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第1行： − #REDIRECT [[Stochastic process]] {{R from other capitalisation}}+ − REDIRECT Stochastic process + − 重定向随机过程+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − +
Moved page from wikipedia:en:Stochastic process (history)
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[[File:BMonSphere.jpg|thumb|A computer-simulated realization of a [[Wiener process|Wiener]] or [[Brownian motion]] process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>]]
Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.
球面上的 Wiener 或 Brownian 运动过程。维纳过程被广泛认为是随机过程概率论研究最多和最核心的维纳过程。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。它们在生物学、化学、生态学、神经科学、物理学、图像处理、信号处理、控制理论、信息理论、计算机科学、密码学和电信学等许多学科都有应用。此外，金融市场表面上的随机变化促使随机过程在金融领域的广泛应用。
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>
Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.
现象的应用和研究反过来激发了新的随机过程的提出。这类随机过程的例子包括路易斯 · 巴舍利耶用来研究巴黎证券交易所价格变化的维纳过程或布朗运动过程，以及 a · k · 埃尔朗用来研究在一定时期内通话次数的泊松过程。这两个随机过程在随机过程理论中被认为是最重要和最核心的，并且在巴舍利耶和 Erlang 之前和之后，在不同的环境和国家被重复和独立地发现。
Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the [[Wiener process]] or Brownian motion process,{{efn|The term ''Brownian motion'' can refer to the physical process, also known as ''Brownian movement'', and the stochastic process, a mathematical object, but to avoid ambiguity this article uses the terms ''Brownian motion process'' or ''Wiener process'' for the latter in a style similar to, for example, Gikhman and Skorokhod<ref name="GikhmanSkorokhod1969">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3}}</ref> or Rosenblatt.<ref name="Rosenblatt1962">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press}}</ref>}} used by [[Louis Bachelier]] to study price changes on the [[Paris Bourse]],<ref name="JarrowProtter2004">{{cite book|last1=Jarrow|first1=Robert|title=A Festschrift for Herman Rubin|last2=Protter|first2=Philip|chapter=A short history of stochastic integration and mathematical finance: the early years, 1880–1970|year=2004|pages=75–80|issn=0749-2170|doi=10.1214/lnms/1196285381|citeseerx=10.1.1.114.632|series=Institute of Mathematical Statistics Lecture Notes - Monograph Series|isbn=978-0-940600-61-4}}</ref> and the [[Poisson process]], used by [[A. K. Erlang]] to study the number of phone calls occurring in a certain period of time.<ref name="Stirzaker2000">{{cite journal|last1=Stirzaker|first1=David|title=Advice to Hedgehogs, or, Constants Can Vary|journal=The Mathematical Gazette|volume=84|issue=500|year=2000|pages=197–210|issn=0025-5572|doi=10.2307/3621649|jstor=3621649}}</ref> These two stochastic processes are considered the most important and central in the theory of stochastic processes,<ref name="doob1953stochasticP46to47"/><ref name="Parzen1999"/><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=32}}</ref> and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.<ref name="JarrowProtter2004"/><ref name="GuttorpThorarinsdottir2012">{{cite journal|last1=Guttorp|first1=Peter|last2=Thorarinsdottir|first2=Thordis L.|title=What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes|journal=International Statistical Review|volume=80|issue=2|year=2012|pages=253–268|issn=0306-7734|doi=10.1111/j.1751-5823.2012.00181.x}}</ref>
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.
随机函数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。随机过程过程和随机过程这两个术语可以互换使用，通常没有特定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。马尔可夫过程，Lévy 过程，高斯过程，随机场，更新过程和分支过程。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"/>
Based on their mathematical properties, stochastic processes can be grouped into various categories, which include [[random walk]]s,<ref name="LawlerLimic2010">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year= 2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1}}</ref> [[Martingale (probability theory)|martingales]],<ref name="Williams1991">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5}}</ref> [[Markov process]]es,<ref name="RogersWilliams2000">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year= 2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7}}</ref> [[Lévy process]]es,<ref name="ApplebaumBook2004">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2}}</ref> [[Gaussian process]]es,<ref>{{cite book|author=Mikhail Lifshits|title=Lectures on Gaussian Processes|url=https://books.google.com/books?id=03m2UxI-UYMC|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-24939-6}}</ref> random fields,<ref name="Adler2010">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA1|year= 2010|publisher=SIAM|isbn=978-0-89871-693-1}}</ref> [[renewal process]]es, and [[branching process]]es.<ref name="KarlinTaylor2012">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year= 2012|publisher=Academic Press|isbn=978-0-08-057041-9}}</ref> The study of stochastic processes uses mathematical knowledge and techniques from [[probability]], [[calculus]], [[linear algebra]], [[set theory]], and [[topology]]<ref name="Hajek2015">{{cite book|author=Bruce Hajek|title=Random Processes for Engineers|url=https://books.google.com/books?id=Owy0BgAAQBAJ|year=2015|publisher=Cambridge University Press|isbn=978-1-316-24124-0}}</ref><ref name="LatoucheRamaswami1999">{{cite book|author1=G. Latouche|author2=V. Ramaswami|title=Introduction to Matrix Analytic Methods in Stochastic Modeling|url=https://books.google.com/books?id=Kan2ki8jqzgC|year=1999|publisher=SIAM|isbn=978-0-89871-425-8}}</ref><ref name="DaleyVere-Jones2007">{{cite book|author1=D.J. Daley|author2=David Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure|url=https://books.google.com/books?id=nPENXKw5kwcC|year= 2007|publisher=Springer Science & Business Media|isbn=978-0-387-21337-8}}</ref> as well as branches of [[mathematical analysis]] such as [[real analysis]], [[measure theory]], [[Fourier analysis]], and [[functional analysis]].<ref name="Billingsley2008">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8}}</ref><ref name="Brémaud2014">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year= 2014|publisher=Springer|isbn=978-3-319-09590-5}}</ref><ref name="Bobrowski2005">{{cite book|author=Adam Bobrowski|title=Functional Analysis for Probability and Stochastic Processes: An Introduction|url=https://books.google.com/books?id=q7dR3d5nqaUC|year= 2005|publisher=Cambridge University Press|isbn=978-0-521-83166-6}}</ref> The theory of stochastic processes is considered to be an important contribution to mathematics<ref name="Applebaum2004">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336–1347}}</ref> and it continues to be an active topic of research for both theoretical reasons and applications.<ref name="BlathImkeller2011">{{cite book|author1=Jochen Blath|author2=Peter Imkeller|author3=Sylvie Rœlly|title=Surveys in Stochastic Processes|url=https://books.google.com/books?id=CyK6KAjwdYkC|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-072-2}}</ref><ref name="Talagrand2014">{{cite book|author=Michel Talagrand|title=Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems|url=https://books.google.com/books?id=tfa5BAAAQBAJ&pg=PR4|year=2014|publisher=Springer Science & Business Media|isbn=978-3-642-54075-2|pages=4–}}</ref><ref name="Bressloff2014VII">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-08488-6|pages=vii–ix}}</ref>
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.
