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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"/>]]

[[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"/>]]

  --[[用户:小趣木木|小趣木木]]（[[用户讨论:小趣木木|讨论]]）图注没有翻译

[[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"/>]]

 

 

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.

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.

球面上的 Wiener 或 Brownian 运动过程。'''<font color="#ff8000"> 维纳过程Wiener process</font>'''被广泛认为是概率论研究最多和最核心的'''<font color="#ff8000"> 随机过程Stochastic processes</font>'''。'''<font color="#ff8000"> 随机过程Stochastic processes</font>'''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'''<font color="#ff8000"> 随机过程Stochastic processes</font>'''在金融领域的广泛应用。

球面上的 <font color="#ff8000"> Wiener维纳</font>或 <font color="#ff8000"> Brownian布朗 运动过程</font>。'''<font color="#ff8000"> 维纳过程Wiener process</font>'''被广泛认为是概率论研究最多和最核心的'''<font color="#ff8000"> 随机过程Stochastic processes</font>'''。'''<font color="#ff8000"> 随机过程</font>'''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'''<font color="#ff8000"> 随机过程</font>'''在金融领域的广泛应用。

  --[[用户:小趣木木|小趣木木]]（[[用户讨论:小趣木木|讨论]]）球面上的 Wiener 或 Brownian 运动过程 首次出现应该翻译出来维纳过程...

   

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>

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>

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[[索引族]]，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”>{引用图书| author1=Iosif Ilyich Gikhman | author2=Anatoly Vladimirovich Skorokhod | title=随机过程理论简介| url=图书https://books.com/？id=q0lo91imeD0C | year=1969 | publisher=Courier Corporation | isbn=978-0-486-693877-3 | page=1}</ref><ref name=“：0”{{引用图书；title=马尔可夫链：从理论到实施和实验；last=Gagniuc | first=Paul A.；出版商=John Wiley&Sons；年=2017年| isbn=978-1-119-387755-3 |位置=NJ NJ NJ[NJ-NJ:从理论到实施到实施和实验；最后=最后=最后随机过程是广泛存在的用作以随机方式变化的系统和现象的[[数学模型]]。{124lossf[author=124lossf]=图书https://books.com/？id=swzybaaqbaj | year=2014 | publisher=Springer | isbn=978-3-319-08488-6}</ref>[[chemistry]]，<ref name=“Kampen2011”>{cite book | author=N.G.Van Kampen | title=物理和化学中的随机过程| url=图书https://books.com/？id=N6II-6HlPxEC | year=2011 | publisher=Elsevier | isbn=978-0-08-047536-3}</ref>[[economic]]，<ref name=“LandeEngen2003”>{引用图书| author1=Russell Lande | author2=Steinar Engen | author3=Bernt Erik S|ther | title=生态学和保护中的随机种群动态| url=图书https://books.com/？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=神经科学中的随机方法| url=图书https://books.com/？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=随机过程：从物理到金融| url=图书https://books.com/？id=owanaaaqbaj | year=2013 | publisher=Springer Science&Business Media | isbn=978-3-319-00327-6}</ref>，[[image processing]]，[[signal processing]]，<ref name=“dougherty999”>{cite book | author=Edward R.Dougherty | title=图像和信号处理的随机过程| url=图书https://books.com/？id=epxdaqaaaj | year=1999 | publisher=SPIE光学工程出版社| isbn=978-0-8194-2513-3}</ref>[[随机控制|控制理论]]，<ref name=“Bertsekas1996”>{cite book | author=Dimitri P.Bertsekas | title=随机最优控制：离散时间情况| url=http://www.athenasc.com/socbook.html|年份=1996 | publisher=Athena Scientific]| isbn=1-886529-03-5}</ref>[[信息理论]]，<ref name=“CoverThomas2012page71”>{cite book | author1=Thomas M.Cover | author2=Joy A.Thomas | title=Elements of Information Theory |网址=图书https://books.com/？id=VWq5GG6ycxMC=PT16 | year=2012 | publisher=John Wiley&Sons | isbn=978-1-118-58577-1 | page=71}</ref>[[computer science]]，<ref name=“Baron2015”>{引用图书|作者=Michael Baron | title=计算机科学家的概率与统计，第二版|网址=图书https://books.com/？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=现代密码学导论：原则和协议| url=https://archive.org/details/Introduction_到\u现代加密|年份=2007年|出版商=CRC按| isbn=978-1-58488-586-3 |页=[https://archive.org/details/Introduction_to_Modern_加密技术/page/n4426]}</ref>和[[telecommunications].<ref name=“BaccelliBlaszczyszyn2009”>{cite book | author1=fraçois Baccelli|author2=Bartlomiej blaszzzyszyn | title=随机几何和无线网络| url=图书https://books.com/？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

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[[索引族]]，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>

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, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.

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, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.

