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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=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

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

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.

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.

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.

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。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。

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"/>

随机或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”。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“/>

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“/>

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.

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.