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随机过程 (查看源代码)

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~~此词条暂由水流心不竞初译，徐培审校。~~

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|keywords=维纳运动,布朗运动,概率论

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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

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[[File:BMonSphere.jpg|thumb|A computer-simulated realization of a [[Wiener process|Wiener]] or [[Brownian motion]] process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>]]

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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="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>]]

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[[文件：BMonSphere.jpg|thumb |计算机模拟在球体表面实现[[Wiener process | Wiener]]或[[Brownian motion]]过程。Wiener过程被广泛认为是概率论中研究最多、最核心的随机过程<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>]]

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在概率论和相关领域，'''随机过程 stochastic process'''或随机过程 random process是一个数学对象，通常定义为随机变量族。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。它们在生物学、化学、生态学、神经科学、物理学、图像处理、信号处理、控制理论、信息理论、计算机科学、密码学和电信学等许多学科都有应用。此外，金融市场表面上的随机变化促进了随机过程在金融领域的广泛应用。

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Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

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~~球面上的 Wiener维纳或 Brownian布朗运动过程。~~'''~~ 维纳过程Wiener~~ process~~~~'''被广泛认为是概率论研究最多和最核心的''' 随机过程Stochastic processes'''。''' 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。它们在生物学、化学、生态学、神经科学、物理学、图像处理、信号处理、控制理论、信息理论、计算机科学、密码学和电信学等许多学科都有应用。此外，金融市场表面上的随机变化促进了''' 随机过程'''在金融领域的广泛应用。

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In [[probability theory]] and related fields, a '''stochastic''' or '''random process''' is a [[mathematical object]] usually defined as a [[Indexed family|family]] of [[random variable]]s. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system [[random]]ly changing over [[time]], such as the growth of a [[bacteria]]l population, an [[electrical current]] fluctuating due to [[thermal noise]], or the movement of a [[gas]] [[molecule]].<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> Stochastic processes are widely used as [[mathematical models]] of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines such as [[biology]],<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> [[chemistry]],<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> [[ecology]],<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> [[neuroscience]]<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>, [[physics]]<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>, [[image processing]], [[signal processing]],<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> [[Stochastic control|control theory]], <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> [[information theory]],<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> [[computer science]],<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> [[cryptography]]<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> and [[telecommunications]].<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> Furthermore, seemingly random changes in [[financial markets]] have motivated the extensive use of stochastic processes in [[finance]].<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>

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在[[概率论]及相关领域中，“随机”或“随机过程”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]]l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]], [[signal processing]],<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> [[Stochastic control|control theory]], <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]].<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]]中看似随机的变化激发了随机过程在[[金融]]中的广泛使用。~~<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>

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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, [[signal processing]],<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> [[Stochastic control|control theory]], <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.<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激发了随机过程在金融中的广泛使用。<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>

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~~各种应用和对现象和研究反过来又启发了新随机过程的提出。这类随机过程的例子包括：维纳过程或布朗运动过程，被路易·巴切利尔~~ Louis Bachelier ~~用来研究巴黎证券交易所的价格变化；泊松过程，被~~ Erlang 用来研究某段时间内发生的电话数量。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存在，它们在路易·巴切利尔路易·巴舍利耶和 Erlang 发现它们之前和之后，在不同的情景和国家被反复、独立地发现。

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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，Gikhman和Skorokhod <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Rosenblatt。<ref name="Rosenblatt1962">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press}}</ref>）使用人[[Louis Bachelier]]为了研究巴黎证券交易所的价格变化，<ref name="JarrowProtter2004">{{cite book|last1=Jarrow|first1=Robert|title=A Festschrift for Herman Rubin|last2=Protter|first2=Philip|chapter=A short history of stochastic integration and mathematical finance: the early years, 1880–1970|year=2004|pages=75–80|issn=0749-2170|doi=10.1214/lnms/1196285381|citeseerx=10.1.1.114.632|series=Institute of Mathematical Statistics Lecture Notes - Monograph Series|isbn=978-0-940600-61-4}}</ref> 以及[[A.K.Erlang]]使用的[[泊松过程]]来研究某段时间内发生的电话号码。<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> 并且在Bachelor和Erlang之前之后在不同的环境和国家被多次独立地发现<ref name="JarrowProtter2004"/><ref name="GuttorpThorarinsdottir2012">{{cite journal|last1=Guttorp|first1=Peter|last2=Thorarinsdottir|first2=Thordis L.|title=What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes|journal=International Statistical Review|volume=80|issue=2|year=2012|pages=253–268|issn=0306-7734|doi=10.1111/j.1751-5823.2012.00181.x}}</ref>。

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Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the [[Wiener process]] or Brownian motion process,{{efn|The term ''Brownian motion'' can refer to the physical process, also known as ''Brownian movement'', and the stochastic process, a mathematical object, but to avoid ambiguity this article uses the terms ''Brownian motion process'' or ''Wiener process'' for the latter in a style similar to, for example, Gikhman and Skorokhod<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> or Rosenblatt.<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>}} used by [[Louis Bachelier]] to study price changes on the [[Paris Bourse]],<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> and the [[Poisson process]], used by [[A. K. Erlang]] to study the number of phone calls occurring in a certain period of time.<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> These two stochastic processes are considered the most important and central in the theory of stochastic processes,<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> and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.<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>

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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]]或布朗运动过程，{efn |术语“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，Gikhman和Skorokhod <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Rosenblatt。~~<ref name="~~Rosenblatt1962~~">{{cite book|author=~~Murray Rosenblatt~~|title=~~Random Processes~~|url=https://~~archive~~.~~org~~/~~details/randomprocesses00rose_0|url-access~~=~~registration~~|year=~~1962~~|publisher=~~Oxford University Press~~}}</ref>~~}} 使用人[[Louis Bachelier]]为了研究[[巴黎证券交易所]]的价格变化，~~<ref name="~~JarrowProtter2004~~">{{cite book|~~last1~~=~~Jarrow~~|~~first1~~=~~Robert~~|~~title~~=~~A Festschrift for Herman Rubin~~|~~last2~~=~~Protter~~|~~first2~~=~~Philip~~|~~chapter~~=~~A short history of stochastic integration and mathematical finance: the early years, 1880–1970~~|~~year~~=~~2004~~|~~pages~~=~~75–80~~|~~issn~~=~~0749-2170~~|~~doi~~=10.~~1214~~/~~lnms/1196285381~~|~~citeseerx~~=~~10.1.1.114.632~~|~~series~~=~~Institute of Mathematical Statistics Lecture Notes - Monograph Series~~|isbn=978-0-~~940600~~-61-4}}</ref> ~~以及[[A.K.Erlang]]使用的[[泊松过程]]来研究某段时间内发生的电话号码。~~<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> ~~并且在Bachelor和Erlang之前之后在不同的环境和国家被多次独立地发现~~<ref name="~~JarrowProtter2004~~"/><ref name="~~GuttorpThorarinsdottir2012~~">{{cite journal|last1=Guttorp|first1=Peter|last2=Thorarinsdottir|first2=Thordis L.|title=What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes|journal=International Statistical Review|volume=80|issue=2|year=2012|pages=253–268|issn=0306-7734|doi=10.1111/j.1751-5823.2012.00181.x}}</~~ref~~>。