一个随机或随机过程可以被定义为一组随机变量的集合，这些随机变量被一些数学集合索引，这意味着随机过程的每个随机变量唯一地与集合中的一个元素相关联。
==Introduction==
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指数集有一个有限或可数的元素数，如一个有限的数字集，一个整数集，或自然数，那么随机过程被称为在离散时间。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，特别是由于指数集是不可数的。如果索引集是整数，或者其中的一些子集，那么随机过程也可以被称为随机序列。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"/>
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.
根据牛津英语词典的研究，英语中随机这个词的早期出现和它现在的意思有关，可以追溯到16世纪，而早期记录的用法开始于14世纪，是一个名词，意思是“浮躁、极速、力量或暴力(在骑马、奔跑、惊人等等)”。这个单词本身来自中世纪法语单词，意思是“速度，匆忙” ，它可能来源于法语动词，意思是“奔跑”或“疾驰”。随机过程这个术语的第一次书面出现早于随机过程，牛津英语词典也把它作为同义词，并在 Francis Edgeworth 1888年发表的一篇文章中使用。
[[File:Wiener process 3d.png|thumb|right|A single computer-simulated '''sample function''' or '''realization''', among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.]]
===Classifications===
The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. Both "collection", while instead of "index set", sometimes the terms "parameter set" though sometimes it is only used when the stochastic process takes real values. while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is n-dimensional Euclidean space \mathbb{R}^n or a manifold. \{X(t)\} or simply as X or X(t), although X(t) is regarded as an abuse of function notation. For example, X(t) or X_t are used to refer to the random variable with the index t, and not the entire stochastic process. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial.
随机过程的定义各不相同，但随机过程通常被定义为由一组随机变量组成的集合。两者都是“集合” ，而不是“索引集合” ，有时使用术语“参数集合” ，但有时只有在随机过程数据库采用真实值时才使用。当索引集被解释为时间时，通常使用随机过程和随机过程，当索引集是 n 维欧氏空间 mathbb { r } ^ n 或流形时，则使用随机场等其他术语。{ x (t)}或简单地作为 x 或 x (t) ，尽管 x (t)被认为是滥用函数表示法。例如，x (t)或 x _ t 用于引用索引为 t 的随机变量，而不是整个随机过程。换句话说，伯努利过程是一系列 iid Bernoulli 随机变量，每次抛硬币都是 Bernoulli 试验的一个例子。
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the [[cardinality]] of the index set and the state space.<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26">{{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|page=26}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|pages=24, 25}}</ref>
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]]'''.<ref name="Billingsley2008page482"/><ref name="Borovkov2013page527">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=527}}</ref> If the index set is some interval of the real line, then time is said to be '''[[continuous time|continuous]]'''. The two types of stochastic processes are respectively referred to as '''discrete-time''' and '''[[continuous-time stochastic process]]es'''.<ref name="KarlinTaylor2012page27"/><ref name="Brémaud2014page120"/><ref name="Rosenthal2006page177">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|pages=177–178}}</ref> 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.<ref name="KloedenPlaten2013page63">{{cite book|author1=Peter E. Kloeden|author2=Eckhard Platen|title=Numerical Solution of Stochastic Differential Equations|url=https://books.google.com/books?id=r9r6CAAAQBAJ=PA1|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-12616-5|page=63}}</ref><ref name="Khoshnevisan2006page153">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=153–155}}</ref> If the index set is the integers, or some subset of them, then the stochastic process can also be called a '''random sequence'''.<ref name="Borovkov2013page527"/>
If the state space is the integers or natural numbers, then the stochastic process is called a '''discrete''' or '''integer-valued stochastic process'''. If the state space is the real line, then the stochastic process is referred to as a '''real-valued stochastic process''' or a '''process with continuous state space'''. If the state space is <math>n</math>-dimensional Euclidean space, then the stochastic process is called a <math>n</math>-'''dimensional vector process''' or <math>n</math>-'''vector process'''.<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26"/>
Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.
随机游动是随机过程，通常定义为欧氏空间中的等价随机变量或随机向量的和，因此它们是在离散时间中变化的过程。但有些人也用这个词来指连续时间中发生变化的过程，特别是在金融领域使用的维纳过程，这种过程导致了一些混淆，从而招致了批评。还有其他各种类型的随机游动，定义它们的状态空间可以是其他数学对象，如格子和群，一般来说，它们被高度研究，在不同学科中有许多应用。
===Etymology===
A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p, or decreases by one with probability 1-p, so the index set of this random walk is the natural numbers, while its state space is the integers. If the p=0.5, this random walk is called a symmetric random walk.
一个经典的随机游走的例子被称为简单随机游走，这是一个以整数为状态空间的离散时间随机过程，它基于一个伯努利过程，其中每个 Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游动发生在整数上，它的值随概率的增加而增加一倍，如 p，或者随概率的减少而减少一倍，因此这种随机游动的指数集是自然数，而它的状态空间是整数。如果 p = 0.5，这种随机游动称为对称随机游动。
The word ''stochastic'' in [[English language|English]] was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a [[Greek language|Greek]] word meaning "to aim at a mark, guess", and the [[Oxford English Dictionary]] gives the year 1662 as its earliest occurrence.<ref name="OxfordStochastic">{{Cite OED|Stochastic}}</ref> 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".<ref name="Sheĭnin2006page5">{{cite book|author=O. B. Sheĭnin|title=Theory of probability and statistics as exemplified in short dictums|url=https://books.google.com/books?id=XqMZAQAAIAAJ|year=2006|publisher=NG Verlag|isbn=978-3-938417-40-9|page=5}}</ref> This phrase was used, with reference to Bernoulli, by [[Ladislaus Bortkiewicz]]<ref name="SheyninStrecker2011page136">{{cite book|author1=Oscar Sheynin|author2=Heinrich Strecker|title=Alexandr A. Chuprov: Life, Work, Correspondence|url=https://books.google.com/books?id=1EJZqFIGxBIC&pg=PA9|year=2011|publisher=V&R unipress GmbH|isbn=978-3-89971-812-6|page=136}}</ref> 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]].<ref name="OxfordStochastic"/> For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term ''stochastischer Prozeß'' was used in German by [[Aleksandr Khinchin]],<ref name="Doob1934"/><ref name="Khintchine1934">{{cite journal|last1=Khintchine|first1=A.|title=Korrelationstheorie der stationeren stochastischen Prozesse|journal=Mathematische Annalen|volume=109|issue=1|year=1934|pages=604–615|issn=0025-5831|doi=10.1007/BF01449156}}</ref> though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.<ref name="Kolmogoroff1931page1">{{cite journal|last1=Kolmogoroff|first1=A.|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung|journal=Mathematische Annalen|volume=104|issue=1|year=1931|page=1|issn=0025-5831|doi=10.1007/BF01457949}}</ref>
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.<ref name="OxfordRandom">{{Cite OED|Random}}</ref>
===Terminology===
The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. The Wiener process is named after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids.
