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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>
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=随机过程| url=图书https://books.com/？id=0mB2CQAAQBAJ |年份=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，[[financial markets]]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”>{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，[[financial markets]]中看似随机的变化激发了随机过程在[[金融]]中的广泛使用
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.