存在一个修正，其具有概率为1的连续样本路径，因此随机过程有一个连续的修改或版本<ref name="Øksendal2003page142" /><ref name="Florescu2014page4722" /><ref name="ApplebaumBook2004page202">{{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=20}}</ref>该定理也可以推广到随机域，因此索引集是<math>n</math>-维欧几里德空间<ref name="Kunita1997page312">{{cite book|author=Hiroshi Kunita|title=Stochastic Flows and Stochastic Differential Equations|url=https://books.google.com/books?id=_S1RiCosqbMC|year=1997|publisher=Cambridge University Press|isbn=978-0-521-59925-2|page=31}}</ref>以及以[https://wiki.swarma.org/index.php/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4 度量空间]为状态空间的随机过程。<ref name="Kallenberg2002page2">{{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|page=35}}</ref>
<center><math>
====难以区分====
两个随机过程<math>X</math>和<math>Y</math>定义在同一概率空间<math>（\Omega，\mathcal{F}，P）</math>上，具有相同的索引集<math>T</math>和集空间<math>S</math>上的两个随机过程如果<center><math>
P(X_t=Y_t \text{ for all } t\in T )=1 ,
P(X_t=Y_t \text{ for all } t\in T )=1 ,
成立，则称为“难以区分的”。<ref name="FrizVictoir2010page571"/><ref name="RogersWilliams2000page130"/>如果两个<math>X</math>和<math>Y</math>是相互修改的，几乎肯定是连续的，那么<math>X</math>和<math>Y</math>是无法区分的。<ref name="JeanblancYor2009page11">{{cite book|author1=Monique Jeanblanc|author2=Marc Yor|author3=Marc Chesney|title=Mathematical Methods for Financial Markets|url=https://books.google.com/books?id=ZhbROxoQ-ZMC|year=2009|publisher=Springer Science & Business Media|isbn=978-1-85233-376-8|page=11}}</ref>
成立，则称为“难以区分的”。<ref name="FrizVictoir2010page5712" /><ref name="RogersWilliams2000page1302" />如果两个<math>X</math>和<math>Y</math>是相互修改的，几乎肯定是连续的，那么<math>X</math>和<math>Y</math>是无法区分的。<ref name="JeanblancYor2009page112">{{cite book|author1=Monique Jeanblanc|author2=Marc Yor|author3=Marc Chesney|title=Mathematical Methods for Financial Markets|url=https://books.google.com/books?id=ZhbROxoQ-ZMC|year=2009|publisher=Springer Science & Business Media|isbn=978-1-85233-376-8|page=11}}</ref>
====可分性====
“可分性”是随机过程的一种性质，它基于与概率测度有关的指标集。假设随机过程或具有不可数指标集的随机场的泛函可以形成随机变量。对于一个随机过程是可分离的，除了其他条件外，它的指标集必须是一个[https://wiki.swarma.org/index.php/%E5%8F%AF%E5%88%86%E7%A6%BB%E7%A9%BA%E9%97%B4 可分离空间]，{efn |术语“可分离”在这里出现了两次，有两种不同的含义，第一种含义来自概率，第二种含义来自拓扑和分析。对于一个随机过程是可分的（概率意义上），它的指标集必须是一个可分空间（在拓扑或分析意义上），除了其他条件。<ref name="Skorokhod2005page932" /><nowiki>}}，这意味着索引集有一个稠密的可数子集。</nowiki><ref name="Adler2010page142" /><ref name="Ito2006page322">{{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|pages=32–33}}</ref>
更精确地说，具有概率空间<math>(\Omega,{\cal F},P)</math>的实值连续时间随机过程<math>X</math>是可分离的，如果它的指数集<math>T</math>有一个稠密的可数子集<math>\Omega_0 \subset \Omega</math>，因此<<math>P(\Omega_0)=0</math>，这样对于每个开集<math>G\subset T</math>和每个闭集<math>F\subset \textstyle R =(-\infty,\infty) </math>，<math>\{ X_t \in F \text{ for all } t \in G\cap U\}</math>和<math>\{ X_t \in F \text{ for all } t \in G\}</math>这两个事件最多在<math>\Omega_0</math>的一个子集上不同。<ref name="GikhmanSkorokhod1969page1502">{{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=150}}</ref><ref name="Todorovic2012page192">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|pages=19–20}}</ref><ref name="Molchanov2005page3402">{{cite book|author=Ilya Molchanov|title=Theory of Random Sets|url=https://books.google.com/books?id=kWEwk1UL42AC|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-892-3|page=340}}</ref>
====独立性 Independence====
可分离性的定义（连续时间实值随机过程的可分性定义可以用其他方式表述。<ref name="Billingsley2008page5262">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=526–527}}</ref><ref name="Borovkov2013page5352">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=535}}</ref>）也可以为其他索引集和状态空间而声明，<ref name="GusakKukush2010page222">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.</ref>例如在随机场的情况下，索引集和状态空间可以是<math>n</math>维欧几里德空间。<ref name="AdlerTaylor2009page72" /><ref name="Adler2010page142" />
随机过程可分性的概念是由约瑟夫-杜布（Joseph Doob），<ref name="Ito2006page322" />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="Billingsley2008page5262" />因此离散时间随机过程总是可分离的。<ref name="Doob1990page562">{{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="Ito2006page322" /><ref name="Todorovic2012page192" /><ref name="Khoshnevisan2006page1552">{{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="Skorokhod2005page932" />
====不相关 Uncorrelatedness====
====独立性====
两个在相同的概率空间<math>(\Omega,\mathcal{F},P)</math>上定义，具有相同索引集<math>T</math>的随机过程<math>X</math>和<math>Y</math>被称为“相互独立”，如果对于所有<math>n \in \mathbb{N}</math>，以及每个特定的<math>t_1,\ldots,t_n \in T</math>，随机向量<math>\left( X(t_1),\ldots,X(t_n) \right)</math> 和<math>\left( Y(t_1),\ldots,Y(t_n) \right)</math>是独立的。<ref name="Lapidoth2">Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.</ref>
====不相关====
两个随机过程<math>\left\{X_t\right\}</math>和<math>\left\{Y_t\right\}</math> 称为“不相关的”的，如果它们的互协方差<math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right]</math>始终为零。<ref name="KunIlPark2">Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3</ref>最后：
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
====独立意味着不相关====
====独立意味着不相关 Independence implies uncorrelatedness====
如果两个随机过程<math>X</math>和<math>Y</math>是独立的，那么它们也是不相关的
如果两个随机过程<math>X</math>和<math>Y</math>是独立的，那么它们也是不相关的
====正交性====