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'''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|author4-link=Yuliya 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对随机变量进行索引。<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线的一个区间索引时，通常使用这两个项。<ref name="GikhmanSkorokhod1969page1"/><ref name="ChaumontYor2012"/>如果随机变量被笛卡尔平面或某些高维欧几里得空间索引，那么随机变量的集合通常被称为'''随机场 random field'''。<ref name="GikhmanSkorokhod1969page1"/><ref name="AdlerTaylor2009page7">{{cite book|author1=Robert J. Adler|author2=Jonathan E. Taylor|title=Random Fields and Geometry|url=https://books.google.com/books?id=R5BGvQ3ejloC|year=2009|publisher=Springer Science & Business Media|isbn=978-0-387-48116-6|pages=7–8}}</ref>随机过程的值并不总是数字，可以是向量或其他数学对象。<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/>

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The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

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~~'''~~<~~font color~~="~~#ff8000~~"> ~~随机函数Random function~~</~~font~~>'''这个术语也用来指随机或随机过程，因为随机过程也可以被解释为函数空间中的随机元素。随机（stochastic）过程和随机（random）过程这两个术语可以互换使用，通常没有专门的数学空间用于对随机变量进行索引。但是，当随机变量被整数或实线的一个区间索引时，通常使用这两个项。随机过程的值并不总是数字，可以是向量或其他数学对象。'''<~~font color~~="~~#ff8000~~"> ~~马尔可夫过程Markov processes，列维过程Lévy processes，高斯过程Gaussian processes，随机场random fields，更新过程renewal processes~~, ~~分支过程branching processes~~</~~font~~>~~'''。随机过程的研究使用了[[概率]]、[[微积分]]、[[线性代数]]、[[集合论]]的数学知识和技术，和~~[[拓扑学]]以及[[数学分析]]的分支，如[[实分析]，[[测度理论]]，[[傅立叶分析]，和[[泛函分析]。随机过程理论被认为是对数学的一个重要贡献，无论从理论上还是应用上，它都一直是一个活跃的研究课题。

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根据随机过程的数学性质，随机过程可以分为不同的类别，包括随机游走，<ref name="LawlerLimic2010">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year= 2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1}}</ref> 鞅（概率论）,<ref name="Williams1991">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5}}</ref> 马尔可夫过程,<ref name="RogersWilliams2000">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year= 2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7}}</ref> Lévy过程,<ref name="ApplebaumBook2004">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2}}</ref> 高斯过程,<ref>{{cite book|author=Mikhail Lifshits|title=Lectures on Gaussian Processes|url=https://books.google.com/books?id=03m2UxI-UYMC|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-24939-6}}</ref> 随机场，<ref name="Adler2010">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA1|year= 2010|publisher=SIAM|isbn=978-0-89871-693-1}}</ref> 更新过程es, 和分支过程.<ref name="KarlinTaylor2012">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year= 2012|publisher=Academic Press|isbn=978-0-08-057041-9}}</ref>。随机过程的研究使用了概率、微积分、线性代数、集合论的数学知识和技术，和[[拓扑学]]以及数学分析的分支，如实分析，测量理论，傅立叶分析，和泛函分析。随机过程理论被认为是对数学的重要贡献<ref name="Applebaum2004">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336–1347}}</ref>，不论由于理论还是应用，它都是一个活跃的研究课题。<ref name="BlathImkeller2011">{{cite book|author1=Jochen Blath|author2=Peter Imkeller|author3=Sylvie Rœlly|title=Surveys in Stochastic Processes|url=https://books.google.com/books?id=CyK6KAjwdYkC|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-072-2}}</ref><ref name="Talagrand2014">{{cite book|author=Michel Talagrand|title=Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems|url=https://books.google.com/books?id=tfa5BAAAQBAJ&pg=PR4|year=2014|publisher=Springer Science & Business Media|isbn=978-3-642-54075-2|pages=4–}}</ref><ref name="Bressloff2014VII">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-08488-6|pages=vii–ix}}</ref>

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The term '''random function''' is also used to refer to a stochastic or random process,<ref name="GusakKukush2010page21">{{cite book|first1=Dmytro|last1=Gusak|first2=Alexander|last2=Kukush|first3=Alexey|last3=Kulik|first4=Yuliya|last4=Mishura|author4-link=Yuliya 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> because a stochastic process can also be interpreted as a random element in a [[function space]].<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> The terms ''stochastic process'' and ''random process'' are used interchangeably, often with no specific [[mathematical space]] for the set that indexes the random variables.<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> But often these two terms are used when the random variables are indexed by the [[integers]] or an [[Interval (mathematics)|interval]] of the [[real line]].<ref name="GikhmanSkorokhod1969page1"/><ref name="ChaumontYor2012"/> If the random variables are indexed by the [[Cartesian plane]] or some higher-dimensional [[Euclidean space]], then the collection of random variables is usually called a [[random field]] instead.<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> The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/>

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术语“随机函数”也用于指随机或随机过程，随机过程的值并不总是数字，可以是向量或其他数学对象。<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/>

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Based on their mathematical properties, stochastic processes can be grouped into various categories, which include [[random walk]]s,<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> [[Martingale (probability theory)|martingales]],<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> [[Markov process]]es,<ref name="RogersWilliams2000">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year= 2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7}}</ref> [[Lévy process]]es,<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> [[Gaussian process]]es,<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> random fields,<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> [[renewal process]]es, and [[branching process]]es.<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> The study of stochastic processes uses mathematical knowledge and techniques from [[probability]], [[calculus]], [[linear algebra]], [[set theory]], and [[topology]]<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> as well as branches of [[mathematical analysis]] such as [[real analysis]], [[measure theory]], [[Fourier analysis]], and [[functional analysis]].<ref name="Billingsley2008">{{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}}</ref><ref name="Brémaud2014">{{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}}</ref><ref name="Bobrowski2005">{{cite book|author=Adam Bobrowski|title=Functional Analysis for Probability and Stochastic Processes: An Introduction|url=https://books.google.com/books?id=q7dR3d5nqaUC|year= 2005|publisher=Cambridge University Press|isbn=978-0-521-83166-6}}</ref> The theory of stochastic processes is considered to be an important contribution to mathematics<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> and it continues to be an active topic of research for both theoretical reasons and applications.<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>

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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]]s，<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）|鞅]],<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]]es,<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]]es,<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]]es,<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]]es, 和 [[分支过程]]es.<ref name="KarlinTaylor2012">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year= 2012|publisher=Academic Press|isbn=978-0-08-057041-9}}</ref>。随机过程的研究使用了[[概率]]、[[微积分]]、[[线性代数]]、[[集合论]]的数学知识和技术，和[[拓扑学]]以及[[数学分析]]的分支，如[[实分析]，[[测量理论]]，[[傅立叶分析]，和[[泛函分析]。随机过程理论被认为是对数学的重要贡献<ref name="Applebaum2004">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336–1347}}</ref>，不论由于理论还是应用，它都是一个活跃的研究课题。<ref name="BlathImkeller2011">{{cite book|author1=Jochen Blath|author2=Peter Imkeller|author3=Sylvie Rœlly|title=Surveys in Stochastic Processes|url=https://books.google.com/books?id=CyK6KAjwdYkC|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-072-2}}</ref><ref name="Talagrand2014">{{cite book|author=Michel Talagrand|title=Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems|url=https://books.google.com/books?id=tfa5BAAAQBAJ&pg=PR4|year=2014|publisher=Springer Science & Business Media|isbn=978-3-642-54075-2|pages=4–}}</ref><ref name="Bressloff2014VII">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-08488-6|pages=vii–ix}}</ref>

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A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.