维纳过程是一个具有平稳和独立增量的随机过程过程，这些增量是基于增量大小的正态分布。维纳过程是以诺伯特 · 维纳的名字命名的，他证明了维纳过程的数学存在性。
The definition of a stochastic process varies,<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> but a stochastic process is traditionally defined as a collection of random variables indexed by some set.<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/> The terms ''random process'' and ''stochastic process'' are considered synonyms and are used interchangeably, without the index set being precisely specified.<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press|page=[https://archive.org/details/randomprocesses00rose_0/page/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref> Both "collection",<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/> or "family" are used<ref name="Parzen1999"/><ref name="Ito2006page13">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|page=13}}</ref> while instead of "index set", sometimes the terms "parameter set"<ref name="Lamperti1977page1"/> or "parameter space"<ref name="AdlerTaylor2009page7"/> are used.
Realizations of Wiener processes (or Brownian motion processes) with drift () and without drift ().
带漂移()和无漂移()的 Wiener 过程(或布朗运动过程)的实现。
The term ''random function'' is also used to refer to a stochastic or random process,<ref name="GikhmanSkorokhod1969page1"/><ref name="Loeve1978">{{cite book|author=M. Loève|title=Probability Theory II|url=https://books.google.com/books?id=1y229yBbULIC|year=1978|publisher=Springer Science & Business Media|isbn=978-0-387-90262-3|page=163}}</ref><ref name="Brémaud2014page133">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=133}}</ref> though sometimes it is only used when the stochastic process takes real values.<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/> This term is also used when the index sets are mathematical spaces other than the real line,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 1</ref> while the terms ''stochastic process'' and ''random process'' are usually used when the index set is interpreted as time,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"/><ref name="Bass2011page1">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=1}}</ref> and other terms are used such as ''random field'' when the index set is <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a [[manifold]].<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state space can be n-dimensional Euclidean space. If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant \mu, which is a real number, then the resulting stochastic process is said to have drift \mu.
在概率论中起着核心作用的维纳过程，通常被认为是最重要的和研究过的随机过程过程，与其他随机过程有联系。它的索引集和状态空间分别为非负数和实数，因此它既有连续索引集又有状态空间。但是这个过程可以定义得更广泛，因此它的状态空间可以是 n 维欧氏空间。如果任何增量的平均值为零，那么由此产生的 Wiener 或 Brownian 运动过程称为零漂过程。如果任意两个时间点的增量的平均值等于时间差乘以某个常数 μ，这是一个实数，那么得到的随机过程就是漂移 μ。
===Notation===
A stochastic process can be denoted, among other ways, by <math>\{X(t)\}_{t\in T} </math>,<ref name="Brémaud2014page120"/> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page408"/> <math>\{X_t\}</math><ref name="Lamperti1977page3">,{{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|page=3}}</ref> <math>\{X(t)\}</math> or simply as <math>X</math> or <math>X(t)</math>, although <math>X(t)</math> is regarded as an [[abuse of notation#Function notation|abuse of function notation]].<ref name="Klebaner2005page55">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=55}}</ref> For example, <math>X(t)</math> or <math>X_t</math> are used to refer to the random variable with the index <math>t</math>, and not the entire stochastic process.<ref name="Lamperti1977page3"/> If the index set is <math>T=[0,\infty)</math>, then one can write, for example, <math>(X_t , t \geq 0)</math> to denote the stochastic process.<ref name="ChaumontYor2012"/>
Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.
几乎可以肯定，Wiener 过程的样本路径在任何地方都是连续的，但是没有可微的地方。它可以看作是简单随机游动的连续形式。这个过程作为其他随机过程的数学极限出现，例如某些随机游动的重新标度，这是 Donsker 定理或不变性原理的主题，也被称为函数中心极限定理。
==Examples==
The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes. It plays a central role in quantitative finance, where it is used, for example, in the Black–Scholes–Merton model. The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.
Wiener 过程是马尔可夫过程、 Lévy 过程和 Gaussian 过程等重要随机过程的一个成员。它在定量金融学中扮演着核心角色，例如，在布莱克-斯科尔斯-默顿模型中就使用了它。这个过程也用于不同的领域，包括大多数自然科学和一些社会科学分支，作为各种随机现象的数学模型。
===Bernoulli process===
{{Main|Bernoulli process}}
One of the simplest stochastic processes is the [[Bernoulli process]],<ref name="Florescu2014page293"/> which is a sequence of [[independent and identically distributed]] (iid) random variables, where each random variable takes either the value one or zero, say one with probability <math>p</math> and zero with probability <math>1-p</math>. This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is <math>p</math> and its value is one, while the value of a tail is zero.<ref name="Florescu2014page301">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=301}}</ref> In other words, a Bernoulli process is a sequence of [[Independent and identically distributed random variables|iid]] Bernoulli random variables,<ref name="BertsekasTsitsiklis2002page273">{{cite book|author1=Dimitri P. Bertsekas|author2=John N. Tsitsiklis|title=Introduction to Probability|url=https://books.google.com/books?id=bcHaAAAAMAAJ|year=2002|publisher=Athena Scientific|isbn=978-1-886529-40-3|page=273}}</ref> where each coin flip is an example of a [[Bernoulli trial]].<ref name="Ibe2013page11">{{cite book|author=Oliver C. Ibe|title=Elements of Random Walk and Diffusion Processes|url=https://books.google.com/books?id=DUqaAAAAQBAJ&pg=PT10|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-61793-9|page=11}}</ref>
The Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes. If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.
泊松过程是一个具有不同形式和定义的随机过程过程。它可以被定义为一个计数过程，这是一个随机过程，代表点或事件的随机数到一定时间。从零到给定时间区间内的过程点数是泊松随机变量，取决于该时间和某些参数。该过程以自然数为状态空间，非负数为索引集。这个过程也被称为泊松计数过程，因为它可以被解释为计数过程的一个例子。齐次泊松过程是一类重要的随机过程，如马尔可夫过程和 Lévy 过程的成员。如果将泊松过程的参数常数替换为 t 的某个非负可积函数，得到的过程称为非齐次或非齐次泊松过程，其点的平均密度不再是常数。泊松过程作为排队论中的一个基本过程，是数学模型中的一个重要过程，它在特定时间窗内随机发生的事件模型中找到了应用。
===Random walk===
Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the n-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.