如果两个随机过程<math>\left\{X_t\right\}</math>和<math>\left\{Y_t\right\}</math>的互相关<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E}[X(t_1) \overline{Y(t_2)}]</math>一直为0，则称为“正交”，形式为<ref name="KunIlPark2" />
====正交性 Orthogonality====
如果两个随机过程<math>\left\{X_t\right\}</math>和<math>\left\{Y_t\right\}</math>的互相关<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E}[X(t_1) \overline{Y(t_2)}]</math>一直为0，则称为“正交”，形式为<ref name=KunIlPark/>
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ orthogonal} \quad \iff \quad \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ orthogonal} \quad \iff \quad \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
====斯科罗霍德空间====
''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="Whitt2006page782">{{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="GusakKukush2010page242">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="Bogachev2007Vol2page532">{{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="Whitt2006page782" /><ref name="Klebaner2005page42">{{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[https://wiki.swarma.org/index.php/Anatoliy_Skorokod Anatoliy Skorokod]引入的Skorokod函数空间，<ref name="Bogachev2007Vol2page532" />通常用字母<math>D</math>表示，<ref name="Whitt2006page782" /><ref name="GusakKukush2010page242" /><ref name="Bogachev2007Vol2page532" /><ref name="Klebaner2005page42" />因此函数空间也被称为空间<math>D</math><ref name="Whitt2006page782" /><ref name="Asmussen2003page4202">{{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="Billingsley2013page1212">{{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="Klebaner2005page42" /><ref name="Billingsley2013page1212" /><ref name="Bass2011page342">{{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="Bogachev2007Vol2page532" /><ref name="Asmussen2003page4202" />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="Billingsley2013page1212" /><ref name="BinghamKiesel2013page1542">{{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>
====规律性====
在随机过程的数学构造中，当讨论和假设随机过程的某些条件以解决可能的构造问题时，使用术语“正则性”。<ref name="Borovkov2013page5322">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=532}}</ref><ref name="Khoshnevisan2006page148to1652">{{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=148–165}}</ref>例如，研究具有不可数索引集的随机过程，假设随机过程服从某种正则条件，例如样本函数是连续的。<ref name="Todorovic2012page222">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|page=22}}</ref><ref name="Whitt2006page792">{{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|page=79}}</ref>
====规律性 Regularity====
在随机过程的数学构造中，当讨论和假设随机过程的某些条件以解决可能的构造问题时，使用术语“正则性”。<ref name="Borovkov2013page532">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=532}}</ref><ref name="Khoshnevisan2006page148to165">{{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=148–165}}</ref>例如，研究具有不可数索引集的随机过程，假设随机过程服从某种正则条件，例如样本函数是连续的。<ref name="Todorovic2012page22">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|page=22}}</ref><ref name="Whitt2006page79">{{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|page=79}}</ref>
==更多示例==
==更多示例==
===马尔可夫过程与链===
===马尔可夫过程与链===
''' 马尔可夫过程 Markov processes '''是一种随机过程，传统上在[[离散时间和连续时间|离散或连续时间]]中，具有马尔可夫特性，即马尔可夫过程的下一个值取决于当前值，但它与随机过程的先前值条件无关。换句话说，给定进程的当前状态，进程在未来的行为与它过去的行为是随机独立的。<ref name="Serfozo2009page2">{{cite book|author=Richard Serfozo|title=Basics of Applied Stochastic Processes|url=https://books.google.com/books?id=JBBRiuxTN0QC|year=2009|publisher=Springer Science & Business Media|isbn=978-3-540-89332-5|page=2}}</ref><ref name="Rozanov2012page58">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=58}}</ref>
''' 马尔可夫过程 Markov processes '''是一种随机过程，传统上在[https://wiki.swarma.org/index.php/%E7%A6%BB%E6%95%A3%E6%97%B6%E9%97%B4%E5%92%8C%E8%BF%9E%E7%BB%AD%E6%97%B6%E9%97%B4 离散或连续时间]中，具有马尔可夫特性，即马尔可夫过程的下一个值取决于当前值，但它与随机过程的先前值条件无关。换句话说，给定进程的当前状态，进程在未来的行为与它过去的行为是随机独立的。<ref name="Serfozo2009page22">{{cite book|author=Richard Serfozo|title=Basics of Applied Stochastic Processes|url=https://books.google.com/books?id=JBBRiuxTN0QC|year=2009|publisher=Springer Science & Business Media|isbn=978-3-540-89332-5|page=2}}</ref><ref name="Rozanov2012page582">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=58}}</ref>
布朗运动过程和泊松过程（一维）都是马尔可夫过程的例子<ref name="Ross1996page235and3582">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|pages=235, 358}}</ref>，整数上的随机游走和赌徒破产问题是离散时间中马尔可夫过程的例子。<ref name="Florescu2014page3732">{{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=373, 374}}</ref><ref name="KarlinTaylor2012page492">{{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=49}}</ref>