一个随机过程可以被定义为一组随机变量的集合，这些随机变量被一些数学集合索引，这意味着随机过程的每个随机变量唯一地与集合中的一个元素相关联。

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==简介==

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~~==Introduction简介==~~

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

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When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.

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当解释为时间时，如果随机过程的索引集有有限个或可数的元素，例如有限的一组数字、一组整数或自然数，则该随机过程称为离散时间的。如果索引集是实数轴的某个区间，则时间被称为连续的。这两类随机过程分别称为 离散时间随机过程和 连续时间随机过程。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。如果索引集是整数或整数的子集，那么随机过程也可以称为 随机序列。Jakob Bernoulli于1713年在拉丁文中首次发表了关于概率的著作Ars Conspectandi，他使用了“Ars consuctandi-sive-Stochastice”一词，该词已被翻译为“推测或随机的艺术”。这个短语是由Ladislaus Bortkiewicz在1917年用德语写下的单词stochastik，意思是随机的。“随机过程”一词最早出现在1934年约瑟夫·杜布的一篇论文中。尽管德语这个词在早些时候就被使用过，例如，安德烈·科尔莫戈洛夫（Andrei Kolmogorov）在1931年就使用过。

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A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.<ref name=~~"Parzen1999"~~/><ref name=~~"GikhmanSkorokhod1969page1"~~/> The set used to index the random variables is called the '''index set'''. Historically, the index set was some [[subset]] of the [[real line]], such as the [[natural numbers]], giving the index set the interpretation of time.<ref name=~~"doob1953stochasticP46to47"~~/> ~~Each random variable in the collection takes values from the same [[mathematical space]] known as the '''state space'''. This~~ state ~~space can be, for example, the integers, the real line or <math>n</math>-dimensional Euclidean space.~~<ref name=~~"doob1953stochasticP46to47"~~/>~~<ref name="GikhmanSkorokhod1969page1"/> An~~ '''increment''' ~~is the amount that a stochastic process changes between two index values, often interpreted as two points in time.~~<ref name=~~"KarlinTaylor2012page27"~~/><ref name=~~"Applebaum2004page1337"~~/> A stochastic process can have many [[Outcome (probability)|outcomes]], due to its randomness, and a single outcome of a stochastic process is called, among other names, a '''sample function''' or '''realization'''.<ref name=~~"Lamperti1977page1"~~/><ref name=~~"RogersWilliams2000page121b"~~/>

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

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According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888.

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

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[[File:Wiener process 3d.png|thumb|right|A single computer-simulated '''sample function''' or '''realization''', among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.]]

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[[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]]

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[[文件：维纳工艺3d.png | thumb | right |单个计算机模拟时间0≤t≤2的三维Wiener或Brownian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]]

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===~~Classifications分类~~===

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===分类===

The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. Both "collection", while instead of "index set", sometimes the terms "parameter set" though sometimes it is only used when the stochastic process takes real values. while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a manifold. <math>\{X(t)\}</math> or simply as <math>X</math> or <math>X(t)</math>, although <math>X(t)</math> is regarded as an abuse of function notation. For example, <math>X(t)</math> or <math>X_t</math> are used to refer to the random variable with the index <math>t</math>, and not the entire stochastic process. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial.

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.

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'''~~~~ 随机游走Random walks~~~~'''是一种随机过程，通常定义为欧氏空间中的等价随机变量或随机向量的和，因此它们是在离散时间中变化的过程。但有些人也用这个词来指连续时间中发生变化的过程，特别是在金融领域使用的维纳过程，这种过程导致了一些混淆，因此招致了批评。还有其他各种类型的'''~~~~ 随机游走~~~~'''，定义它们的状态空间可以是其他数学对象，如格子和群。一般来说，它们被高度研究，在不同学科中有许多应用。

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''' 随机游走Random walks'''是一种随机过程，通常定义为欧氏空间中的等价随机变量或随机向量的和，因此它们是在离散时间中变化的过程。但有些人也用这个词来指连续时间中发生变化的过程，特别是在金融领域使用的维纳过程，这种过程导致了一些混淆，因此招致了批评。还有其他各种类型的''' 随机游走'''，定义它们的状态空间可以是其他数学对象，如格子和群。一般来说，它们被高度研究，在不同学科中有许多应用。

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A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If the <math>p=0.5</math>, this random walk is called a symmetric random walk.

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一个经典的'''~~~~ 随机游走~~~~'''的例子被称为'''~~~~ 简单随机游走SimpleRandom walk~~~~'''，这是一个以整数为状态空间的离散时间随机过程，它基于一个'''~~~~伯努利过程Bernoulli process~~~~'''，其中每个伯努利Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游走发生在整数上，它的值随概率<math>p</math>的增加而增加1，或随概率<math>1-p</math>的减少而减少1，所以这种'''~~~~ 随机游走~~~~'''的索引集是自然数，而它的状态空间是整数。如果 <math>p=0.5</math>，这种随机漫步称为'''~~~~ 对称随机游走Symmetric Random walk~~~~'''。

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一个经典的''' 随机游走'''的例子被称为''' 简单随机游走SimpleRandom walk'''，这是一个以整数为状态空间的离散时间随机过程，它基于一个'''伯努利过程Bernoulli process'''，其中每个伯努利Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游走发生在整数上，它的值随概率<math>p</math>的增加而增加1，或随概率<math>1-p</math>的减少而减少1，所以这种''' 随机游走'''的索引集是自然数，而它的状态空间是整数。如果 <math>p=0.5</math>，这种随机漫步称为''' 对称随机游走Symmetric Random walk'''。

The word ''stochastic'' in [[English language|English]] was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a [[Greek language|Greek]] word meaning "to aim at a mark, guess", and the [[Oxford English Dictionary]] gives the year 1662 as its earliest occurrence.<ref name="OxfordStochastic">{{Cite OED|Stochastic}}</ref> In his work on probability ''Ars Conjectandi'', originally published in Latin in 1713, [[Jakob Bernoulli]] used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics".<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> This phrase was used, with reference to Bernoulli, by [[Ladislaus Bortkiewicz]]<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> who in 1917 wrote in German the word ''stochastik'' with a sense meaning random. The term ''stochastic process'' first appeared in English in a 1934 paper by [[Joseph Doob]].<ref name="OxfordStochastic"/> For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term ''stochastischer Prozeß'' was used in German by [[Aleksandr Khinchin]],<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> though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.<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>

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The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. The Wiener process is named after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids.

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'''~~~~ 维纳过程Wiener process~~~~'''是一个具有平稳和独立增量的随机过程，这些增量是基于增量大小的正态分布。维纳过程是以诺伯特 · 维纳的名字命名的，他证明了维纳过程的数学存在性。

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''' 维纳过程Wiener process'''是一个具有平稳和独立增量的随机过程，这些增量是基于增量大小的正态分布。维纳过程是以诺伯特 · 维纳的名字命名的，他证明了维纳过程的数学存在性。

The definition of a stochastic process varies,<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> but a stochastic process is traditionally defined as a collection of random variables indexed by some set.<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/> The terms ''random process'' and ''stochastic process'' are considered synonyms and are used interchangeably, without the index set being precisely specified.<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> Both "collection",<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/> or "family" are used<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> while instead of "index set", sometimes the terms "parameter set"<ref name="Lamperti1977page1"/> or "parameter space"<ref name="AdlerTaylor2009page7"/> are used.

