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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=图书https://books.com/？id=7bu8jgeaaj | 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=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
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.
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,Lévy 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。
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"/>