在实际线上定义的泊松过程可以被解释为随机过程过程，以及其他随机对象。但是，它可以定义在 n 维欧氏空间或其他数学空间，在那里，它经常被解释为一个随机集或随机计数测度，而不是一个随机过程。但是人们注意到泊松过程并没有得到应有的重视，部分原因是泊松过程通常只考虑实线，而不考虑其他数学空间。
{{Main|Random walk}}
[[Random walks]] are stochastic processes that are usually defined as sums of [[iid]] random variables or random vectors in Euclidean space, so they are processes that change in discrete time.<ref name="Klenke2013page347">{{cite book|author=Achim Klenke|title=Probability Theory: A Comprehensive Course|url=https://books.google.com/books?id=aqURswEACAAJ|year=2013|publisher=Springer|isbn=978-1-4471-5362-7|pages=347}}</ref><ref name="LawlerLimic2010page1">{{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|page=1}}</ref><ref name="Kallenberg2002page136">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date= 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=136}}</ref><ref name="Florescu2014page383">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=383}}</ref><ref name="Durrett2010page277">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=277}}</ref> But some also use the term to refer to processes that change in continuous time,<ref name="Weiss2006page1">{{cite book|last1=Weiss|first1=George H.|title=Encyclopedia of Statistical Sciences|chapter=Random Walks|year=2006|doi=10.1002/0471667196.ess2180.pub2|page=1|isbn=978-0471667193}}</ref> particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism.<ref name="Spanos1999page454">{{cite book|author=Aris Spanos|title=Probability Theory and Statistical Inference: Econometric Modeling with Observational Data|url=https://books.google.com/books?id=G0_HxBubGAwC|year=1999|publisher=Cambridge University Press|isbn=978-0-521-42408-0|page=454}}</ref> There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.<ref name="Weiss2006page1"/><ref name="Klebaner2005page81">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=81}}</ref>
A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If the <math>p=0.5</math>, this random walk is called a symmetric random walk.<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>
A stochastic process is defined as a collection of random variables defined on a common probability space (\Omega, \mathcal{F}, P), where \Omega is a sample space, \mathcal{F} is a \sigma-algebra, and P is a probability measure; and the random variables, indexed by some set T, all take values in the same mathematical space S, which must be measurable with respect to some \sigma-algebra \Sigma.
一个随机过程是定义在一个公共的概率空间上的随机变量的集合(Omega，mathcal { f } ，p) ，其中 Omega 是一个样本空间，mathcal { f }是一个 Sigma-algebra，p 是一个 Sigma-algebra; 而随机变量，由一些集合 t 索引，都在相同的数学空间 s 中取值，这些值对于一些 Sigma-Sigma 代数必须是可测量的。
<center><math>
< 中心 > < 数学 >
===Wiener process===
\{X(t):t\in T \}.
{ x (t) : t in t }.
{{Main|Wiener process}}
</math></center>
[数学中心]
The Wiener process is a stochastic process with stationary and [[independent increments]] that are [[normally distributed]] based on the size of the increments.<ref name="RogersWilliams2000page1">{{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|page=1}}</ref><ref name="Klebaner2005page56">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=56}}</ref> The Wiener process is named after [[Norbert Wiener]], who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for [[Brownian movement]] in liquids.<ref name="Brush1968page1">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=1–2|issn=0003-9519|doi=10.1007/BF00328110}}</ref><ref name="Applebaum2004page1338">{{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=1338}}</ref><ref name="Applebaum2004page1338"/><ref name="GikhmanSkorokhod1969page21">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=21}}</ref>
Historically, in many problems from the natural sciences a point t\in T had the meaning of time, so X(t) is a random variable representing a value observed at time t. A stochastic process can also be written as \{X(t,\omega):t\in T \} to reflect that it is actually a function of two variables, t\in T and \omega\in \Omega.
从历史上看，在自然科学的许多问题中，t 中的一个点 t 具有时间的意义，因此 x (t)是一个随机变量，代表在时间 t 观测到的一个值。随机过程也可以写成 t 中的{ x (t，ω) : t，以反映它实际上是一个双变量的函数，t 中的 t 和 ω 中的 ω。
[[File:DriftedWienerProcess1D.svg|thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).]]
There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a S^T-valued random variable, where S^T is the space of all the possible S-valued functions of t\in T that map from the set T into the space S. of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set T the interpretation of time. such as the Cartesian plane R^2 or n-dimensional Euclidean space, where an element t\in T can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.
还有其他的方法来考虑随机过程，上面的定义被认为是传统的定义。例如，随机过程可以被解释或定义为 s ^ t 值随机变量，其中 s ^ t 是 t 中所有可能的 s 值函数的空间，这些函数从集合 t 映射到随机过程空间 s。这个集合通常是实数直线的一些子集，比如自然数或者区间，给予集合 t 时间的解释。例如笛卡尔平面 r ^ 2或 n 维欧氏空间，其中 t 中的元素 t 可以表示空间中的一个点。但是一般来说，当指标集是有序的时候，对于随机过程可能有更多的结果和定理。
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=29}}</ref><ref name="Florescu2014page471">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=471}}</ref><ref name="KarlinTaylor2012page21">{{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|pages=21, 22}}</ref><ref name="KaratzasShreve2014pageVIII">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=VIII}}</ref><ref name="RevuzYor2013pageIX">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=IX|author1-link=Daniel Revuz}}</ref> Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.<ref name="Rosenthal2006page186">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|page=186}}</ref> But the process can be defined more generally so its state space can be <math>n</math>-dimensional Euclidean space.<ref name="Klebaner2005page81"/><ref name="KarlinTaylor2012page21"/><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=33}}</ref> If the [[mean]] of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant <math> \mu</math>, which is a real number, then the resulting stochastic process is said to have drift <math> \mu</math>.<ref name="Steele2012page118">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=118}}</ref><ref name="MörtersPeres2010page1"/><ref name="KaratzasShreve2014page78">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=78}}</ref>
The mathematical space S of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, n-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.
随机过程的数学空间 s 称为状态空间。这个数学空间可以用整数、实线、 n 维欧氏空间、复平面或更抽象的数学空间来定义。状态空间使用元素定义，这些元素反映了随机过程可以采用的不同值。
[[Almost surely]], a sample path of a Wiener process is continuous everywhere but [[nowhere differentiable function|nowhere differentiable]]. It can be considered as a continuous version of the simple random walk.<ref name="Applebaum2004page1337">{{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|page=1337}}</ref><ref name="MörtersPeres2010page1">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|pages=1, 3}}</ref> The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,<ref name="KaratzasShreve2014page61">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=61}}</ref><ref name="Shreve2004page93">{{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|page=93}}</ref> which is the subject of [[Donsker's theorem]] or invariance principle, also known as the functional central limit theorem.<ref name="Kallenberg2002page225and260">{{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=225, 260}}</ref><ref name="KaratzasShreve2014page70">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=70}}</ref><ref name="MörtersPeres2010page131">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=131}}</ref>
The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.<ref name="RogersWilliams2000page1"/><ref name="Applebaum2004page1337"/> The process also has many applications and is the main stochastic process used in stochastic calculus.<ref name="Klebaner2005">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7}}</ref><ref name="KaratzasShreve2014page">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2}}</ref> It plays a central role in quantitative finance,<ref name="Applebaum2004page1341">{{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|page=1341}}</ref><ref name="KarlinTaylor2012page340">{{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|page=340}}</ref> where it is used, for example, in the Black–Scholes–Merton model.<ref name="Klebaner2005page124">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=124}}</ref> The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.<ref name="Steele2012page29"/><ref name="KaratzasShreve2014page47">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=47}}</ref><ref name="Wiersema2008page2">{{cite book|author=Ubbo F. Wiersema|title=Brownian Motion Calculus|url=https://books.google.com/books?id=0h-n0WWuD9cC|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-02171-2|page=2}}</ref>
A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if \{X(t,\omega):t\in T \} is a stochastic process, then for any point \omega\in\Omega, the mapping
样本函数是随机过程的单一结果，所以它是由每个随机过程的随机变量的单一可能值构成的。更确切地说，如果 t 中的{ x (t，ω) : t }是随机过程，那么对于 ω 中的任意点 ω，映射
<center><math>
< 中心 > < 数学 >
===Poisson process===
X(\cdot,\omega): T \rightarrow S,
X (cdot，omega) : t,
{{Main|Poisson process}}
</math></center>
[数学中心]
is called a sample function, a realization, or, particularly when T is interpreted as time, a sample path of the stochastic process \{X(t,\omega):t\in T \}. This means that for a fixed \omega\in\Omega, there exists a sample function that maps the index set T to the state space S. or path.