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

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带漂移()和无漂移()的 '''~~~~ 维纳过程~~~~'''(或布朗运动过程)的实现。

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带漂移()和无漂移()的 ''' 维纳过程'''(或布朗运动过程)的实现。

The term ''random function'' is also used to refer to a stochastic or random process,<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> though sometimes it is only used when the stochastic process takes real values.<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/> This term is also used when the index sets are mathematical spaces other than the real line,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 1</ref> while the terms ''stochastic process'' and ''random process'' are usually used when the index set is interpreted as time,<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> and other terms are used such as ''random field'' when the index set is <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a [[manifold]].<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>

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术语'''~~~~ “随机函数”Random function~~~~'''也用于指随机或随机过程，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Loeve1978”>{cite book | author=M.Loève|title=Probability Theory II | url=https://books.google.com/books？id=1y229ybulic | year=1978 | publisher=Springer Science&Business Media | isbn=978-0-387-90262-3 | page=163}</ref><ref name=“Brémaud2014page133”>{cite book |作者=Pierre Brémaud | title=Fourier Analysis and randocial Processes |网址=https://books.google.com/books？id=dP2JBAAAQBAJ&pg=PA1 | year=2014 | publisher=Springer | isbn=978-3-319-09590-5 | page=133}</ref>尽管有时它只在随机过程取实值时使用。<ref name=“Lamperti1977page1”/><ref name=“Ito2006page13”/>当索引集是数学空间而不是实线时，也使用这个术语，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“gusakkush2010page1”>{harvxt | Gusak | Kukush | Kulik | Mishura | 2010}，p.1</ref>，而术语“随机过程”和“随机过程”通常在指数集被解释为时间时使用，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“GusakKukush2010page1”/><ref name=“Bass2011page1”>{引用图书|作者=Richard F.Bass | title=随机过程| url=https://books.google.com/books？id=Ll0T7PIkcKMC | year=2011 | publisher=Cambridge University Press | isbn=978-1-139-50147-7 | page=1}</ref>和其他术语，例如当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或[[流形]].<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/><ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/>name=“adlertaylor2009第7页”/>

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术语''' “随机函数”Random function'''也用于指随机或随机过程，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Loeve1978”>{cite book | author=M.Loève|title=Probability Theory II | url=https://books.google.com/books？id=1y229ybulic | year=1978 | publisher=Springer Science&Business Media | isbn=978-0-387-90262-3 | page=163}</ref><ref name=“Brémaud2014page133”>{cite book |作者=Pierre Brémaud | title=Fourier Analysis and randocial Processes |网址=https://books.google.com/books？id=dP2JBAAAQBAJ&pg=PA1 | year=2014 | publisher=Springer | isbn=978-3-319-09590-5 | page=133}</ref>尽管有时它只在随机过程取实值时使用。<ref name=“Lamperti1977page1”/><ref name=“Ito2006page13”/>当索引集是数学空间而不是实线时，也使用这个术语，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“gusakkush2010page1”>{harvxt | Gusak | Kukush | Kulik | Mishura | 2010}，p.1</ref>，而术语“随机过程”和“随机过程”通常在指数集被解释为时间时使用，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“GusakKukush2010page1”/><ref name=“Bass2011page1”>{引用图书|作者=Richard F.Bass | title=随机过程| url=https://books.google.com/books？id=Ll0T7PIkcKMC | year=2011 | publisher=Cambridge University Press | isbn=978-1-139-50147-7 | page=1}</ref>和其他术语，例如当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或[[流形]].<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/><ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/>name=“adlertaylor2009第7页”/>

Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state space can be <math>n</math>-dimensional Euclidean space. If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant <math> \mu</math>, which is a real number, then the resulting stochastic process is said to have drift <math> \mu</math>.

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在概率论中起着核心作用的'''~~~~ 维纳过程~~~~'''，通常被认为是最重要的和研究过的随机过程，与其他随机过程有联系。它的索引集和状态空间分别为非负数和实数，因此它既有连续索引集又有状态空间。但是这个过程可以定义得更广泛，因此它的状态空间可以是<math>n</math>维的'''~~~~ 欧氏空间Euclidean space~~~~'''。如果增量的平均值为零，那么由此产生的维纳Wiener或布朗Brownian运动过程称为具有零漂移。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>，即一个实数，那么得到的随机过程就具有<math> \mu</math>漂移。

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在概率论中起着核心作用的''' 维纳过程'''，通常被认为是最重要的和研究过的随机过程，与其他随机过程有联系。它的索引集和状态空间分别为非负数和实数，因此它既有连续索引集又有状态空间。但是这个过程可以定义得更广泛，因此它的状态空间可以是<math>n</math>维的''' 欧氏空间Euclidean space'''。如果增量的平均值为零，那么由此产生的维纳Wiener或布朗Brownian运动过程称为具有零漂移。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>，即一个实数，那么得到的随机过程就具有<math> \mu</math>漂移。

===Notation符号===

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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.

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几乎可以肯定，'''~~~~ 维纳过程Wiener process~~~~'''的样本路径在任何地方都是连续的，但是没有可微的地方。它可以看作是简单随机游走的连续形式。这个过程作为其他随机过程的数学极限出现，例如某些随机游走的重新标度，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。~~

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几乎可以肯定，''' 维纳过程Wiener process'''的样本路径在任何地方都是连续的，但是没有可微的地方。它可以看作是简单随机游走的连续形式。这个过程作为其他随机过程的数学极限出现，例如某些随机游走的重新标度，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。

==Examples示例==

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The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes. It plays a central role in quantitative finance, where it is used, for example, in the Black–Scholes–Merton model. The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.

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'''~~~~ 维纳过程Wiener process~~~~'''是马尔可夫过程、列维Lévy 过程和高斯Gaussian 过程等重要随机过程的一个成员。它在定量金融学中扮演着核心角色，例如，在'''~~~~ 布莱克-斯科尔斯-默顿模型Black–Scholes–Merton model~~~~'''中就使用了它。这个过程也用于不同的领域，包括大多数自然科学和一些社会科学分支，作为各种随机现象的数学模型。

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''' 维纳过程Wiener process'''是马尔可夫过程、列维Lévy 过程和高斯Gaussian 过程等重要随机过程的一个成员。它在定量金融学中扮演着核心角色，例如，在''' 布莱克-斯科尔斯-默顿模型Black–Scholes–Merton model'''中就使用了它。这个过程也用于不同的领域，包括大多数自然科学和一些社会科学分支，作为各种随机现象的数学模型。

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==='''~~~~ Bernoulli process伯努利过程~~~~'''===

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===''' Bernoulli process伯努利过程'''===

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The Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes. If the parameter constant of the Poisson process is replaced with some non-negative integrable function of <math>t</math>, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.