马尔可夫链是一种具有离散状态空间或离散索引集（通常表示时间）的马尔可夫过程，但马尔可夫链的精确定义各不相同<ref name="Asmussen2003page72">{{cite book|url=https://books.google.com/books?id=BeYaTxesKy0C|title=Applied Probability and Queues|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=7|author=Søren Asmussen}}</ref>例如，通常将马尔可夫链定义为具有可数状态空间的[https://wiki.swarma.org/index.php/%E8%BF%9E%E7%BB%AD%E5%8F%98%E9%87%8F 离散或连续时间]中的马尔可夫过程（因此不管时间的性质），<ref name="Parzen1999page1882">{{cite book|url=https://books.google.com/books?id=0mB2CQAAQBAJ|title=Stochastic Processes|year=2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|page=188|author=Emanuel Parzen}}</ref><ref name="KarlinTaylor2012page292">{{cite book|url=https://books.google.com/books?id=dSDxjX9nmmMC|title=A First Course in Stochastic Processes|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=29, 30|author1=Samuel Karlin|author2=Howard E. Taylor}}</ref><ref name="Lamperti1977chap62">{{cite book|url=https://books.google.com/books?id=Pd4cvgAACAAJ|title=Stochastic processes: a survey of the mathematical theory|publisher=Springer-Verlag|year=1977|isbn=978-3-540-90275-1|pages=106–121|author=John Lamperti}}</ref><ref name="Ross1996page174and2312">{{cite book|url=https://books.google.com/books?id=ImUPAQAAMAAJ|title=Stochastic processes|publisher=Wiley|year=1996|isbn=978-0-471-12062-9|pages=174, 231|author=Sheldon M. Ross}}</ref>但通常将马尔可夫链定义为在可数状态空间或连续状态空间中具有离散时间（因此与状态空间无关），<ref name="Asmussen2003page72" />有人认为，现在倾向于使用具有离散时间的马尔可夫链的第一个定义，尽管Joseph Doob和Kai Lai Chung等研究人员使用了第二个定义。<ref name="MeynTweedie20092">{{cite book|author1=Sean Meyn|author2=Richard L. Tweedie|title=Markov Chains and Stochastic Stability|url=https://books.google.com/books?id=Md7RnYEPkJwC|year=2009|publisher=Cambridge University Press|isbn=978-0-521-73182-9|page=19}}</ref>
===鞅 Martingale===
马尔可夫过程是一类重要的随机过程，在许多领域有着广泛的应用。<ref name="LatoucheRamaswami19992" /><ref name="KarlinTaylor2012page472">{{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=47}}</ref>例如，它们是一种称为Markov chain Monte Carlo的一般随机模拟方法的基础，该方法用于模拟具有特定概率分布的随机对象，并在贝叶斯统计中得到应用。<ref name="RubinsteinKroese2011page2252">{{cite book|author1=Reuven Y. Rubinstein|author2=Dirk P. Kroese|title=Simulation and the Monte Carlo Method|url=https://books.google.com/books?id=yWcvT80gQK4C|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-21052-9|page=225}}</ref><ref name="GamermanLopes20062">{{cite book|author1=Dani Gamerman|author2=Hedibert F. Lopes|title=Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition|url=https://books.google.com/books?id=yPvECi_L3bwC|year=2006|publisher=CRC Press|isbn=978-1-58488-587-0}}</ref>
马尔可夫特性的概念最初是针对连续和离散时间的随机过程，但它也适用于其它指标集，如<math>n</math>维欧氏空间，这导致随机变量的集合被称为马尔可夫随机场。<ref name="Rozanov2012page612">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=61}}</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=27}}</ref><ref name="Bremaud2013page2532">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=253}}</ref>
===鞅===
'''鞅（Martingale）'''是一个离散时间或连续时间的随机过程，其性质是在给定过程的当前值和所有过去值的情况下，每个未来值的条件期望值等于当前值。在离散时间中，如果此属性适用于下一个值，则它适用于所有未来值。鞅的精确数学定义需要另外两个条件与过滤的数学概念相结合，这与随时间推移增加可用信息的直觉有关。鞅通常被定义为实值，<ref name="Klebaner2005page652">{{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=65}}</ref><ref name="KaratzasShreve2014page112">{{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=11}}</ref><ref name="Williams1991page932" /> 但他们也可以是复数或<ref name="Doob1990page2922">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref>或更一般的取值。<ref name="Pisier20162">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref>
对称随机游动和维纳过程（具有零漂移）分别是离散时间和连续时间的鞅的例子。<ref name="Klebaner2005page652" /><ref name="KaratzasShreve2014page112" />对于一个独立且同分布随机变量的序列<math>X_1, X_2, X_3, \dots</math>且平均值为零，由连续部分和<math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> 构成的随机过程是一个离散时间鞅<ref name="Steele2012page122">{{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|pages=12, 13}}</ref>，离散时间鞅推广了独立随机变量的部分和的概念。<ref name="HallHeyde2014page22">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</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="KaratzasShreve2014page112" />也可以从其他鞅中构建鞅。<ref name="Steele2012page122" />例如，有基于鞅的鞅Wiener过程，形成连续时间鞅。<ref name="Klebaner2005page652" /><ref name="Steele2012page1152">{{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>
数学上的鞅形式化了公平博弈的概念，<ref name="Ross1996page2952">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=295}}</ref>它们最初的开发目的是表明不可能赢得一场公平的比赛。<ref name="Steele2012page112">{{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=11}}</ref>但现在它们被用于许多概率领域，这是研究它们的主要原因之一。<ref name="Williams1991page932" /><ref name="Steele2012page112" /><ref name="Kallenberg2002page962">{{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=96}}</ref>许多概率问题已经通过在问题中找到鞅并加以研究而得到解决。<ref name="Steele2012page3712">{{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=371}}</ref>在给定鞅矩的条件下，鞅会收敛，因此经常使用鞅得到收敛结果，这主要是由于鞅收敛定理。<ref name="HallHeyde2014page22" /><ref name="Steele2012page222">{{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=22}}</ref><ref name="GrimmettStirzaker2001page3362">{{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=336}}</ref>
鞅在统计学中有许多应用，但有人指出，它的使用和应用并不像它在统计学领域那样广泛，尤其是统计推断。<ref name="GlassermanKou20062">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref>他们在排队论和棕榈微积分等概率论领域找到了应用<ref name="BaccelliBremaud20132">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref>以及其他领域，如经济学。<ref name="HallHeyde2014pageX2">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski20062" />
===莱维过程 Lévy process===
===莱维过程===
'''莱维过程'''是随机过程的一种类型，可以看作是连续时间中随机游动的推广。<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref>这些过程在金融、流体力学等领域有着广泛的应用。<ref name="Applebaum2004page1336">{{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}}</ref><ref name="ApplebaumBook2004page69">{{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=69}}</ref>这些过程和过程的独立性被称为平稳过程的主要特征。换句话说，一个随机过程<math>X</math>是一个Lévy过程，如果对非负数<math>n</math>，<math>0\leq t_1\leq \dots \leq t_n</math>，当<math>n-1</math>递增