被称为样例函数，一个实现，或者，特别是当 t 被解释为时间时，随机过程{ x (t，omega) : t in t }的样例路径。这意味着，对于 Omega 中的一个固定 ω，存在一个示例函数，该函数将索引集 t 映射到状态空间 s 或路径。
The Poisson process is a stochastic process that has different forms and definitions.<ref name="Tijms2003page1">{{cite book|author=Henk C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=eBeNngEACAAJ|year=2003|publisher=Wiley|isbn=978-0-471-49881-0|pages=1, 2}}</ref><ref name="DaleyVere-Jones2006chap2">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|pages=19–36}}</ref> It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.<ref name="Tijms2003page1"/>
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.<ref name="Tijms2003page1"/><ref name="PinskyKarlin2011">{{cite book|author1=Mark A. Pinsky|author2=Samuel Karlin|title=An Introduction to Stochastic Modeling|url=https://books.google.com/books?id=PqUmjp7k1kEC|year=2011|publisher=Academic Press|isbn=978-0-12-381416-6|page=241}}</ref> The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.<ref name="Applebaum2004page1337"/>
An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if \{X(t):t\in T \} is a stochastic process with state space S and index set T=[0,\infty), then for any two non-negative numbers t_1\in [0,\infty) and t_2\in [0,\infty) such that t_1\leq t_2, the difference X_{t_2}-X_{t_1} is a S-valued random variable known as an increment.
一个随机过程的增量是同一个随机过程的两个随机变量之间的差。对于一个索引集可以被解释为时间的随机过程，增量是随机过程在一定时间段内的变化量。例如，如果{ x (t) : t 在 t }中是一个状态空间 s 和索引集 t = [0，infty)的随机过程，那么对于任意两个非负数 t _ 1在[0，infty)和 t _ 2在[0，infty)中，使得 t _ 1 leq t _ 2，差 x _ { t _ 2}-x _ { t _ 1}是一个 s 值随机变量，称为增量。
The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.<ref name="Kingman1992page38">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=38}}</ref><ref name="DaleyVere-Jones2006page19">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=19}}</ref> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of <math>t</math>, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.<ref name="Kingman1992page22">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=22}}</ref> Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.<ref name="KarlinTaylor2012page118">{{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|pages=118, 119}}</ref><ref name="Kleinrock1976page61">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems00klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1|page=[https://archive.org/details/queueingsystems00klei/page/61 61]}}</ref>
For a measurable subset B of S^T, the pre-image of X gives
对于 s ^ t 的一个可测子集 b，x 的前象给出
<center><math>
< 中心 > < 数学 >
Defined on the real line, the Poisson process can be interpreted as a stochastic process,<ref name="Applebaum2004page1337"/><ref name="Rosenblatt1962page94">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press|page=[https://archive.org/details/randomprocesses00rose_0/page/94 94]}}</ref> among other random objects.<ref name="Haenggi2013page10and18">{{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5|pages=10, 18}}</ref><ref name="ChiuStoyan2013page41and108">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41, 108}}</ref> But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces,<ref name="Kingman1992page11">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=11}}</ref> where it is often interpreted as a random set or a random counting measure, instead of a stochastic process.<ref name="Haenggi2013page10and18"/><ref name="ChiuStoyan2013page41and108"/> In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons.<ref name="Stirzaker2000"/><ref name="Streit2010page1">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC&pg=PA11|year=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|page=1}}</ref> But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.<ref name="Streit2010page1"/><ref name="Kingman1992pagev">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=v}}</ref>
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
X ^ {-1}(b) = { Omega: x (Omega) in b } ,
</math></center>
[数学中心]
==Definitions==
so the law of a X can be written as:
所以 x 的定律可以写成:
===Stochastic process===
A stochastic process is defined as a collection of random variables defined on a common [[probability space]] <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a [[sample space]], <math>\mathcal{F}</math> is a <math>\sigma</math>-[[Sigma-algebra|algebra]], and <math>P</math> is a [[probability measure]]; and the random variables, indexed by some set <math>T</math>, all take values in the same mathematical space <math>S</math>, which must be [[measurable]] with respect to some <math>\sigma</math>-algebra <math>\Sigma</math>.<ref name="Lamperti1977page1"/>
For a stochastic process X with law \mu, its finite-dimensional distributions are defined as:
对于具有 μ 定律的随机过程 x，其有限维分布定义为:
In other words, for a given probability space <math>(\Omega, \mathcal{F}, P)</math> and a measurable space <math>(S,\Sigma)</math>, a stochastic process is a collection of <math>S</math>-valued random variables, which can be written as:<ref name="Florescu2014page293">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=293}}</ref>
<center><math>
< 中心 > < 数学 >
<center><math>
\mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1},
Mu _ { t _ 1，dots，t _ n } = p circ (x ({ t _ 1}) ，dots，x ({ t _ n })) ^ {-1} ,
\{X(t):t\in T \}.
</math></center>
[数学中心]
</math></center>
where n\geq 1 is a counting number and each set t_i is a non-empty finite subset of the index set T, so each t_i\subset T, which means that t_1,\dots,t_n is any finite collection of subsets of the index set T.
其中 n geq1是一个计数数，每个集合 ti 是指数集 t 的一个非空有限子集，因此每个 t i 子集 t，意味着 t _ 1，点，t _ n 是指数集 t 的任意有限子集。
Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>.<ref name="Borovkov2013page528">{{cite book|author=Alexander A. Borovkov|authorlink=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=528}}</ref> A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.<ref name="Lamperti1977page1"/><ref name="LindgrenRootzen2013page11">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>
For any measurable subset C of the n-fold Cartesian power S^n=S\times\dots \times S, the finite-dimensional distributions of a stochastic process X can be written as: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.
对于 n 次笛卡尔幂 s ^ n = s 乘以点 s 的任意可测子集 c，随机过程 x 的有限维分布可以写成: 但是平稳性的概念也存在于点过程和随机场，其中指数集不解释为时间。
There are other ways to consider a stochastic process, with the above definition being considered the traditional one.<ref name="RogersWilliams2000page121">{{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|pages=121, 122}}</ref><ref name="Asmussen2003page408">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=408}}</ref> For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued [[Function (mathematics)|functions]] of <math>t\in T</math> that [[Map (mathematics)|map]] from the set <math>T</math> into the space <math>S</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/>
When the index set T can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.