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'''~~~~ 泊松过程Poisson process~~~~'''是一个具有不同形式和定义的随机过程。它可以被定义为一个计数过程，这是一个随机过程，代表到某个时间，点或事件的随机数。从零到给定时间区间内的过程点数是泊松随机变量，取决于该时间和某些参数。该过程以自然数为状态空间，非负数为索引集。这个过程也被称为泊松计数过程，因为它可以被解释为计数过程的一个例子。'''~~~~ 齐次泊松过程Homogeneous Poisson process~~~~'''是一类重要的随机过程，如马尔可夫过程和 Lévy 过程的成员。如果将泊松过程的参数常数替换为 < math > t </math > 的非负可积函数，则得到的过程称为'''~~~~ 非齐次或非齐次泊松过程Inhomogeneous or nonhomogeneous Poisson process~~~~'''，其点的平均密度不再是常数。泊松过程作为排队论中的一个基本过程，是数学模型中的一个重要过程，它在特定时间窗内随机发生的事件模型中找到了应用。

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''' 泊松过程Poisson process'''是一个具有不同形式和定义的随机过程。它可以被定义为一个计数过程，这是一个随机过程，代表到某个时间，点或事件的随机数。从零到给定时间区间内的过程点数是泊松随机变量，取决于该时间和某些参数。该过程以自然数为状态空间，非负数为索引集。这个过程也被称为泊松计数过程，因为它可以被解释为计数过程的一个例子。''' 齐次泊松过程Homogeneous Poisson process'''是一类重要的随机过程，如马尔可夫过程和 Lévy 过程的成员。如果将泊松过程的参数常数替换为 < math > t </math > 的非负可积函数，则得到的过程称为''' 非齐次或非齐次泊松过程Inhomogeneous or nonhomogeneous Poisson process'''，其点的平均密度不再是常数。泊松过程作为排队论中的一个基本过程，是数学模型中的一个重要过程，它在特定时间窗内随机发生的事件模型中找到了应用。

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==='''~~~~ 随机游走Random walk~~~~'''===

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===''' 随机游走Random walk'''===

Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.

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A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If the <math>p=0.5</math>, this random walk is called a symmetric random walk.<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>

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'''~~~~ 随机游走Random walk~~~~'''的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，例如其值随概率<math>p</math>增加1，，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<ref name=“Gut2012page88”>{cite book | author=Allan Gut | title=Probability:a Graduate Course=https://books.google.com/books？id=XDFA-n|M5hMC | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4614-4708-5 | page=88}</ref><ref name=“grimmetttstirzaker2001page71”>{引用图书| author1=Geoffrey Grimmett | author2=David Stirzaker | title=概率和随机过程| url=https://books.google.com/books？id=G3ig-0M4wSIC |年份=2001 | publisher=OUP Oxford | isbn=978-0-19-857222-0 | page=71}</ref>

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''' 随机游走Random walk'''的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，例如其值随概率<math>p</math>增加1，，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<ref name=“Gut2012page88”>{cite book | author=Allan Gut | title=Probability:a Graduate Course=https://books.google.com/books？id=XDFA-n|M5hMC | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4614-4708-5 | page=88}</ref><ref name=“grimmetttstirzaker2001page71”>{引用图书| author1=Geoffrey Grimmett | author2=David Stirzaker | title=概率和随机过程| url=https://books.google.com/books？id=G3ig-0M4wSIC |年份=2001 | publisher=OUP Oxford | isbn=978-0-19-857222-0 | page=71}</ref>

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

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< 中心 > < 数学 >

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==='''~~~~ Wiener process维纳过程~~~~'''===

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===''' Wiener process维纳过程'''===

\{X(t):t\in T \}.

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The Wiener process is a stochastic process with stationary and [[independent increments]] that are [[normally distributed]] based on the size of the increments.<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> The Wiener process is named after [[Norbert Wiener]], who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for [[Brownian movement]] in liquids.<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>

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'''~~~~ Wiener process维纳过程~~~~'''是一个随机过程，具有平稳的[[独立的增量]]并且基于增量的大小是[[正态分布的].<ref name=“RogersWilliams2000page1”>{cite book | author1=L.C.G.Rogers | author2=David Williams | title=扩散、马尔可夫过程和鞅：第1卷，基金会网址=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=随机微积分及其应用简介|网址=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=56}</ref>维纳过程是以[[Norbert Wiener]]命名的，他证明了它的数学存在性，但是这个过程也被称为布朗运动过程或仅仅是布朗运动，因为它是液体中[[布朗运动]]的模型科学|卷=5 |议题=1 |年份=1968 |页数=1-2 | issn=0003-9519 | doi=10.1007/BF00328110}}</ref><ref name=“applebauma2004page1338”{{{引用杂志| last1=Applebaum | first1=David | title=Lévy过程：从概率到金融和量子群的概率到金融和量子群| journal=Na从概率到金融和量子群| journal=通知AMS | volume=51 | volume=11；年份=2004 |页数=1338}</ref><refname=“Applebaum2004page1338”/><ref name=“GikhmanSkorokhod1969page21”>{cite book | author1=Iosif Ilyich Gikhman | author2=Anatoly Vladimirovich skorokod | title=随机过程理论简介| url=https://books.google.com/books？id=yJyLzG7N7r8C&pg=PR2 |年份=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=21}</ref>

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''' Wiener process维纳过程'''是一个随机过程，具有平稳的[[独立的增量]]并且基于增量的大小是[[正态分布的].<ref name=“RogersWilliams2000page1”>{cite book | author1=L.C.G.Rogers | author2=David Williams | title=扩散、马尔可夫过程和鞅：第1卷，基金会网址=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=随机微积分及其应用简介|网址=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=56}</ref>维纳过程是以[[Norbert Wiener]]命名的，他证明了它的数学存在性，但是这个过程也被称为布朗运动过程或仅仅是布朗运动，因为它是液体中[[布朗运动]]的模型科学|卷=5 |议题=1 |年份=1968 |页数=1-2 | issn=0003-9519 | doi=10.1007/BF00328110}}</ref><ref name=“applebauma2004page1338”{{{引用杂志| last1=Applebaum | first1=David | title=Lévy过程：从概率到金融和量子群的概率到金融和量子群| journal=Na从概率到金融和量子群| journal=通知AMS | volume=51 | volume=11；年份=2004 |页数=1338}</ref><refname=“Applebaum2004page1338”/><ref name=“GikhmanSkorokhod1969page21”>{cite book | author1=Iosif Ilyich Gikhman | author2=Anatoly Vladimirovich skorokod | title=随机过程理论简介| url=https://books.google.com/books？id=yJyLzG7N7r8C&pg=PR2 |年份=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=21}</ref>

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

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Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.<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|author1-link=Daniel Revuz}}</ref> Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.<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> But the process can be defined more generally so its state space can be <math>n</math>-dimensional Euclidean space.<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> If the [[mean]] of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant <math> \mu</math>, which is a real number, then the resulting stochastic process is said to have drift <math> \mu</math>.<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>

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'''~~~~ Wiener process维纳过程~~~~'''在概率论中起着中心作用，通常被认为是最重要和研究的随机过程，并与其他随机过程联系在一起微积分与金融应用|网址=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=29}</ref><ref name=“florescu214page471”>{cite book |作者=Ionut Florescu | title=概率与随机过程|网址=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=随机过程的第一门课程| url=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | pages=21，22}</ref><ref name=“karatzarshreeve2014pageviii”{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991；publisher=Springer | isbn=978-1-4612-0949-2 | page=VIII}</ref><ref name=“RevuzYor2013pageIX”>{cite book | author1=Daniel Revuz | author2=Marc Yor| title=连续鞅和布朗运动| url=https://books.google.com/books？id=oybncaaqbaj | year=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-06400-9 | page=IX | author1 link=Daniel Revuz}</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=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=“klebaner205page81”/><ref name=“KarlinTaylor2012page21”/><ref>{cite book | author1=Donald L。Snyder | author2=Michael I.Miller | title=时空中的随机点过程| url=https://books.google.com/books？id=c_3UBwAAQBAJ | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4612-3166-0 | page=33}</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<math>\mu</math><ref name=“Steele2012page118”>{cite book | author=J.Michael Steele | title=随机微积分和金融应用程序| url=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=118}</ref><ref name=“MörtersPeres2010page1”/><ref name=“Karatzasshreeve2014page78”>{cite book | author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机演算| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 |年份=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=78}</ref>