'''莱维过程'''是随机过程的一种类型，可以看作是连续时间中随机游动的推广。<ref name="Applebaum2004page13372" /><ref name="Bertoin1998pageVIII2">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref>这些过程在金融、流体力学等领域有着广泛的应用。<ref name="Applebaum2004page13362">{{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}}</ref><ref name="ApplebaumBook2004page692">{{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=69}}</ref>这些过程和过程的独立性被称为平稳过程的主要特征。换句话说，一个随机过程<math>X</math>是一个莱维过程，如果对非负数<math>n</math>，<math>0\leq t_1\leq \dots \leq t_n</math>，当<math>n-1</math>递增<center><math>
<center><math>
X_{t_2}-X_{t_1}, \dots , X_{t_{n-1}}-X_{t_n},
X_{t_2}-X_{t_1}, \dots , X_{t_{n-1}}-X_{t_n},
它们彼此独立，每个增量的分布只取决于时间的差异。<ref name="Applebaum2004page1337"/>
它们彼此独立，每个增量的分布只取决于时间的差异。<ref name="Applebaum2004page13372" />
===随机场===
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="AdlerTaylor2009page72" />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="GikhmanSkorokhod1969page12" /><ref name="Lamperti1977page12" /><ref name="KoralovSinai2007page1712">{{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="ApplebaumBook2004page192">{{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===
===随机 Random field===
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="ChiuStoyan2013page1092">{{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="ChiuStoyan2013page1082">{{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="Haenggi2013page102">{{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-Jones2006page1942">{{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="CoxIsham1980page32">{{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="CoxIsham1980page32" />
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"/>
<br>
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="Kingman1992page82">{{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="MollerWaagepetersen2003page72">{{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="KarlinTaylor2012page312">{{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="Schmidt2014page992">{{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-Jones2002">{{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="CoxIsham19802">{{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>
==历史 History==
==历史 History==
===早期概率论 Early probability theory===
===早期概率论 Early probability theory===
概率论起源于机会博弈，它有着悠久的历史，有些游戏在几千年前就已经开始了，<ref name=":12">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|location=US|pages=1–2}}</ref><ref name="David19552">{{cite journal|last1=David|first1=F. N.|title=Studies in the History of Probability and Statistics I. Dicing and Gaming (A Note on the History of Probability)|journal=Biometrika|volume=42|issue=1/2|pages=1–15|year=1955|issn=0006-3444|doi=10.2307/2333419|jstor=2333419}}</ref>但人们很少从概率的角度对其进行分析。<ref name=":12" /><ref name="Maistrov2014page12">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=1}}</ref>1654年通常被认为是概率论的诞生，当时法国数学家皮埃尔-费马和布莱斯-帕斯卡尔在一个赌博问题的激励下，就概率问题进行了书面通信。<ref name=":12" /><ref name="Seneta2006page12">{{cite book|last1=Seneta|first1=E.|title=Encyclopedia of Statistical Sciences|chapter=Probability, History of|year=2006|doi=10.1002/0471667196.ess2065.pub2|page=1|isbn=978-0471667193}}</ref><ref name="Tabak2014page24to262">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|pages=24–26}}</ref>但是在更早的时候，就有关于赌博游戏概率的数学工作，比如吉罗拉莫·卡尔达诺（Gerolamo Cardano）16世纪写作的“Liber de Ludo Aleae”，在他死后于1663年发表。<ref name=":12" /><ref name="Bellhouse20052">{{cite journal|last1=Bellhouse|first1=David|title=Decoding Cardano's Liber de Ludo Aleae|journal=Historia Mathematica|volume=32|issue=2|year=2005|pages=180–202|issn=0315-0860|doi=10.1016/j.hm.2004.04.001|doi-access=free}}</ref>
继卡尔达诺之后，雅各布·伯努利（Jakob Bernoulli，也被称为James Bernoulli或Jacques Bernoulli<ref name="Hald2005page2212">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=221}}</ref>）写了Ars conjuctandi，这在概率论史上被认为是重大事件。<ref name=":12" />伯努利的书出版于1713年，也是在他死后出版的，这本书激发了许多数学家研究概率。<ref name=":12" /><ref name="Maistrov2014page562">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=56}}</ref><ref name="Tabak2014page372">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|page=37}}</ref> 但是，尽管一些著名的数学家对概率论做出了贡献，比如皮埃尔-西蒙-拉普拉斯（Pierre-Simon Laplace,）、亚伯拉罕-德-莫伊夫（Abraham de Moivre）、卡尔-高斯（Carl Gauss）、西蒙-泊阿松 （Siméon Poisson）和帕夫努蒂·切比雪夫（Pafnuty Chebyshev），<ref name="Chung19982">{{cite journal|last1=Chung|first1=Kai Lai|title=Probability and Doob|journal=The American Mathematical Monthly|volume=105|issue=1|pages=28–35|year=1998|issn=0002-9890|doi=10.2307/2589523|jstor=2589523}}</ref><ref name="Bingham20002">{{cite journal|last1=Bingham|first1=N.|title=Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov|journal=Biometrika|volume=87|issue=1|year=2000|pages=145–156|issn=0006-3444|doi=10.1093/biomet/87.1.145}}</ref>大多数数学界人士直到20世纪，才认为概率论是数学的一部分（一个显著的例外是俄罗斯的圣彼得堡学派，在那里，以切比雪夫为首的数学家研究概率论<ref name="BenziBenzi20072">{{cite journal|last1=Benzi|first1=Margherita|last2=Benzi|first2=Michele|last3=Seneta|first3=Eugene|title=Francesco Paolo Cantelli. b. 20 December 1875 d. 21 July 1966|journal=International Statistical Review|volume=75|issue=2|year=2007|page=128|issn=0306-7734|doi=10.1111/j.1751-5823.2007.00009.x}}</ref>）。<ref name="Chung19982" /><ref name="BenziBenzi20072" /><ref name="Doob19962">{{cite journal|last1=Doob|first1=Joseph L.|title=The Development of Rigor in Mathematical Probability (1900-1950)|journal=The American Mathematical Monthly|volume=103|issue=7|pages=586–595|year=1996|issn=0002-9890|doi=10.2307/2974673|jstor=2974673}}</ref><ref name="Cramer19762">{{cite journal|last1=Cramer|first1=Harald|title=Half a Century with Probability Theory: Some Personal Recollections|journal=The Annals of Probability|volume=4|issue=4|year=1976|pages=509–546|issn=0091-1798|doi=10.1214/aop/1176996025|doi-access=free}}</ref>