现象的应用和研究反过来激发了新的随机过程的提出。这类随机过程的例子包括路易斯 · 巴舍利耶用来研究巴黎证券交易所价格变化的'''<font color="#ff8000"> 维纳过程Wiener process</font>''或'''<font color="#ff8000"> 布朗运动过程Brownian motion process</font>''，以及 a · k · 埃尔朗用来研究在一定时期内通话次数的'''<font color="#ff8000"> 泊松过程Poisson process</font>'''。这两个随机过程在随机过程理论中被认为是最重要和最核心的，并且在巴舍利耶和 Erlang 之前和之后，在不同的环境和国家被重复和独立地发现。

现象的应用和研究反过来激发了新的随机过程的提出。这类随机过程的例子包括路易斯 · 巴舍利耶用来研究巴黎证券交易所价格变化的'''<font color="#ff8000"> 维纳过程Wiener process</font>''或'''<font color="#ff8000"> 布朗运动过程Brownian motion process</font>''，以及 a · k · 埃尔朗用来研究在一定时期内通话次数的'''<font color="#ff8000"> 泊松过程Poisson process</font>'''。这两个随机过程在随机过程理论中被认为是最重要和最核心的，并且在巴舍利耶和 Erlang 之前和之后，在不同背景和国家多次独立地被发现。

 

 

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>

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>

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=随机过程理论导论| url=图书https://books.com/？id=yJyLzG7N7r8C |年份=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3}</ref>或Rosenblatt。<ref name=“Rosenblatt1962”>{引用图书|作者=Murray Rosenblatt | title=Random Processes | url=https://archive.org/details/randomprocess00rose\u 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=随机积分和数学金融学简史：早期，1880-1970 |年份=2004 |页数=75–80 | issn=0749-2170 | doi=10.1214/lnms/1196285381 | citeserx=10.1.1.114.632 |系列=数理统计研究所讲座笔记-专著系列| isbn=978-0-940600-61-4}</ref>和[[Poisson过程]]，被[[A.K.Erlang]]用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，或，常数可以变化| journal=The mathematic 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=时空中的随机点过程| url=图书https://books.com/？id=c_3UBwAAQBAJ | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4612-3166-0 | page=32}}</ref>并且在Bachelier和Erlang前后被反复独立地发现，在不同的环境和国家。<ref name=“JarrowProtter2004”/><ref name=“GuttorpThorarinsdottir2012”>{cite journal | last1=Guttorp | first1=Peter | last2=Thorarinsdottir | first2=Thordis L.| title=离散混沌、Quenouille过程和Sharp Markov属性发生了什么？随机点过程的一些历史| 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>

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>。

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.

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.

'''<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的数学知识和技术，从概率，微积分，线性代数，集合论，拓扑，以及数学分析的分支，如实分析，测度理论，傅立叶变换家族中的关系，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。

'''<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。

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"/>

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"/>

术语“随机函数”也用于指随机或随机过程，[[参考资料][[[参考名称]古斯库库库库库库库库库库什2010年第21页第21页][引用图书| first1=first1=Dmytro；last1=Gusak | first2=Alexander |最后2=Kukukush；first3=Alexey |最后3=Kulik | first4=Yuliya |最后4=Mishura | author4 link=Yuliya Mishura | first5=Andrey | last5=Pilipenko | title=随机过程理论：随机过程理论：应用金融数学和风险理论应用124；网址=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”{引用图书|作者=Valeriy skorokord | title=概率论的基本原理和应用| 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解释为[[函数空间]]中的随机元素数学理论|网址=https://books.google.com/books？id=pd4cvgaacaj | year=1977 | publisher=Springer Verlag | isbn=978-3-540-90275-1 | pages=1–2}</ref>术语“随机过程”和“随机过程”可以互换使用，对于索引随机变量的集合，通常没有特定的[[数学空间]]=https://books.google.com/books？id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | pages=24-25}</ref name=“ChaumontYor2012”>{cite book | author1=Lo|Chaumont | author2=Marc Yor | title=practicess in Probability:A Guided Tour from Theory to Random，Via condition | 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>但是当随机变量由[[实线]]的[[整数]]或[[区间（数学）|区间]]索引时，通常使用这两个术语。<ref name=“GikhmanSkorokhod1969page1”/><ref name=“ChaumontYor2012”/>变量由[[笛卡尔平面]]或更高维的[[欧几里德空间]]索引，通常称之为Jonathan{124ora@random Fields=124hoj=random-field[author2name=random-field]=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”/>