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''' Wiener process维纳过程'''在概率论中起着中心作用，通常被认为是最重要和研究的随机过程，并与其他随机过程联系在一起微积分与金融应用|网址=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=29}</ref><ref name=“florescu214page471”>{cite book |作者=Ionut Florescu | title=概率与随机过程|网址=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=随机过程的第一门课程| url=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | pages=21，22}</ref><ref name=“karatzarshreeve2014pageviii”{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991；publisher=Springer | isbn=978-1-4612-0949-2 | page=VIII}</ref><ref name=“RevuzYor2013pageIX”>{cite book | author1=Daniel Revuz | author2=Marc Yor| title=连续鞅和布朗运动| url=https://books.google.com/books？id=oybncaaqbaj | year=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-06400-9 | page=IX | author1 link=Daniel Revuz}</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=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=“klebaner205page81”/><ref name=“KarlinTaylor2012page21”/><ref>{cite book | author1=Donald L。Snyder | author2=Michael I.Miller | title=时空中的随机点过程| url=https://books.google.com/books？id=c_3UBwAAQBAJ | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4612-3166-0 | page=33}</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<math>\mu</math><ref name=“Steele2012page118”>{cite book | author=J.Michael Steele | title=随机微积分和金融应用程序| url=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=118}</ref><ref name=“MörtersPeres2010page1”/><ref name=“Karatzasshreeve2014page78”>{cite book | author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机演算| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 |年份=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=78}</ref>

The mathematical space <math>S</math> of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, <math>n</math>-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.

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[[Almost surely]], a sample path of a Wiener process is continuous everywhere but [[nowhere differentiable function|nowhere differentiable]]. It can be considered as a continuous version of the simple random walk.<ref name="Applebaum2004page1337">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|page=1337}}</ref><ref name="MörtersPeres2010page1">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|pages=1, 3}}</ref> The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,<ref name="KaratzasShreve2014page61">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=61}}</ref><ref name="Shreve2004page93">{{cite book|author=Steven E. Shreve|title=Stochastic Calculus for Finance II: Continuous-Time Models|url=https://books.google.com/books?id=O8kD1NwQBsQC|year=2004|publisher=Springer Science & Business Media|isbn=978-0-387-40101-0|page=93}}</ref> which is the subject of [[Donsker's theorem]] or invariance principle, also known as the functional central limit theorem.<ref name="Kallenberg2002page225and260">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=225, 260}}</ref><ref name="KaratzasShreve2014page70">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=70}}</ref><ref name="MörtersPeres2010page131">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=131}}</ref>

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[[几乎可以肯定]]，'''~~~~ Wiener process维纳过程~~~~'''的样本路径处处连续，但[[无处可微函数|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过程：从概率到金融和量子群| journal=AMS的通知| volume=51 | issue=11 | year=2004|page=1337}</ref name=“MörtersPeres2010page1”>{citebook | author1=Peter Mörters | author2=Yuval Peres | title=布朗运动|网址=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=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=61}</ref><ref name=“Shreve2004page93”>{cite book |作者=Steven E.Shreve | title=金融随机微积分II：连续时间模型| url=https://books.google.com/books？id=O8kD1NwQBsQC | year=2004 | publisher=Springer Science&Business Media | isbn=978-0-387-40101-0 | page=93}</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=https://books.google.com/books？id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | pages=225260}}</ref><ref name=“karatzarshreve2014page70”{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机演算| 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=布朗运动| url=https://books.google.com/books？id=e-TbA-dSrzYC |年=2010 | publisher=剑桥大学出版社| isbn=978-1-139-48657-6 | page=131}</ref>

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[[几乎可以肯定]]，''' Wiener process维纳过程'''的样本路径处处连续，但[[无处可微函数|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过程：从概率到金融和量子群| journal=AMS的通知| volume=51 | issue=11 | year=2004|page=1337}</ref name=“MörtersPeres2010page1”>{citebook | author1=Peter Mörters | author2=Yuval Peres | title=布朗运动|网址=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=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=61}</ref><ref name=“Shreve2004page93”>{cite book |作者=Steven E.Shreve | title=金融随机微积分II：连续时间模型| url=https://books.google.com/books？id=O8kD1NwQBsQC | year=2004 | publisher=Springer Science&Business Media | isbn=978-0-387-40101-0 | page=93}</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=https://books.google.com/books？id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | pages=225260}}</ref><ref name=“karatzarshreve2014page70”{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机演算| 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=布朗运动| url=https://books.google.com/books？id=e-TbA-dSrzYC |年=2010 | publisher=剑桥大学出版社| isbn=978-1-139-48657-6 | page=131}</ref>

The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.<ref name="RogersWilliams2000page1"/><ref name="Applebaum2004page1337"/> The process also has many applications and is the main stochastic process used in stochastic calculus.<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> It plays a central role in quantitative finance,<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> where it is used, for example, in the Black–Scholes–Merton model.<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> The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.<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>

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'''~~~~ Wiener process维纳过程~~~~'''是一些重要的随机过程家族的成员，包括马尔可夫过程，Lévy过程和高斯过程。<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=随机微积分简介应用程序| url=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7}</ref><ref name=“KaratzasShreve2014page”>{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2}</ref>它在数量金融中起着核心作用，{{本刊|从概率到金融金融和量子集团的过程〈124; journal从概率到金融和量子群的群| journal=journal=金融金融和量子群群| journal=noticof the AMS | volume=51 | issue=11 |年=2004年2004年| page=1341}}</ref><ref name=“KarlinTaylor2012page340 340{引用书〈author1=author1=Samuel Karlin | author1=Samuel Karlin；author2=Howard2=Howarde.Taylor | Howarde.Taylor标题=第一门课程随机过程| url=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical 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 Rastic Calculation with Applications |网址=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=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 |作者=Ubbo F.Wiersema | title=布朗运动演算| url=https://books.google.com/books？id=0h-n0WWuD9cC |年=2008 | publisher=John Wiley&Sons | isbn=978-0-470-02171-2 | page=2}</ref>

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''' Wiener process维纳过程'''是一些重要的随机过程家族的成员，包括马尔可夫过程，Lévy过程和高斯过程。<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=随机微积分简介应用程序| url=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7}</ref><ref name=“KaratzasShreve2014page”>{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2}</ref>它在数量金融中起着核心作用，{{本刊|从概率到金融金融和量子集团的过程〈124; journal从概率到金融和量子群的群| journal=journal=金融金融和量子群群| journal=noticof the AMS | volume=51 | issue=11 |年=2004年2004年| page=1341}}</ref><ref name=“KarlinTaylor2012page340 340{引用书〈author1=author1=Samuel Karlin | author1=Samuel Karlin；author2=Howard2=Howarde.Taylor | Howarde.Taylor标题=第一门课程随机过程| url=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical 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 Rastic Calculation with Applications |网址=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=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 |作者=Ubbo F.Wiersema | title=布朗运动演算| url=https://books.google.com/books？id=0h-n0WWuD9cC |年=2008 | publisher=John Wiley&Sons | isbn=978-0-470-02171-2 | page=2}</ref>

A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, the mapping

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A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).