===统计力学 Statistical mechanics===
在物理科学领域，科学家们在19世纪发展了[https://wiki.swarma.org/index.php/%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6 统计力学]学科。在这个学科中，物理系统，例如装满气体的容器，可以从数学上看作或处理为许多运动粒子的集合。尽管有些科学家（比如鲁道夫·克劳修斯（Rudolf Clausius））试图将随机性纳入统计物理学，但大部分工作没有或几乎没有随机性。<ref name="Truesdell1975page222">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=22–23|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1967page1502">{{cite journal|last1=Brush|first1=Stephen G.|title=Foundations of statistical mechanics 1845?1915|journal=Archive for History of Exact Sciences|volume=4|issue=3|year=1967|pages=150–151|issn=0003-9519|doi=10.1007/BF00412958}}</ref>
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这种情况在1859年发生了变化，当时詹姆斯·克拉克·麦克斯韦 James Clerk Maxwell对该领域做出了重大贡献，更具体地说，他提出了气体的动力学理论，假设气体粒子以随机速度向随机方向移动。<ref name="Truesdell1975page312">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=31–32|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush19582">{{cite journal|last1=Brush|first1=S.G.|title=The development of the kinetic theory of gases IV. Maxwell|journal=Annals of Science|volume=14|issue=4|year=1958|pages=243–255|issn=0003-3790|doi=10.1080/00033795800200147}}</ref>气体动力学理论和统计物理在19世纪下半叶继续发展，其工作主要由克劳修斯，路德维希·玻尔兹曼和[https://wiki.swarma.org/index.php/%E7%BA%A6%E8%A5%BF%E4%BA%9A%C2%B7%E5%A8%81%E6%8B%89%E5%BE%B7%C2%B7%E5%90%89%E5%B8%83%E6%96%AF 约西亚·威拉德·吉布斯]完成，这些工作后来对阿尔伯特·爱因斯坦的布朗运动数学模型产生了影响。<ref name="Brush1968page152">{{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=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>
===测度论与概率论===
1900年在巴黎举行的国际数学家大会上，大卫-希尔伯特（David Hilbert）提出了一份数学问题的清单，其中他的第六个问题要求对物理和概率的公理化进行数学处理。<ref name="Bingham20002" />大约在20世纪初，数学家们发展了测度论，这是研究函数积分的数学分支，其中两位创始人是法国数学家亨利-勒贝斯格（Henri Lebesgue）和米尔-博莱尔É（mile Borel。）1925年，另一位法国数学家P保罗·莱维（aul Lévy出）版了第一本使用测度论思想的概率论书籍。<ref name="Bingham20002" />
20世纪20年代，苏联的数学家们对概率论做出了重大贡献，比如谢尔盖-伯恩斯坦（Sergei Bernstein），亚历山大-辛钦（Aleksandr Khinchin，Khinchin这个名字也用英语写成（或音译成）Khintchine。<ref name="Doob19342" />）和安德烈-科尔莫戈罗夫（Andrei Kolmogorov）。<ref name="Cramer19762" /> 科尔莫戈罗夫于1984年发表了他的一次尝试，为概率论提出一个基于度量理论的数学基础。<ref name="KendallBatchelor1990page332">{{cite journal|last1=Kendall|first1=D. G.|last2=Batchelor|first2=G. K.|last3=Bingham|first3=N. H.|last4=Hayman|first4=W. K.|last5=Hyland|first5=J. M. E.|last6=Lorentz|first6=G. G.|last7=Moffatt|first7=H. K.|last8=Parry|first8=W.|last9=Razborov|first9=A. A.|last10=Robinson|first10=C. A.|last11=Whittle|first11=P.|title=Andrei Nikolaevich Kolmogorov (1903–1987)|journal=Bulletin of the London Mathematical Society|volume=22|issue=1|year=1990|page=33|issn=0024-6093|doi=10.1112/blms/22.1.31}}</ref>在20世纪30年代初，辛钦和科尔莫戈罗夫成立了概率研讨会，金-斯卢茨基（Eugene Slutsky等）、尼古拉-斯米尔诺夫（Nikolai Smirnov）等研究人员参加了这些研讨会，<ref name="Vere-Jones2006page12">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=1|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>辛钦首次给出了随机变量的数学定义，即由实数索引的一组随机变量。<ref name="Doob19342" /><ref name="Vere-Jones2006page42">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=4|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>（Doob在引用Khinchin时，使用了“机会变量”这个词，它曾经是“随机变量”的替代词。<ref name="Snell20052">{{cite journal|last1=Snell|first1=J. Laurie|title=Obituary: Joseph Leonard Doob|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|page=251|issn=0021-9002|doi=10.1239/jap/1110381384|doi-access=free}}</ref>）
===现代概率论的诞生 Birth of modern probability theory===
1933年，德烈-科尔莫戈罗夫（Andrei Kolmogorov在）德国出版了一本关于概率论基础的书，名为《概率计算的基本概念》，后来翻译成英文，1950年出版，作为概率论的基础。这本书的出版现在被广泛认为是现代概率论的诞生，从此概率论和随机过程理论成为数学的一部分。<ref name="Bingham20002" /><ref name="Cramer19762" />
在科尔莫戈罗夫的书出版后，辛钦和科尔莫戈洛夫以及其他数学家如约瑟夫-杜布（Joseph Doob）、威廉-费勒（William Feller）、莫里斯-弗雷谢（Maurice Fréchet）、保罗-莱维（Paul Lévy）、沃尔夫冈-多布林（Wolfgang Doeblin）和哈拉尔-克拉梅尔（Harald Cramér）<ref name="Bingham20002" /><ref name="Cramer19762" />，都在概率论和随机过程方面做了进一步的基础工作，
===测度论与概率论 Measure theory and probability theory===
几十年后，克拉梅尔（Cramér）把20世纪30年代称为“数学概率论的英雄时期”。<ref name="Cramer19762" />第二次世界大战很大程度上中断了概率论的发展。例如，大战导致费勒从瑞典迁移到美国<ref name="Cramer19762" />，以及现在被认为是随机过程先驱的多布林的去世。<ref name="Lindvall19912">{{cite journal|last1=Lindvall|first1=Torgny|title=W. Doeblin, 1915-1940|journal=The Annals of Probability|volume=19|issue=3|year=1991|pages=929–934|issn=0091-1798|doi=10.1214/aop/1176990329|doi-access=free}}</ref>[[File:Joseph_Doob.jpg|thumb|right|数学家约瑟夫-杜布（Joseph Doob）在随机过程理论方面做了早期的工作，做出了基本贡献，尤其是在鞅理论方面。<ref name="Getoor20092">{{cite journal|last1=Getoor|first1=Ronald|title=J. L. Doob: Foundations of stochastic processes and probabilistic potential theory|journal=The Annals of Probability|volume=37|issue=5|year=2009|page=1655|issn=0091-1798|doi=10.1214/09-AOP465|arxiv=0909.4213|bibcode=2009arXiv0909.4213G|s2cid=17288507}}</ref><ref name="Snell20052" />他的书《随机过程》被在概率论领域具有很高的影响力。<ref name="Bingham20052">{{cite journal|last1=Bingham|first1=N. H.|title=Doob: a half-century on|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|pages=257–266|issn=0021-9002|doi=10.1239/jap/1110381385|doi-access=free}}</ref>|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:Joseph_Doob.jpg]]