当指数集 t 可以解释为时间时，如果一个随机过程的有限维分布在时间平移下是不变的，则称其为稳定的。这种类型的随机过程可以用来描述一个处于稳定状态但仍然经历随机波动的物理系统。只有当随机变量是同分布的时候，一系列随机变量才会形成一个平稳的随机过程。Khinchin 提出了广义平稳性的相关概念，广义的协方差平稳性或平稳性又有其他名称。
===Index set===
The set <math>T</math> is called the '''index set'''<ref name="Parzen1999"/><ref name="Florescu2014page294"/> or '''parameter set'''<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{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|pages=93, 94}}</ref> of the stochastic process. Often this set is some subset of the [[real line]], such as the [[natural numbers]] or an interval, giving the set <math>T</math> the interpretation of time.<ref name="doob1953stochasticP46to47"/> In addition to these sets, the index set <math>T</math> can be other linearly ordered sets or more general mathematical sets,<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=482}}</ref> such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space.<ref name="KarlinTaylor2012page27">{{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|page=27}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=25}}</ref> But in general more results and theorems are possible for stochastic processes when the index set is ordered.<ref name="Skorokhod2005page104">{{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=104}}</ref>
A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration \{\mathcal{F}_t\}_{t\in T} , on a probability space (\Omega, \mathcal{F}, P) is a family of sigma-algebras such that \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} for all s \leq t, where t, s\in T and \leq denotes the total order of the index set T.
过滤是一个增加序列的 sigma-代数定义关于一些概率空间和一个索引集，有一些总序关系，例如在情况下的索引集是一些子集的实数。更正式地说，如果一个随机过程有一个总序的索引集，那么在一个概率空间(Omega，mathcal { f } ，p)上的一个过滤{ mathcal { f } _ t } _ { t in t }是一个 σ 代数族，使得对于所有的 s leq t，数学{ f } _ t 子序列 q { f } ，其中 t，s in t 和 leq 表示索引集 t 的总序列。
=== State space ===
The [[mathematical space]] <math>S</math> of a stochastic process is called its ''state space''. This mathematical space can be defined using [[integer]]s, [[real line]]s, <math>n</math>-dimensional [[Euclidean space]]s, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="Florescu2014page294">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=294, 295}}</ref><ref name="Brémaud2014page120">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=120}}</ref>
A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process X that has the same index set T, set space S, and probability space (\Omega,{\cal F},P) as another stochastic process Y is said to be a modification of Y if for all t\in T the following
随机过程的一个修改是另一个随机过程，这是密切相关的原始随机过程。更确切地说，如果一个随机过程 x 与另一个随机过程 y 的索引集 t、索引空间 s 和索引概率空间(Omega，{ cal f } ，p)相同，那么这个 x 就是 y 的修改，如果 t 中的所有 t 都是 y 的话
<center><math>
< 中心 > < 数学 >
===Sample function===
P(X_t=Y_t)=1 ,
P (x _ t = y _ t) = 1,
A '''sample function''' is a single [[Outcome (probability)|outcome]] of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.<ref name="Lamperti1977page1"/><ref name="Florescu2014page296">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=296}}</ref> More precisely, if <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, the [[Map (mathematics)|mapping]]
</math></center>
[数学中心]
<center><math>
holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be stochastically equivalent or equivalent.
持有。两个相互修正的随机过程具有相同的有限维定律，随机等价或等价。
X(\cdot,\omega): T \rightarrow S,
</math></center>
Instead of modification, the term version is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse. The theorem can also be generalized to random fields so the index set is n-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces.
不同的概率空间可以定义不同的两个随机过程，因此两个相互修正的过程，在后一种意义上也是相互修正的过程，但不是相反。这个定理也可以推广到随机场，因此指标集是 n 维欧氏空间，也可以推广到以度量空间为状态空间的随机过程。
is called a sample function, a '''realization''', or, particularly when <math>T</math> is interpreted as time, a '''sample path''' of the stochastic process <math>\{X(t,\omega):t\in T \}</math>.<ref name="RogersWilliams2000page121b">{{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|pages=121–124}}</ref> This means that for a fixed <math>\omega\in\Omega</math>, there exists a sample function that maps the index set <math>T</math> to the state space <math>S</math>.<ref name="Lamperti1977page1"/> Other names for a sample function of a stochastic process include '''trajectory''', '''path function'''<ref name="Billingsley2008page493">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=493}}</ref> or '''path'''.<ref name="Øksendal2003page10">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=10}}</ref>
===Increment===
Two stochastic processes X and Y defined on the same probability space (\Omega,\mathcal{F},P) with the same index set T and set space S are said be indistinguishable if the following
在同一概率空间上定义的具有相同指数集 t 和集空间 s 的随机过程 x 和 y，如果下列情况，则无法区分
An '''increment''' of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if <math>\{X(t):t\in T \}</math> is a stochastic process with state space <math>S</math> and index set <math>T=[0,\infty)</math>, then for any two non-negative numbers <math>t_1\in [0,\infty)</math> and <math>t_2\in [0,\infty)</math> such that <math>t_1\leq t_2</math>, the difference <math>X_{t_2}-X_{t_1}</math> is a <math>S</math>-valued random variable known as an increment.<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/> When interested in the increments, often the state space <math>S</math> is the real line or the natural numbers, but it can be <math>n</math>-dimensional Euclidean space or more abstract spaces such as [[Banach space]]s.<ref name="Applebaum2004page1337"/>
<center><math>
< 中心 > < 数学 >
P(X_t=Y_t \text{ for all } t\in T )=1 ,
P (x _ t = y _ t text { for all } t in t) = 1,
===Further definitions===
</math></center>
[数学中心]
holds.
持有。
====Law====
For a stochastic process <math>X\colon\Omega \rightarrow S^T</math> defined on the probability space <math>(\Omega, \mathcal{F}, P)</math>, the '''law''' of stochastic process <math>X</math> is defined as the [[Pushforward measure|image measure]]:
<center><math>
Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space,{{efn|The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic sense), in addition to other conditions.
可分性是随机过程的一个属性，基于它的索引集与机率量测的关系。假设随机过程泛函或具有不可数指标集的随机场泛函可以形成随机变量。对于可分离的随机过程，除了其他条件外，它的索引集必须是可分离的空间。对于一个可分的随机过程集(在概率意义上) ，它的指数集必须是一个可分的空间(在拓扑或分析意义上) ，除了其他条件。
\mu=P\circ X^{-1},
</math></center>
More precisely, a real-valued continuous-time stochastic process X with a probability space (\Omega,{\cal F},P) is separable if its index set T has a dense countable subset U\subset T and there is a set \Omega_0 \subset \Omega of probability zero, so P(\Omega_0)=0, such that for every open set G\subset T and every closed set F\subset \textstyle R =(-\infty,\infty) , the two events \{ X_t \in F \text{ for all } t \in G\cap U\} and \{ X_t \in F \text{ for all } t \in G\} differ from each other at most on a subset of \Omega_0.