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马尔可夫链是一种具有离散状态空间或离散指标集(通常表示时间)的马尔可夫过程，但是马尔可夫链的精确定义是变化的。例如，通常将'''~~~~ 马尔可夫链Markov chain~~~~'''定义为离散或连续时间中具有可数状态空间的马尔可夫过程(因此不考虑时间的性质) ，但也通常将'''~~~~ 马尔可夫链Markov chain~~~~'''定义为在可数或连续状态空间中具有离散时间的马尔可夫链(因此不考虑状态空间)。

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马尔可夫链是一种具有离散状态空间或离散指标集(通常表示时间)的马尔可夫过程，但是马尔可夫链的精确定义是变化的。例如，通常将''' 马尔可夫链Markov chain'''定义为离散或连续时间中具有可数状态空间的马尔可夫过程(因此不考虑时间的性质) ，但也通常将''' 马尔可夫链Markov chain'''定义为在可数或连续状态空间中具有离散时间的马尔可夫链(因此不考虑状态空间)。

</math></center>

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Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.

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'''~~~~ 马尔可夫过程Markov processes~~~~'''是一类重要的随机过程，在许多领域有着广泛的应用。例如，它们是一种通用的随机模拟方法的基础，这种方法被称为'''~~~~ 马尔科夫蒙特卡洛模拟法Markov chain MonteCarlo~~~~'''，用于模拟具有特定概率分布的随机目标，并已在贝叶斯统计中得到应用。

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''' 马尔可夫过程Markov processes'''是一类重要的随机过程，在许多领域有着广泛的应用。例如，它们是一种通用的随机模拟方法的基础，这种方法被称为''' 马尔科夫蒙特卡洛模拟法Markov chain MonteCarlo'''，用于模拟具有特定概率分布的随机目标，并已在贝叶斯统计中得到应用。

====Stationarity稳定性====

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A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.

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'''~~~~ 鞅Martingale~~~~'''是一个离散时间或连续时间的随机过程，其特性是，在给定过程的当前值和所有过去值的任何时刻，每个未来值的条件期望都等于当前值。在离散时间中，如果此属性对下一个值有效，则对所有未来值都有效。'''~~~~ 鞅Martingale~~~~'''的精确数学定义需要两个其他条件加上过滤的数学概念，这与随着时间的推移增加可用信息的直觉有关。'''~~~~ 鞅Martingale~~~~'''通常被定义为实值的，但是它们也可以取复值，甚至是更一般的值。

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''' 鞅Martingale'''是一个离散时间或连续时间的随机过程，其特性是，在给定过程的当前值和所有过去值的任何时刻，每个未来值的条件期望都等于当前值。在离散时间中，如果此属性对下一个值有效，则对所有未来值都有效。''' 鞅Martingale'''的精确数学定义需要两个其他条件加上过滤的数学概念，这与随着时间的推移增加可用信息的直觉有关。''' 鞅Martingale'''通常被定义为实值的，但是它们也可以取复值，甚至是更一般的值。

</math></center>

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A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

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在离散时间和连续时间中，'''~~~~ 对称随机游动Symmetric random walk~~~~'''和维纳Wiener 过程(带零漂移)都是'''~~~~ 鞅Martingale~~~~'''的例子。在这方面，离散'''~~~~ 鞅Martingale~~~~'''推广了独立随机变量部分和的概念。

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在离散时间和连续时间中，''' 对称随机游动Symmetric random walk'''和维纳Wiener 过程(带零漂移)都是''' 鞅Martingale'''的例子。在这方面，离散''' 鞅Martingale'''推广了独立随机变量部分和的概念。

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Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.

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也可以通过适当的变换从随机过程中产生'''~~~~ 鞅Martingale~~~~'''，这是齐次泊松过程(在实线上)产生一个被称为补偿泊松过程的'''~~~~ 鞅Martingale~~~~'''的情形。

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也可以通过适当的变换从随机过程中产生''' 鞅Martingale'''，这是齐次泊松过程(在实线上)产生一个被称为补偿泊松过程的''' 鞅Martingale'''的情形。

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Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments

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'''~~~~ 鞅Martingales~~~~'''在统计学中有许多应用，但有人指出，鞅的使用和应用并不象在统计学领域，特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性，因此它们被称为具有平稳增量和独立增量的过程。换句话说，如果对于 <math>n</math> 非负数，<math>0\leq t_1\leq \dots \leq t_n</math> ，相应的 <math>n-1</math> 递增值是一个列维 Lévy 过程

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''' 鞅Martingales'''在统计学中有许多应用，但有人指出，鞅的使用和应用并不象在统计学领域，特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性，因此它们被称为具有平稳增量和独立增量的过程。换句话说，如果对于 <math>n</math> 非负数，<math>0\leq t_1\leq \dots \leq t_n</math> ，相应的 <math>n-1</math> 递增值是一个列维 Lévy 过程

A [[Filtration (probability theory)|filtration]] is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some [[total order]] relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math>\{\mathcal{F}_t\}_{t\in T} </math>, on a probability space <math>(\Omega, \mathcal{F}, P)</math> is a family of sigma-algebras such that <math> \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} </math> for all <math>s \leq t</math>, where <math>t, s\in T</math> and <math>\leq</math> denotes the total order of the index set <math>T</math>.<ref name="Florescu2014page294"/> With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process <math>X_t</math> at <math>t\in T</math>, which can be interpreted as time <math>t</math>.<ref name="Florescu2014page294"/><ref name="Williams1991page93"/> The intuition behind a filtration <math>\mathcal{F}_t</math> is that as time <math>t</math> passes, more and more information on <math>X_t</math> is known or available, which is captured in <math>\mathcal{F}_t</math>, resulting in finer and finer partitions of <math>\Omega</math>.<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>

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In 1905 Karl Pearson coined the term random walk while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. and is an example of a random walk with absorbing barriers. Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods, and then more detailed solutions were presented by Jakob Bernoulli and Abraham de Moivre.

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1905年，卡尔 · 皮尔森在提出一个描述平面上随机漫步的问题时，创造了'''~~~~ 随机漫步Random walk~~~~'''这个术语，这个问题的动机是生物学中的一个应用，但是这种涉及随机漫步的问题已经在其他领域得到了研究。几个世纪前研究过的某些赌博问题可以被认为是涉及随机漫步的问题。这是一个带有吸收屏障的随机漫步的例子。和 Huyens 都给出了这个问题的数值解，但没有详细介绍他们的方法，然后 Jakob Bernoulli 和亚伯拉罕·棣莫弗提供了更详细的解。

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1905年，卡尔 · 皮尔森在提出一个描述平面上随机漫步的问题时，创造了''' 随机漫步Random walk'''这个术语，这个问题的动机是生物学中的一个应用，但是这种涉及随机漫步的问题已经在其他领域得到了研究。几个世纪前研究过的某些赌博问题可以被认为是涉及随机漫步的问题。这是一个带有吸收屏障的随机漫步的例子。和 Huyens 都给出了这个问题的数值解，但没有详细介绍他们的方法，然后 Jakob Bernoulli 和亚伯拉罕·棣莫弗提供了更详细的解。

====Orthogonality正交性====

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另一个问题是，连续时间过程的泛函依赖于指数集中无法计算的点数，因此某些事件的概率可能无法很好地定义。可分性保证了无穷维分布决定样本函数的性质，它要求样本函数本质上是由指数集中的稠密可数点集上的值决定的。此外，如果随机过程是可分的，那么指数集上不可数个点的泛函是可测的，并且可以研究它们的概率。对于任意度量空间作为状态空间的连续时间随机过程。为了构造这样一个随机过程，我们假设随机过程的样本函数属于某个适当的函数空间，这个空间通常是由所有右连续函数和左极限组成的 Skorokhod 空间。这种方法现在比可分离性假设更常用，但是基于这种方法的随机过程可自动分离。

