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随机过程 (查看源代码)

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在[[概率论]]及相关领域中，'''随机过程 stochastic process'''（或random process）是一个数学对象，通常被定义为随机变量的集合，给出对一个随机过程的解释，该过程表示某个系统随机的数值随时间的变化，例如细菌种群的增长，电流由于热噪声而波动，或者一个气体分子的运动。<ref name="doob1953stochasticP46to47">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=7Bu8jgEACAAJ|year=1990|publisher=Wiley|pages=46, 47}}</ref><ref name="Parzen1999">{{cite book|author=Emanuel Parzen|title=Stochastic Processes|url=https://books.google.com/books?id=0mB2CQAAQBAJ|year= 2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|pages=7, 8}}</ref><ref name="GikhmanSkorokhod1969page1">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=q0lo91imeD0C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=1}}</ref><ref name=":0">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|location= NJ|pages=1–235}}</ref>随机过程被广泛用作以随机方式变化的系统和现象的数学模型。它们在许多学科都有应用，比如生物学<ref name="Bressloff2014">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ|year=2014|publisher=Springer|isbn=978-3-319-08488-6}}</ref>，[[化学]] <ref name="Kampen2011">{{cite book|author=N.G. Van Kampen|title=Stochastic Processes in Physics and Chemistry|url=https://books.google.com/books?id=N6II-6HlPxEC|year=2011|publisher=Elsevier|isbn=978-0-08-047536-3}}</ref> 生态学,<ref name="LandeEngen2003">{{cite book|author1=Russell Lande|author2=Steinar Engen|author3=Bernt-Erik Sæther|title=Stochastic Population Dynamics in Ecology and Conservation|url=https://books.google.com/books?id=6KClauq8OekC|year=2003|publisher=Oxford University Press|isbn=978-0-19-852525-7}}</ref> [[神经科学]]<ref name="LaingLord2010">{{cite book|author1=Carlo Laing|author2=Gabriel J Lord|title=Stochastic Methods in Neuroscience|url=https://books.google.com/books?id=RaYSDAAAQBAJ|year=2010|publisher=OUP Oxford|isbn=978-0-19-923507-0}}</ref>, 物理学<ref name="PaulBaschnagel2013">{{cite book|author1=Wolfgang Paul|author2=Jörg Baschnagel|title=Stochastic Processes: From Physics to Finance|url=https://books.google.com/books?id=OWANAAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-319-00327-6}}</ref>, 图像处理, [[signal processing]],<ref name="Dougherty1999">{{cite book|author=Edward R. Dougherty|title=Random processes for image and signal processing|url=https://books.google.com/books?id=ePxDAQAAIAAJ|year=1999|publisher=SPIE Optical Engineering Press|isbn=978-0-8194-2513-3}}</ref> [[Stochastic control|control theory]], <ref name="Bertsekas1996">{{cite book|author=Dimitri P. Bertsekas|title=Stochastic Optimal Control: The Discrete-Time Case|url=http://www.athenasc.com/socbook.html|year=1996|publisher=Athena Scientific]|isbn=1-886529-03-5}}</ref> [[信息论]],<ref name="CoverThomas2012page71">{{cite book|author1=Thomas M. Cover|author2=Joy A. Thomas|title=Elements of Information Theory|url=https://books.google.com/books?id=VWq5GG6ycxMC=PT16|year=2012|publisher=John Wiley & Sons|isbn=978-1-118-58577-1|page=71}}</ref> 计算机科学,<ref name="Baron2015">{{cite book|author=Michael Baron|title=Probability and Statistics for Computer Scientists, Second Edition|url=https://books.google.com/books?id=CwQZCwAAQBAJ|year=2015|publisher=CRC Press|isbn=978-1-4987-6060-7|page=131}}</ref> 密码学<ref>{{cite book|author1=Jonathan Katz|author2=Yehuda Lindell|title=Introduction to Modern Cryptography: Principles and Protocols|url=https://archive.org/details/Introduction_to_Modern_Cryptography|year=2007|publisher=CRC Press|isbn=978-1-58488-586-3|page=[https://archive.org/details/Introduction_to_Modern_Cryptography/page/n44 26]}}</ref> 和电信.<ref name="BaccelliBlaszczyszyn2009">{{cite book|author1=François Baccelli|author2=Bartlomiej Blaszczyszyn|title=Stochastic Geometry and Wireless Networks|url=https://books.google.com/books?id=H3ZkTN2pYS4C|year=2009|publisher=Now Publishers Inc|isbn=978-1-60198-264-3}}</ref> 此外，金融市场中看似随机的变化激发了随机过程在金融中的广泛使用。<ref name="Steele2001">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=H06xzeRQgV4C|year=2001|publisher=Springer Science & Business Media|isbn=978-0-387-95016-7}}</ref><ref name="MusielaRutkowski2006">{{cite book|author1=Marek Musiela|author2=Marek Rutkowski|title=Martingale Methods in Financial Modelling|url=https://books.google.com/books?id=iojEts9YAxIC|year= 2006|publisher=Springer Science & Business Media|isbn=978-3-540-26653-2}}</ref><ref name="Shreve2004">{{cite book|author=Steven E. Shreve|title=Stochastic Calculus for Finance II: Continuous-Time Models|url=https://books.google.com/books?id=O8kD1NwQBsQC|year=2004|publisher=Springer Science & Business Media|isbn=978-0-387-40101-0}}</ref>