更精确地说，如果指数集 t 有一个稠密可数子集 u 子集 t 且存在一个概率为零的随机过程 ω，则实值连续时刻具有概率空间(ω，{ cal f } ，p)的 x 是可分的,所以 p (Omega _ 0) = 0，对于每个开集 g 子集 t 和每个闭集 f 子集 textstyle r = (- infty，infty) ，两个事件{ x _ t 在 f text { for all } t 在 g 开头 u }和{ x _ t 在 f text { for all } t 在 Omega _ 0的子集上最多不同。
where <math>P</math> is a probability measure, the symbol <math>\circ </math> denotes function composition and <math>X^{-1}</math> is the pre-image of the measurable function or, equivalently, the <math>S^T</math>-valued random variable <math>X</math>, where <math>S^T</math> is the space of all the possible <math>S</math>-valued functions of <math>t\in T</math>, so the law of a stochastic process is a probability measure.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/><ref name="FrizVictoir2010page571"/><ref name="Resnick2013page40">{{cite book|author=Sidney I. Resnick|title=Adventures in Stochastic Processes|url=https://books.google.com/books?id=VQrpBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4612-0387-2|pages=40–41}}</ref>
The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be n-dimensional Euclidean space. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.
可分性的定义也适用于其他的索引集和状态空间，例如在随机场的情况下，索引集和状态空间都可以是 n 维欧氏空间。Doob 的一个定理，有时也被称为 Doob 的可分性定理，说任何实值连续时间随机过程都有一个可分的修正。这个定理的版本也存在于更一般的索引集和状态空间的随机过程，而不是实线。
For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives
<center><math>
Two stochastic processes \left\{X_t\right\} and \left\{Y_t\right\} are called uncorrelated if their cross-covariance \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right] is zero for all times. Formally:
如果左{ x _ t 右}和左{ y _ t 右}的互协方差操作数名{ k }{ mathbf { x } mathbf { y }(t _ 1，t _ 2) = 操作数名{ e }左[ x (t _ 1)-mu _ x (t _ 1)右]左(y (t _ 2)-mu _ y (t _ 2)右]始终为零，则称左{ x _ t 右}和左{ y _ t 右}是不相关的。形式上:
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
</math></center>
\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2.
左{ x _ t 右} ，左{ y _ t 右}文本{ uncorrelated }四边形函数名{ k }{ mathbf { x } mathbf { y }(t _ 1，t _ 2) = 0对所有 t _ 1，t _ 2。
so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>
<center><math>
\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).
If two stochastic processes X and Y are independent, then they are also uncorrelated. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression continue à droite, limite à gauche, due to the functions being right-continuous with left limits. A Skorokhod function space, introduced by Anatoliy Skorokhod, The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, D[0,1] denotes the space of càdlàg functions defined on the unit interval [0,1].
如果两个随机过程 x 和 y 是独立的，那么它们也是不相关的。这种函数称为 càdlàg 或 cadlag 函数，由法语表达式 continue à droite，limite à gauche 的首字母缩写而来，因为这些函数是右连续的，有左限制。由 Anatoliy Skorokhod 引入的 Skorokhod 函数空间，这个函数空间的符号也可以包括定义所有 càdlàg 函数的区间，因此，例如，d [0,1]表示在单位区间[0,1]上定义的 càdlà g 函数的空间。
</math></center>
Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.
Skorokhod 函数空间是随机过程理论中的常用空间，因为它经常假定连续时间随机过程的样本函数属于 Skorokhod 空间。
The law of a stochastic process or a random variable is also called the '''probability law''', '''probability distribution''', or the '''distribution'''.<ref name="Borovkov2013page528"/><ref name="FrizVictoir2010page571"/><ref name="Whitt2006page23">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|page=23}}</ref><ref name="ApplebaumBook2004page4">{{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|page=4}}</ref><ref name="RevuzYor2013page10">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=10}}</ref>
====Finite-dimensional probability distributions====
In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues. For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.
在随机过程的数学构造的背景下，当讨论和假设一个随机过程的某些条件来解决可能的构造问题时，使用术语正则性。例如，为了研究具有不可数指标集的随机过程，我们假设随机过程函数遵守某种类型的正则性条件，如样本函数是连续的。
{{Main|Finite-dimensional distribution}}
For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distributions''' are defined as:
<center><math>
\mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1},
</math></center>
where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <math>T</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page123">{{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=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=123}}</ref>
Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.
马尔可夫过程是随机过程，传统上在离散或连续时间，具有马尔可夫性，这意味着马尔可夫过程的下一个值取决于当前值，但它是有条件地独立于以前的价值随机过程。换句话说，考虑到过程的当前状态，过程在未来的行为随机地独立于过去的行为。
For any measurable subset <math>C</math> of the <math>n</math>-fold [[Cartesian power]] <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as:<ref name="Lamperti1977page1"/>
The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.
布朗运动过程和一维泊松过程都是连续时间马氏过程的例子，而整数上的随机游动和赌徒破产问题都是离散时间马氏过程的例子。
<center><math>
\mu_{t_1,\dots,t_n}(C) =P \Big(\big\{\omega\in \Omega: \big( X_{t_1}(\omega), \dots, X_{t_n}(\omega) \big) \in C \big\} \Big).
A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).
马尔可夫链是一种具有离散状态空间或离散指标集(通常表示时间)的马尔可夫过程，但是马尔可夫链的精确定义是变化的。例如，通常将马尔可夫链定义为离散或连续时间中具有可数状态空间的马尔可夫过程(因此不考虑时间的性质) ，但也通常将马尔可夫链定义为在可数或连续状态空间中具有离散时间的马尔可夫链(因此不考虑状态空间)。
</math></center>
The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.<ref name="Rosenthal2006page177"/>
Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.
马尔可夫过程是一类重要的随机过程，在许多领域有着广泛的应用。例如，它们是一种通用的随机模拟方法的基础，这种方法被称为马尔科夫蒙特卡洛模拟法，用于模拟具有特定概率分布的随机目标，并已在贝叶斯统计中得到应用。
====Stationarity====
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as n-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.
马尔可夫性的概念最初是针对连续时间和离散时间的随机过程，但这一性质已经适用于其他指标集，如 n 维欧氏空间，这导致了被称为马尔可夫随机场的随机变量集合。
{{Main|Stationary process}}
'''Stationarity''' is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if <math>X</math> is a stationary stochastic process, then for any <math>t\in T</math> the random variable <math>X_t</math> has the same distribution, which means that for any set of <math>n</math> index set values <math>t_1,\dots, t_n</math>, the corresponding <math>n</math> random variables
<center><math>
X_{t_1}, \dots X_{t_n},
A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.
鞅是一个离散时间或连续时间的随机过程，其特性是，在给定过程的当前值和所有过去值的任何时刻，每个未来值的条件期望都等于当前值。在离散时间中，如果此属性对下一个值有效，则对所有未来值都有效。鞅的精确数学定义需要两个其他条件加上过滤的数学概念，这与随着时间的推移增加可用信息的直觉有关。鞅通常被定义为实值的，但是它们也可以是复值的，甚至是更一般的。
</math></center>
all have the same [[probability distribution]]. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.<ref name="Lamperti1977page6">{{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=6 and 7}}</ref><ref name="GikhmanSkorokhod1969page4">{{cite book|author1=Iosif I. Gikhman |author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=4}}</ref> But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.<ref name="Lamperti1977page6"/><ref name="Adler2010page14">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA263|year=2010|publisher=SIAM|isbn=978-0-89871-693-1|pages=14, 15}}</ref><ref name="ChiuStoyan2013page112">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=112}}</ref>
A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.