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”/>

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>

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>

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 |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}</ref>[[Markov process]]es，<ref name=“rogerswillams2000”>{引用图书| author1=L.C.G.Rogers | author2=David Williams | title=扩散，马尔可夫过程，和鞅：第一卷，基础|网址=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 |作者=David Applebaum | title=Lévy过程和随机微积分| 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=关于高斯过程的讲座| url=https://books.google.com/books？id=03m2UxI UYMC | year=2012 | publisher=Springer Science&Business Media | isbn=978-3-642-24939-6}</ref>随机域，<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，和[[branching process]]es.<ref name=“KarlinTaylor2012”>{引用图书| 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}</ref>随机过程的研究使用了[[概率]]、[[微积分]]、[[线性代数]]、[[集理论]]的数学知识和技术，和[[topology]]<ref name=“Hajek2015”>{cite book | author=Bruce Hajek | title=Random Processes for Engineers |网址=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=随机建模中的矩阵分析方法简介| url=https://books.google.com/books？id=Kan2ki8jqzgC | year=1999 | publisher=SIAM | isbn=978-0-89871-425-8}</ref><ref name=“DaleyVere-Jones 2007”>{引用图书| author1=D.J.Daley | author2=David Vere-Jones | title=点过程理论导论：第二卷：一般理论与结构| url=https://books.google.com/books？id=nPENXKw5kwcC | year=2007 | publisher=Springer Science&Business Media | isbn=978-0-387-21337-8}</ref>以及[[数学分析]]的分支，如[[真实分析]，[[测量理论]]，[[傅立叶分析]，和[[功能分析].<ref name=“Billingsley2008”>{cite book | author=Patrick Billingsley | title=Probability and Measure |网址=https://books.google.com/books？id=QyXqOXyxEeIC | year=2008 | publisher=Wiley India私人有限公司| isbn=978-81-265-1771-8}</ref><ref name=“Brémaud2014”>{cite book | author=Pierre Brémaud | title=Fourier Analysis and Random Processes |网址=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=概率与随机过程的函数分析：简介|网址=https://books.google.com/books？id=q7dR3d5nqaUC | year=2005 | publisher=Cambridge University Press | isbn=978-0-521-83166-6}</ref>随机过程理论被认为是对数学的重要贡献AMS | volume=51 | issue=11 | year=2004 | pages=1336–1347}</ref>并且由于理论原因和应用，它仍然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 random Processes | url=https://books.google.com/books？id=CyK6KAjwdYkC | year=2011 | publisher=欧洲数学学会| isbn=978-3-03719-072-2}</ref><ref name=“Talagrand2014”>{引用图书|作者=Michel Talagrand | title=随机过程的上下界：现代方法和经典问题| url=https://books.google.com/books？id=tfa5baaqbaj&pg=PR4 | year=2014 | publisher=Springer Science&Business Media | isbn=978-3-642-54075-2 | pages=4–}</ref><ref name=“Bressloff2014VII”>{引用图书|作者=Paul C.Bressloff | title=细胞生物学中的随机过程|网址=https://books.google.com/books？id=SWZYBAAQBAJ&pg公司

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>

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.

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.

当被解释为时间时，如果一个随机过程的索引集的元素数量有限或可数，如一个有限的数字集，一个整数集，或自然数集，那么随机过程被称为在离散时间内随机。如果索引集是实线的某个区间，那么时间就是连续的。这两类随机过程分别称为离散时间过程和连续时间过程。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，特别是由于索引集是不可数的。如果索引集是整数，或者整数的一些子集，那么随机过程也可以被称为'''<font color="#ff8000"> 随机序列Random sequence</font>'''。雅各布 · 伯努利在1713年以拉丁文出版的《猜测概率论》一书中使用了“猜测随机论”这个短语，这个短语被翻译成了“猜测或推测的艺术”。1917年，拉迪斯劳斯·博特基威茨在德语中写下了“随机”一词，意思是随机。1934年，Joseph Doob 在一篇论文中首次提到随机过程这个词。尽管这个德语术语早在1931年就被安德烈 · 科尔莫哥罗夫使用过。

当被解释为时间时，如果一个随机过程的索引集的元素数量有限或可数，如一个有限的数字集，一个整数集，或自然数集，那么随机过程被称为在离散时间内随机。如果索引集是实线的某个区间，那么时间就是连续的。这两类随机过程分别称为离散时间过程和连续时间过程。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，特别是由于索引集是不可数的。如果索引集是整数，或者整数的一些子集，那么随机过程也可以被称为'''<font color="#ff8000"> 随机序列Random sequence</font>'''。雅各布 · 伯努利在1713年以拉丁文出版的《猜测概率论》一书中使用了“猜测随机论”这个短语，这个短语被翻译成了“猜测或推测的艺术”。1917年，拉迪斯劳斯·博特基威茨在德语中写下了“随机”一词，意思是随机。1934年，Joseph Doob 在一篇论文中首次提到随机过程这个词。尽管这个德语术语早在1931年就被安德烈 · 科尔莫哥罗夫使用过。

  --~~~如果一个随机过程的索引集的元素数量有限或可数，如一个有限的数字集，一个整数集，或自然数集，那么随机过程被称为在离散时间内随机。如果索引集是实线的某个区间，那么时间就是连续的   修改润色语言  “实线的某个区间”→在实数轴上的某个区间上

   --~~~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，特别是由于索引集是不可数的。 可进一步修改

--[[用户:水流心不竞|水流心不竞]]（[[用户讨论:水流心不竞|讨论]]）如果一个随机过程的索引集的元素数量有限或可数，如一个有限的数字集，一个整数集，或自然数集，那么随机过程被称为在离散时间内随机。如果索引集是实线的某个区间，那么时间就是连续的   修改润色语言  “实线的某个区间”→在实数轴上的某个区间上

   --[[用户:水流心不竞|水流心不竞]]（[[用户讨论:水流心不竞|讨论]]）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，特别是由于索引集是不可数的。 可进一步修改

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"/>

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"/>