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==='''~~~~ 马尔可夫过程与链Markov processes and chains~~~~'''===

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===''' 马尔可夫过程与链Markov processes and chains'''===

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Markov processes are stochastic processes, traditionally in [[Discrete time and continuous time|discrete or continuous time]], that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.<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>

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'''~~~~ 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 randocial 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 |作者=Y.A.Rozanov | title=Markov Random Fields| url=https://books.google.com/books？id=wguecaaqbaj | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4613-8190-7 | page=58}</ref>

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''' 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 randocial 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 |作者=Y.A.Rozanov | title=Markov Random Fields| url=https://books.google.com/books？id=wguecaaqbaj | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4613-8190-7 | page=58}</ref>

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes<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> in continuous time, while [[random walk]]s on the integers and the [[gambler's ruin]] problem are examples of Markov processes in discrete time.<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>

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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”>{引用图书|作者=Y.A.Rozanov | title=Markov随机场 |网址=https://books.google.com/books？id=wguecaaqbaj | 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=时空中的随机点过程| url=https://books.google.com/books？id=c_3UBwAAQBAJ | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4612-3166-0 | page=27}</ref><ref name=“bremaud2013 page253”>{cite book |作者=Pierre Bremaud | title=Markov Chains:Gibbs Fields，montecarlo Simulation，and Queues |网址=https://books.google.com/books？id=jrpvwwaaqbaj |年份=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4757-3124-8 | page=253}</ref>

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==='''~~~~鞅Martingale~~~~'''===

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==='''鞅Martingale'''===

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A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,<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> or even more general.<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>

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'''~~~~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Martingale~~~~'''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Martingale~~~~'''通常被定义为实值，<ref name=“Klebaner2005page65”>{cite book | author=Fima C.Klebaner | title=随机微积分及其应用简介=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=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=11}</ref><ref name=“Williams1991page93”>{引用图书|作者=David Williams | title=Probability with鞅| url=https://books.google.com/books？id=e9saZ0YSi AC | year=1991 | publisher=Cambridge University Press | isbn=978-0-521-40605-5 | pages=93，94}</ref>但是它们也可以是复杂值<ref name=“Doob1990page292”>{cite book | author=Joseph L.Doob | title=randouses | url=https://books.google.com/books？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=2929293}</ref>或更一般的。<ref name=“Pisier2016”>{cite book | author=Gilles Pisier | title=Banach空格中的鞅| url=https://books.google.com/books？id=n3JNDAAAQBAJ&pg=PR4 | year=2016 | publisher=Cambridge University Press | isbn=978-1-316-67946-3}</ref>

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'''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Martingale'''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Martingale'''通常被定义为实值，<ref name=“Klebaner2005page65”>{cite book | author=Fima C.Klebaner | title=随机微积分及其应用简介=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=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=11}</ref><ref name=“Williams1991page93”>{引用图书|作者=David Williams | title=Probability with鞅| url=https://books.google.com/books？id=e9saZ0YSi AC | year=1991 | publisher=Cambridge University Press | isbn=978-0-521-40605-5 | pages=93，94}</ref>但是它们也可以是复杂值<ref name=“Doob1990page292”>{cite book | author=Joseph L.Doob | title=randouses | url=https://books.google.com/books？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=2929293}</ref>或更一般的。<ref name=“Pisier2016”>{cite book | author=Gilles Pisier | title=Banach空格中的鞅| url=https://books.google.com/books？id=n3JNDAAAQBAJ&pg=PR4 | year=2016 | publisher=Cambridge University Press | isbn=978-1-316-67946-3}</ref>

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A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.<ref name="Klebaner2005page65"/><ref name="KaratzasShreve2014page11"/> For a [[sequence]] of [[independent and identically distributed]] random variables <math>X_1, X_2, X_3, \dots</math> with zero mean, the stochastic process formed from the successive partial sums <math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> is a discrete-time martingale.<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> In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.<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>

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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Martingale~~~~'''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Martingale~~~~'''<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>

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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Martingale'''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Martingale'''<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>

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the ''compensated Poisson process''.<ref name="KaratzasShreve2014page11"/> Martingales can also be built from other martingales.<ref name="Steele2012page12"/> For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.<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>

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通过应用适当的变换，也可以从随机过程中产生'''~~~~鞅Martingale~~~~'''，这就是齐次泊松过程（在实线上）的情形，其结果是一个称为“补偿泊松过程”的鞅。<ref name=“karatzashreserve2014page11”/>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=随机微积分与金融应用=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=115}</ref>

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通过应用适当的变换，也可以从随机过程中产生'''鞅Martingale'''，这就是齐次泊松过程（在实线上）的情形，其结果是一个称为“补偿泊松过程”的鞅。<ref name=“karatzashreserve2014page11”/>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=随机微积分与金融应用=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=115}</ref>

Martingales mathematically formalize the idea of a fair game,<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> and they were originally developed to show that it is not possible to win a fair game.<ref name="Steele2012page11"/> But now they are used in many areas of probability, which is one of the main reasons for studying them.<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> Many problems in probability have been solved by finding a martingale in the problem and studying it.<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> Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to [[martingale convergence theorem]]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>

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Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<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> They have found applications in areas in probability theory such as queueing theory and Palm calculus<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> and other fields such as economics<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"/>

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'''~~~~鞅Martingale~~~~'''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，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=排队论的元素：Palm鞅演算和随机递归| url=https://books.google.com/books？id=dh3pcaaqbaj&pg=PR2 | year=2013 | publisher=Springer科学与商业媒体| isbn=978-3-662-11657-9}</ref>。以及其他领域，如经济学<ref name=“HallHeyde2014pageX”>{cite book | author1=P.Hall | author2=C.C.Heyde | title=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=x}</ref>和金融。<ref name=“Musielarukowski2006”/>

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'''鞅Martingale'''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，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=排队论的元素：Palm鞅演算和随机递归| url=https://books.google.com/books？id=dh3pcaaqbaj&pg=PR2 | year=2013 | publisher=Springer科学与商业媒体| isbn=978-3-662-11657-9}</ref>。以及其他领域，如经济学<ref name=“HallHeyde2014pageX”>{cite book | author1=P.Hall | author2=C.C.Heyde | title=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=x}</ref>和金融。<ref name=“Musielarukowski2006”/>

===Lévy process莱维过程===

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Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.<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> These processes have many applications in fields such as finance, fluid mechanics, physics and biology.<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> The main defining characteristics of these processes are their stationarity and independence properties, so they were known as ''processes with stationary and independent increments''. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments

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'''~~~~ 莱维Lévy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”>{引用图书|作者=Jean Bertoin | title=莱维过程 |网址=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=Cambridge University Press | isbn=978-0-521-64632-1 | page=viii}}</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

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''' 莱维Lévy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”>{引用图书|作者=Jean Bertoin | title=莱维过程 |网址=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=Cambridge University Press | isbn=978-0-521-64632-1 | page=viii}}</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

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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

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~~Category:Stochastic models~~

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~~类别: 随机模型~~

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~~Category:Statistical data types~~

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~~类别: 统计数据类型~~

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~~<noinclude>~~

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~~This page was moved from [[wikipedia:en:Stochastic process]]. Its edit history can be viewed at [[随机过程/edithistory]]</noinclude>~~

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~~此页摘自[[维基百科：英语：随机过程]]。其编辑历史记录可以在[[随efor过程/edithistory]]]</noinclude>查阅~~

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[[Category:~~待整理页面~~]]

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[[Category:随机模型]]

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[[Category:统计数据类型]]

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