===二战后的随机过程===
第二次世界大战后，概率论和随机过程的研究得到了数学家的更多关注，在概率论和数学的许多领域做出了重大贡献，并开创了新的领域统计学。<ref name="Cramer19762" /><ref name="Meyer20092">{{cite journal|last1=Meyer|first1=Paul-André|title=Stochastic Processes from 1950 to the Present|journal=Electronic Journal for History of Probability and Statistics|volume=5|issue=1|year=2009|pages=1–42}}</ref>从20世纪40年代开始，伊藤清司（Kiyosi Itô）发表了发展随机微积分领域的论文，其中设计基于维纳或布朗运动过程的随机积分和随机[https://wiki.swarma.org/index.php/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B 微分方程]。<ref name="Ito1998Prize2">{{cite journal|title=Kiyosi Itô receives Kyoto Prize|journal=Notices of the AMS|volume=45|issue=8|year=1998|pages=981–982}}</ref>
同样从20世纪40年代开始，随机过程（尤其是鞅）与数学领域的势理论之间建立了联系，角谷静夫（Shizuo Kakutani）的早期思想和后来约瑟夫（Joseph Doob）的工作都是如此。<ref name="Meyer20092" />在20世纪50年代尔伯特-亨特G（ilbert Hunt）完成了被创性的进一步工作，他把马尔科夫过程和势能理论联系起来，这对莱维过程的理论产生了重大影响，并使人们对用伊藤开发的方法研究马尔科夫过程产生了更多兴趣。<ref name="JarrowProtter20042" /><ref name="Bertoin1998pageVIIIandIX2">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii and ix}}</ref><ref name="Steele2012page1762">{{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=176}}</ref>
1953年，约瑟夫-杜布（Joseph Doob）出版了《随机过程》一书，这本书对随机过程理论产生了重大影响，并强调了测度理论在概率论中的重要性。<ref name="Meyer20092" /><ref name="Bingham20052" />杜布还要发展了鞅理论，后来Paul-André Meyer也作出了重大贡献。早期的工作是由谢尔盖-伯恩斯坦（Sergei Bernstein）、保罗-莱维（Paul Lévy）和让-维尔（Jean Ville）进行的，后者对随机过程采用了“鞅”一词。<ref name="HallHeyde2014page12">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|pages=1, 2}}</ref><ref name="Dynkin19892">{{cite journal|last1=Dynkin|first1=E. B.|title=Kolmogorov and the Theory of Markov Processes|journal=The Annals of Probability|volume=17|issue=3|year=1989|pages=822–832|issn=0091-1798|doi=10.1214/aop/1176991248|doi-access=free}}</ref>鞅理论中的方法已成为解决各种概率问题的常用方法。研究马尔可夫过程的技术和理论被开发出来，然后被应用于鞅上。反之，从鞅理论中也建立了处理马尔可夫过程的方法。<ref name="Meyer20092" />
概率的其他领域也被发展和用于随机过程的研究，其中一个主要方法是大偏差理论。<ref name="Meyer20092" /> 该理论在统计物理等领域有许多应用，其核心思想至少可以追溯到20世纪30年代。在20世纪60年代和70年代后期，苏联的亚历山大·温策尔（Alexander Wentzell）和美国的门罗-D-唐斯克（Monroe D.Donsker）以及斯里尼瓦萨-瓦拉丹（Srinivasa Varadhan）完成了基础工作，<ref name="Ellis1995page982">{{cite journal|last1=Ellis|first1=Richard S.|title=An overview of the theory of large deviations and applications to statistical mechanics|journal=Scandinavian Actuarial Journal|volume=1995|issue=1|year=1995|page=98|issn=0346-1238|doi=10.1080/03461238.1995.10413952}}</ref>后来瓦拉丹获得了2007年阿贝尔奖。<ref name="RaussenSkau20082">{{cite journal|last1=Raussen|first1=Martin|last2=Skau|first2=Christian|title=Interview with Srinivasa Varadhan|journal=Notices of the AMS|volume=55|issue=2|year=2008|pages=238–246}}</ref>在上世纪90年代和21世纪，Schramm-Loewner演化<ref name="HenkelKarevski2012page1132">{{cite book|author1=Malte Henkel|author2=Dragi Karevski|title=Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution|url=https://books.google.com/books?id=fnCQWd0GEZ8C&pg=PA113|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-27933-1|page=113}}</ref>和粗略路径<ref name="FrizVictoir2010page5712" />理论被引入和发展来研究概率论中的随机过程和其他数学对象，这促使菲尔兹奖分别在2008年被授予德林-维尔纳W（endelin Werner）<ref name="Werner2004Fields2">{{cite journal|title=2006 Fields Medals Awarded|journal=Notices of the AMS|volume=53|issue=9|year=2015|pages=1041–1044}}</ref>，在2014年被授予马丁-海勒（Martin Hairer）<ref name="Hairer2004Fields2">{{cite journal|last1=Quastel|first1=Jeremy|title=The Work of the 2014 Fields Medalists|journal=Notices of the AMS|volume=62|issue=11|year=2015|pages=1341–1344}}</ref>。
随机过程理论仍然是研究的焦点，每年都有关于随机过程的国际会议。<ref name="BlathImkeller20112" /><ref name="Applebaum2004page13362" />
===特定随机过程的发现===
虽然辛钦在1930年代给出了随机过程的数学定义<ref>Doob, Joseph (1934). "Stochastic Processes and Statistics". ''Proceedings of the National Academy of Sciences of the United States of America''. '''20''' (6): 376–379. Bibcode:1934PNAS...20..376D. doi:10.1073/pnas.20.6.376. PMC 1076423. <nowiki>PMID 16587907</nowiki></ref><ref>Vere-Jones, David (2006). "Khinchin, Aleksandr Yakovlevich". ''Encyclopedia of Statistical Sciences''. p. 4. doi:10.1002/0471667196.ess6027.pub2. ISBN 978-0471667193</ref>，但在不同的环境中已经发现了具体的随机过程，如布朗运动过程和泊松过程<ref>Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: the early years, 1880–1970". ''A Festschrift for Herman Rubin''. Institute of Mathematical Statistics Lecture Notes - Monograph Series. pp. 75–80. CiteSeerX 10.1.1.114.632. doi:10.1214/lnms/1196285381. ISBN <bdi>978-0-940600-61-4</bdi>. ISSN 0749-2170.</ref><ref>Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes". ''International Statistical Review''. '''80''' (2): 253–268. doi:10.1111/j.1751-5823.2012.00181.x. ISSN 0306-7734.</ref>。一些随机过程的族，如点过程或更新过程，有着漫长而复杂的历史，可以追溯到几个世纪前<ref>D.J. Daley; D. Vere-Jones (2006). ''An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods''. Springer Science & Business Media. pp. 1–4. ISBN <bdi>978-0-387-21564-8</bdi>.</ref>。
==== 伯努利过程 ====