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在[[概率论]]及相关领域中，'''随机过程 stochastic process'''（或random process）是一个数学对象，通常被定义为随机变量的集合，给出对一个随机过程的解释，该过程表示某个系统随机的数值随时间的变化，例如细菌种群的增长，电流由于热噪声而波动，或者一个气体分子的运动。<ref name="doob1953stochasticP46to47">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=7Bu8jgEACAAJ|year=1990|publisher=Wiley|pages=46, 47}}</ref><ref name="Parzen1999">{{cite book|author=Emanuel Parzen|title=Stochastic Processes|url=https://books.google.com/books?id=0mB2CQAAQBAJ|year= 2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|pages=7, 8}}</ref><ref name="GikhmanSkorokhod1969page1">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=q0lo91imeD0C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=1}}</ref><ref name=":0">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|location= NJ|pages=1–235}}</ref>随机过程被广泛用作以随机方式变化的系统和现象的数学模型。它们在许多学科都有应用，比如生物学<ref name="Bressloff2014">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ|year=2014|publisher=Springer|isbn=978-3-319-08488-6}}</ref>，[[化学]] <ref name="Kampen2011">{{cite book|author=N.G. Van Kampen|title=Stochastic Processes in Physics and Chemistry|url=https://books.google.com/books?id=N6II-6HlPxEC|year=2011|publisher=Elsevier|isbn=978-0-08-047536-3}}</ref> 生态学,<ref name="LandeEngen2003">{{cite book|author1=Russell Lande|author2=Steinar Engen|author3=Bernt-Erik Sæther|title=Stochastic Population Dynamics in Ecology and Conservation|url=https://books.google.com/books?id=6KClauq8OekC|year=2003|publisher=Oxford University Press|isbn=978-0-19-852525-7}}</ref> 神经科学<ref name="LaingLord2010">{{cite book|author1=Carlo Laing|author2=Gabriel J Lord|title=Stochastic Methods in Neuroscience|url=https://books.google.com/books?id=RaYSDAAAQBAJ|year=2010|publisher=OUP Oxford|isbn=978-0-19-923507-0}}</ref>, 物理学<ref name="PaulBaschnagel2013">{{cite book|author1=Wolfgang Paul|author2=Jörg Baschnagel|title=Stochastic Processes: From Physics to Finance|url=https://books.google.com/books?id=OWANAAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-319-00327-6}}</ref>, 图像处理, [[signal processing]],<ref name="Dougherty1999">{{cite book|author=Edward R. Dougherty|title=Random processes for image and signal processing|url=https://books.google.com/books?id=ePxDAQAAIAAJ|year=1999|publisher=SPIE Optical Engineering Press|isbn=978-0-8194-2513-3}}</ref> [[Stochastic control|control theory]], <ref name="Bertsekas1996">{{cite book|author=Dimitri P. Bertsekas|title=Stochastic Optimal Control: The Discrete-Time Case|url=http://www.athenasc.com/socbook.html|year=1996|publisher=Athena Scientific]|isbn=1-886529-03-5}}</ref> [[信息论]],<ref name="CoverThomas2012page71">{{cite book|author1=Thomas M. Cover|author2=Joy A. Thomas|title=Elements of Information Theory|url=https://books.google.com/books?id=VWq5GG6ycxMC=PT16|year=2012|publisher=John Wiley & Sons|isbn=978-1-118-58577-1|page=71}}</ref> 计算机科学,<ref name="Baron2015">{{cite book|author=Michael Baron|title=Probability and Statistics for Computer Scientists, Second Edition|url=https://books.google.com/books?id=CwQZCwAAQBAJ|year=2015|publisher=CRC Press|isbn=978-1-4987-6060-7|page=131}}</ref> 密码学<ref>{{cite book|author1=Jonathan Katz|author2=Yehuda Lindell|title=Introduction to Modern Cryptography: Principles and Protocols|url=https://archive.org/details/Introduction_to_Modern_Cryptography|year=2007|publisher=CRC Press|isbn=978-1-58488-586-3|page=[https://archive.org/details/Introduction_to_Modern_Cryptography/page/n44 26]}}</ref> 和电信.<ref name="BaccelliBlaszczyszyn2009">{{cite book|author1=François Baccelli|author2=Bartlomiej Blaszczyszyn|title=Stochastic Geometry and Wireless Networks|url=https://books.google.com/books?id=H3ZkTN2pYS4C|year=2009|publisher=Now Publishers Inc|isbn=978-1-60198-264-3}}</ref> 此外，金融市场中看似随机的变化激发了随机过程在金融中的广泛使用。<ref name="Steele2001">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=H06xzeRQgV4C|year=2001|publisher=Springer Science & Business Media|isbn=978-0-387-95016-7}}</ref><ref name="MusielaRutkowski2006">{{cite book|author1=Marek Musiela|author2=Marek Rutkowski|title=Martingale Methods in Financial Modelling|url=https://books.google.com/books?id=iojEts9YAxIC|year= 2006|publisher=Springer Science & Business Media|isbn=978-3-540-26653-2}}</ref><ref name="Shreve2004">{{cite book|author=Steven E. Shreve|title=Stochastic Calculus for Finance II: Continuous-Time Models|url=https://books.google.com/books?id=O8kD1NwQBsQC|year=2004|publisher=Springer Science & Business Media|isbn=978-0-387-40101-0}}</ref>

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

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

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