伯努利过程可以作为抛出有偏的硬币的数学模型，它可能是第一个被研究的随机过程<ref name="Florescu2014page3012" />。这个过程是一连串独立的伯努利试验<ref>Bertsekas, Dimitri P.; Tsitsiklis, John N. (2002). ''Introduction to Probability''. Athena Scientific. p. 273. ISBN <bdi>978-1-886529-40-3</bdi>.</ref>，伯努利是以杰克-伯努利（Jackob Bernoulli）的名字命名的，他用这些试验来研究机会博弈，包括克里斯蒂安-惠更斯（Christiaan Huygens）早先提出和研究的概率问题<ref name="Hald2005page2212" />。伯努利的工作，包括伯努利过程，于1713年发表在他的《猜想论》一书中。
<br>
==== 随机游走过程 ====
1905年，卡尔-皮尔逊（Karl Pearson）在提出一个描述平面上随机行走的问题时，创造了随机游走这个术语，其动机是在生物学中的应用，但这种涉及随机行走的问题在其他领域已经得到研究。几个世纪前研究的某些赌博问题也可以被视为涉及随机漫步的问题。<ref name="Weiss2006page12" /><ref>Joel Louis Lebowitz (1984). ''Nonequilibrium phenomena II: from stochastics to hydrodynamics''. North-Holland Pub. pp. 8–10. ISBN <bdi>978-0-444-86806-0</bdi>.</ref>例如，被称为 "赌徒的毁灭"的问题是基于一个简单的随机行走<ref name="KarlinTaylor20122" /><ref name="Florescu2014page3732" />，是一个具有吸收障碍的随机行走的例子。<ref name="Seneta2006page12" /><ref name="Ibe2013page112" />帕斯卡尔、费马和惠恩斯都给出了这个问题的数值解决方案，但没有详细说明他们的方法<ref name="Hald2005page2212" />，然后雅各布-伯努利（Jakob Bernoulli）和亚伯拉罕-德莫伊夫尔（ Abraham de Moivre）提出了更详细的解决方案。
对于n维整数格中的随机行走，乔治·波利亚（George Pólya）在1919年和1921年发表了工作，他研究了对称随机行走返回到格子中先前位置的概率。波利亚表明，一个对称的随机行走，在格子的任何方向前进的概率都是相同的，在一维和二维中会以概率1返回到格子中的前一个位置，但在三维或更高维度中概率为零<ref>Ionut Florescu (2014). ''Probability and Stochastic Processes''. John Wiley & Sons. p. 385. ISBN <bdi>978-1-118-59320-2</bdi>.</ref><ref>Barry D. Hughes (1995). ''Random Walks and Random Environments: Random walks''. Clarendon Press. p. 111. ISBN <bdi>978-0-19-853788-5</bdi>.</ref>。
==== 维纳过程 ====
维纳过程或布朗运动过程起源于不同的领域，包括统计学、金融学和物理学<ref name="JarrowProtter20042" />。1880年，托瓦尔·蒂勒（Thorvald Thiele）写了一篇关于最小二乘法的论文，他用这个过程来研究时间序列分析中模型的误差。<ref>Thiele, Thorwald N. (1880). "Om Anvendelse af mindste Kvadraterbs Methode i nogle Tilfælde, hvoren Komplikation af visse Slags uensartede tilfældige Fejlkilder giver Fejleneen "systematisk" Karakter". ''Kongelige Danske Videnskabernes Selskabs Skrifter''. Series 5 (12): 381–408.</ref><ref>Hald, Anders (1981). "T. N. Thiele's Contributions to Statistics". ''International Statistical Review / Revue Internationale de Statistique''. '''49''' (1): 1–20. doi:10.2307/1403034. ISSN 0306-7734. JSTOR 1403034</ref><ref name=":2">Lauritzen, Steffen L. (1981). "Time Series Analysis in 1880: A Discussion of Contributions Made by T.N. Thiele". ''International Statistical Review / Revue Internationale de Statistique''. '''49''' (3): 319–320. doi:10.2307/1402616. ISSN 0306-7734. JSTOR 1402616.</ref>这项工作现在被认为是卡尔曼滤波的统计方法的早期发现，但这项工作在很大程度上被忽略了。人们认为蒂勒的论文中的观点太过先进，以至于当时更广泛的数学和统计学界都不理解。<ref name=":2" />
法国数学家路易斯·巴施里耶 (Louis Bachelier）在他1900年的论文中使用了维纳过程<ref>Bachelier, Luis (1900). "Théorie de la spéculation" (PDF). ''Ann. Sci. Éc. Norm. Supér.'' Serie 3, 17: 21–89. doi:10.24033/asens.476</ref><ref>Bachelier, Luis (1900). "The Theory of Speculation". ''Ann. Sci. Éc. Norm. Supér''. Serie 3, 17: 21–89 (Engl. translation by David R. May, 2011). doi:10.24033/asens.476</ref>以模拟巴黎证券交易所的价格变化，<ref>Courtault, Jean-Michel; Kabanov, Yuri; Bru, Bernard; Crepel, Pierre; Lebon, Isabelle; Le Marchand, Arnaud (2000). "Louis Bachelier on the Centenary of Theorie de la Speculation" (PDF). ''Mathematical Finance''. '''10''' (3): 339–353. doi:10.1111/1467-9965.00098. ISSN 0960-1627.</ref>但他并不知道蒂勒的工作。<ref name="JarrowProtter20042" />有人猜测巴施里耶从朱尔斯·雷格诺尔（Jules Regnault）的随机漫步模型中汲取了灵感，但Bachelier并没有引用他的话<ref name=":3">Jovanovic, Franck (2012). "Bachelier: Not the forgotten forerunner he has been depicted as. An analysis of the dissemination of Louis Bachelier's work in economics" (PDF). ''The European Journal of the History of Economic Thought''. '''19''' (3): 431–451. doi:10.1080/09672567.2010.540343. ISSN 0967-2567. S2CID 154003579.</ref>，而巴施里耶的论文现在被认为是金融数学领域的先驱。
人们普遍认为，巴施里耶的作品没有获得什么关注，被遗忘了几十年，直到20世纪50年代被伦纳德-萨维奇重新发现，然后在1964年被翻译成英文后变得更加流行。但这项工作在数学界从未被遗忘，因为巴施里耶在1912年出版了一本书，详细介绍了他的想法，<ref name=":3" />这本书被包括杜布、费勒<ref name=":3" />和科尔莫戈罗夫<ref name="JarrowProtter20042" />在内的数学家所引用。这本书仍然被引用着，但后来从1960年代开始，当经济学家开始引用巴施里耶的工作时，巴施里耶的原始论文开始被引用得多于他的书<ref name=":3" />。
1905年，阿尔伯特-爱因斯坦发表了一篇论文，他研究了布朗运动或运动的物理观察，通过使用气体动力学理论的思想来解释液体中颗粒的看似随机的运动。爱因斯坦推导出一个被称为扩散方程的微分方程，用于描述在某一空间区域找到一个粒子的概率。在爱因斯坦发表第一篇关于布朗运动的论文后不久，马里安-斯莫鲁奇夫斯基（ Marian Smoluchowski）发表了他引用爱因斯坦的作品，但他写道，他通过使用不同的方法独立得出了同等的结果<ref>Brush, Stephen G. (1968). "A history of random processes". ''Archive for History of Exact Sciences''. '''5''' (1): 25. doi:10.1007/BF00328110. ISSN 0003-9519. S2CID 117623580.</ref>。
爱因斯坦的工作，以及让-佩兰获得的实验结果，后来在20世纪20年代启发了诺伯特-维纳<ref>Brush, Stephen G. (1968). "A history of random processes". ''Archive for History of Exact Sciences''. '''5''' (1): 1–36. doi:10.1007/BF00328110. ISSN 0003-9519. S2CID 117623580.</ref>，他使用珀西-丹尼尔发展的一种度量理论和傅里叶分析来证明维纳过程作为一个数学对象的存在<ref name="JarrowProtter20042" />。
==== 泊松过程 ====
泊松过程是以西梅翁·泊松（Siméon Poisson）的名字命名的，因为其定义涉及泊松分布，但泊松从未研究过这个过程<ref name="Stirzaker20002" /><ref>D.J. Daley; D. Vere-Jones (2006). ''An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods''. Springer Science & Business Media. pp. 8–9. ISBN <bdi>978-0-387-21564-8</bdi>.</ref>。有许多关于泊松过程的早期使用或发现的说法<ref name="Stirzaker20002" /><ref name="GuttorpThorarinsdottir20122" />。在20世纪初，泊松过程会在不同的情况下独立出现<ref name="Stirzaker20002" /><ref name="GuttorpThorarinsdottir20122" />。1903年在瑞典，菲利普-伦德伯格（ Filip Lundberg）发表了一篇论文，其中包含的工作现在被认为是基本的和开创性的，他提出用一个同质泊松过程来模拟保险索赔<ref>Embrechts, Paul; Frey, Rüdiger; Furrer, Hansjörg (2001). "Stochastic processes in insurance and finance". ''Stochastic Processes: Theory and Methods''. Handbook of Statistics. Vol. 19. p. 367. doi:10.1016/S0169-7161(01)19014-0. ISBN <bdi>978-0444500144</bdi>. ISSN 0169-7161.</ref><ref>Cramér, Harald (1969). "Historical review of Filip Lundberg's works on risk theory". ''Scandinavian Actuarial Journal''. '''1969''' (sup3): 6–12. doi:10.1080/03461238.1969.10404602. ISSN 0346-1238.</ref>。
另一项发现发生在1909年的丹麦，当时尔朗（A.K.Erlang）在为有限时间间隔内的来电数量开发一个数学模型时得出了泊松分布。尔朗当时并不知道泊松的早期工作，他假设每个时间间隔内到达的电话数量是相互独立的。然后，他发现了极限情况，这实际上是将泊松分布重塑为二项分布的极限<ref name="Stirzaker20002" />。
1910年，欧内斯特-卢瑟福（Ernest Rutherford ）和汉斯-盖格（Hans Geiger ）发表了关于α粒子计数的实验结果。在他们工作的推动下，哈里-贝特曼（Harry Bateman）研究了计数问题，并推导出泊松概率作为一组微分方程的解，从而独立发现了泊松过程。<ref name="Stirzaker20002" /> 在这之后，有许多关于泊松过程的研究和应用，但其早期历史很复杂，生物学家、生态学家、工程师和各种物理科学家在众多领域对该过程的各种应用已经说明了这一点<ref name="Stirzaker20002" />。
==== 马尔可夫过程 ====
马尔可夫过程和马尔可夫链是以安德烈-马尔可夫命名的，他在20世纪初研究了马尔可夫链<ref name=":4">Gagniuc, Paul A. (2017). ''Markov Chains: From Theory to Implementation and Experimentation''. NJ: John Wiley & Sons. pp. 2–8. ISBN <bdi>978-1-119-38755-8</bdi>.</ref>。马尔可夫对研究独立随机序列的扩展很感兴趣<ref name=":4" />。在他于1906年发表的第一篇关于马尔可夫链的论文中，马尔可夫表明，在某些条件下，马尔可夫链的平均结果会收敛到一个固定的数值向量，因此证明了一个没有独立假设的弱大数法则，<ref name=":5">Gagniuc, Paul A. (2017). ''Markov Chains: From Theory to Implementation and Experimentation''. NJ: John Wiley & Sons. pp. 1–235. ISBN <bdi>978-1-119-38755-8</bdi>.</ref><ref name=":6">Charles Miller Grinstead; James Laurie Snell (1997). ''Introduction to Probability''. American Mathematical Soc. pp. 464–466. ISBN <bdi>978-0-8218-0749-1</bdi>.</ref> <ref name=":7">Pierre Bremaud (2013). ''Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues''. Springer Science & Business Media. p. ix. ISBN <bdi>978-1-4757-3124-8</bdi>.</ref><ref name=":8">Hayes, Brian (2013). "First links in the Markov chain". ''American Scientist''. '''101''' (2): 92–96. doi:10.1511/2013.101.92.</ref> ，而独立假设曾被普遍认为是这种数学定律成立的条件之一<ref name=":8" />。 马尔可夫后来用马尔可夫链来研究亚历山大-普希金写的《欧仁-奥涅金》中元音的分布，并证明了这种链的中心极限定理<ref name=":5" /><ref name=":6" />。
1912年，庞加莱（Poincaré）研究了有限群上的马尔科夫链，目的是研究重排。马尔科夫链的其他早期应用包括保罗（Paul）和塔季扬娜-埃伦费斯特（Tatyana Ehrenfest）在1907年提出的扩散模型，以及弗朗西斯-高尔顿（Francis Galton）和亨利-威廉-沃森（Henry William Watson）在1873年提出的分支过程，比马尔科夫的工作更早。<ref name=":6" /> <ref name=":7" />在高尔顿和沃森的工作之后，人们发现他们的分支过程在大约三十年前就被伊雷内-朱尔·比内梅（Irénée-Jules Bienaymé）独立发现和研究了<ref>Seneta, E. (1998). "I.J. Bienaymé [1796-1878]: Criticality, Inequality, and Internationalization". ''International Statistical Review / Revue Internationale de Statistique''. '''66''' (3): 291–292. doi:10.2307/1403518. ISSN 0306-7734. JSTOR 1403518.</ref>。从1928年开始，莫里斯·弗雷歇（Maurice Fréchet）开始对马尔科夫链感兴趣，最终导致他在1938年发表了一篇关于马尔科夫链的详细研究<ref name=":6" /><ref>Bru, B.; Hertz, S. (2001). "Maurice Fréchet". ''Statisticians of the Centuries''. pp. 331–334. doi:10.1007/978-1-4613-0179-0_71. ISBN <bdi>978-0-387-95283-3</bdi>.</ref>。
安德雷·柯尔莫哥洛夫（Andrei Kolmogorov）在1931年的一篇论文中发展了早期连续时间马尔可夫过程的大部分理论<ref name="Cramer19762" /><ref name="KendallBatchelor1990page332" />。柯尔莫哥洛夫部分受到了路易斯·巴施里耶（Louis Bachelier）1900年关于股票市场波动的工作以及诺伯特·维纳（Norbert Wiener）关于爱因斯坦布朗运动模型的工作的启发。<ref name="KendallBatchelor1990page332" /><ref>Marc Barbut; Bernard Locker; Laurent Mazliak (2016). ''Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters''. Springer London. p. 5. ISBN <bdi>978-1-4471-7262-8</bdi>.</ref> 他引入并研究了一组被称为扩散过程的特殊马尔可夫过程，然后在其上推导了一组描述该过程的微分方程<ref name="KendallBatchelor1990page332" /><ref>Valeriy Skorokhod (2005). ''Basic Principles and Applications of Probability Theory''. Springer Science & Business Media. p. 146. ISBN <bdi>978-3-540-26312-8</bdi>.</ref>。独立于柯尔莫哥洛夫的工作，悉尼-查普曼（Sydney Chapman）在1928年的一篇论文中，以比柯尔莫哥洛夫更不严格的数学方式，在研究布朗运动时得出了一个方程<ref>Bernstein, Jeremy (2005). "Bachelier". ''American Journal of Physics''. '''73''' (5): 398–396. Bibcode:2005AmJPh..73..395B. doi:10.1119/1.1848117. ISSN 0002-9505.</ref>，现在称为查普曼-科尔莫戈罗夫方程。这些微分方程现在被称为科尔莫戈罗夫方程<ref>William J. Anderson (2012). ''Continuous-Time Markov Chains: An Applications-Oriented Approach''. Springer Science & Business Media. p. vii. ISBN <bdi>978-1-4612-3038-0</bdi>.</ref>或科尔莫戈罗夫-查普曼方程<ref>Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W.; Razborov, A. A.; Robinson, C. A.; Whittle, P. (1990). "Andrei Nikolaevich Kolmogorov (1903–1987)". ''Bulletin of the London Mathematical Society''. '''22''' (1): 57. doi:10.1112/blms/22.1.31. ISSN 0024-6093.</ref>。对马尔科夫过程的基础作出重大贡献的其他数学家包括从1930年代开始的威廉-费勒（William Feller），后来又有从1950年代开始的尤金-代金（Eugene Dynkin）。
==== 莱维过程 ====
像维纳过程和泊松过程一样的莱维过程由1930年带学习他们的保罗·莱维（Paul Lévy）命名<ref name="Applebaum2004page13362" />，但是它们与无限可分分布的联系可以追溯到20世纪20年代<ref name="Bertoin1998pageVIII2" />。这一结果后来由莱维于1934年在更一般的条件下推导出来，然后辛钦（Khinchin）在1937年独立地给出了这一特征函数的另一种形式<ref name="Cramer19762" /><ref>David Applebaum (2004). ''Lévy Processes and Stochastic Calculus''. Cambridge University Press. p. 67. ISBN <bdi>978-0-521-83263-2</bdi>.</ref>。除了莱维、辛钦和柯尔莫哥洛夫之外，福内梯B（runo de Finetti）和伊藤清（Kiyosi Itô）也对莱维过程的理论做出了早期的基本贡献<ref name="Bertoin1998pageVIII2" />。
==参考文献==
==参考文献==
<references/>
<references />
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