在离散时间和连续时间中，对称随机游动和 Wiener 过程(带零漂)都是鞅的例子。在这方面，离散鞅推广了独立随机变量部分和的概念。
When the index set <math>T</math> can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations.<ref name="Lamperti1977page6"/> The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same.<ref name="Doob1990page94">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=94–96}}</ref> A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.<ref name="Lamperti1977page6"/>
Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.
也可以通过适当的变换从随机过程中产生鞅，这是齐次泊松过程(在实线上)产生一个被称为补偿泊松过程的鞅的情形。
A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process <math>X</math> is said to be stationary in the wide sense, then the process <math>X</math> has a finite second moment for all <math>t\in T</math> and the covariance of the two random variables <math>X_t</math> and <math>X_{t+h}</math> depends only on the number <math>h</math> for all <math>t\in T</math>.<ref name="Doob1990page94"/><ref name="Florescu2014page298">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=298, 299}}</ref> [[Aleksandr Khinchin|Khinchin]] introduced the related concept of '''stationarity in the wide sense''', which has other names including '''covariance stationarity''' or '''stationarity in the broad sense'''.<ref name="Florescu2014page298"/><ref name="GikhmanSkorokhod1969page8">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=8}}</ref>
Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.
鞅数学形式化了公平游戏的概念，它们最初是为了证明不可能赢得公平游戏而开发的。通过在问题中找到一个鞅并研究它，已经解决了许多概率问题。由于鞅收敛定理的存在，在给定矩的一些条件下，鞅会收敛，因此常用它们来推导收敛结果。
====Filtration====
Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process X is a Lévy process if for n non-negatives numbers, 0\leq t_1\leq \dots \leq t_n, the corresponding n-1 increments
鞅在统计学中有许多应用，但有人指出，鞅的使用和应用并不象在统计学领域，特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性，因此它们被称为具有平稳增量和独立增量的过程。换句话说，一个随机过程 x 是一个 Lévy 过程，如果对 n 个非负数，0 leq t _ 1 leq dots leq t _ n，相应的 n-1增量
A [[Filtration (probability theory)|filtration]] is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some [[total order]] relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math>\{\mathcal{F}_t\}_{t\in T} </math>, on a probability space <math>(\Omega, \mathcal{F}, P)</math> is a family of sigma-algebras such that <math> \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} </math> for all <math>s \leq t</math>, where <math>t, s\in T</math> and <math>\leq</math> denotes the total order of the index set <math>T</math>.<ref name="Florescu2014page294"/> With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process <math>X_t</math> at <math>t\in T</math>, which can be interpreted as time <math>t</math>.<ref name="Florescu2014page294"/><ref name="Williams1991page93"/> The intuition behind a filtration <math>\mathcal{F}_t</math> is that as time <math>t</math> passes, more and more information on <math>X_t</math> is known or available, which is captured in <math>\mathcal{F}_t</math>, resulting in finer and finer partitions of <math>\Omega</math>.<ref name="Klebaner2005page22">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|pages=22–23}}</ref><ref name="MörtersPeres2010page37">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=37}}</ref>
<center><math>
< 中心 > < 数学 >
X_{t_2}-X_{t_1}, \dots , X_{t_{n-1}}-X_{t_n},
2}-x _ { t _ 1} ，点，x _ { t _ { n-1}-x _ { t _ n } ,
====Modification====
</math></center>
[数学中心]
A '''modification''' of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process <math>X</math> that has the same index set <math>T</math>, set space <math>S</math>, and probability space <math>(\Omega,{\cal F},P)</math> as another stochastic process <math>Y</math> is said to be a modification of <math>Y</math> if for all <math>t\in T</math> the following
are all independent of each other, and the distribution of each increment only depends on the difference in time. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.
都是相互独立的，每个增量的分布只取决于时间的差异。如果随机过程的具体定义要求索引集是实线的一个子集，那么随机场可以被认为是随机过程的推广。
<center><math>
P(X_t=Y_t)=1 ,
</math></center>
holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law<ref name="RogersWilliams2000page130">{{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=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=130}}</ref> and they are said to be '''stochastically equivalent''' or '''equivalent'''.<ref name="Borovkov2013page530">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>
A point process is a collection of points randomly located on some mathematical space such as the real line, n-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or n-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.
点过程是在一些数学空间(如实直线、 n 维欧氏空间或更多的抽象空间)上随机定位的点的集合。有时，词汇点过程并不是首选，因为历史上词汇过程表示某个系统在时间上的演变，所以点过程也称为随机点场。一个点过程有不同的解释，比如随机计数测度或随机集合。有些作者把点过程和随机过程过程看作是两个不同的对象，例如，点过程是一个随机的对象，它起源于或与随机过程过程相关联，尽管有人指出点过程和随机过程之间的区别并不清楚。它对应于随机过程术语中的索引集。}实线或 n 维欧几里德空间。在点过程理论中研究了更新和计数过程等其他随机过程。
Instead of modification, the term '''version''' is also used,<ref name="Adler2010page14"/><ref name="Klebaner2005page48">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=48}}</ref><ref name="Øksendal2003page14">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=14}}</ref><ref name="Florescu2014page472">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=472}}</ref> however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.<ref name="RevuzYor2013page18">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|pages=18–19}}</ref><ref name="FrizVictoir2010page571"/>
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the [[Kolmogorov continuity theorem]] says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.<ref name="Øksendal2003page14"/><ref name="Florescu2014page472"/><ref name="ApplebaumBook2004page20">{{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|page=20}}</ref> The theorem can also be generalized to random fields so the index set is <math>n</math>-dimensional Euclidean space<ref name="Kunita1997page31">{{cite book|author=Hiroshi Kunita|title=Stochastic Flows and Stochastic Differential Equations|url=https://books.google.com/books?id=_S1RiCosqbMC|year=1997|publisher=Cambridge University Press|isbn=978-0-521-59925-2|page=31}}</ref> as well as to stochastic processes with [[metric spaces]] as their state spaces.<ref name="Kallenberg2002page">{{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|page=35}}</ref>
Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago, but very little analysis on them was done in terms of probability. The year 1654 is often considered the birth of probability theory when French mathematicians Pierre Fermat and Blaise Pascal had a written correspondence on probability, motivated by a gambling problem. But there was earlier mathematical work done on the probability of gambling games such as Liber de Ludo Aleae by Gerolamo Cardano, written in the 16th century but posthumously published later in 1663.
概率论游戏起源于机会游戏，这种游戏有着悠久的历史，有些游戏在几千年前就已经开始玩了，但是很少从概率的角度对它们进行分析。1654年通常被认为是概率论的诞生，当时法国数学家 Pierre Fermat 和 Blaise Pascal 因为一个赌博问题写了一封关于概率的信。但是在赌博游戏的可能性方面，早期的数学工作已经完成，比如吉罗拉莫·卡尔达诺的 Liber de Ludo Aleae，写于16世纪，但死后于1663年出版。
====Indistinguishable====
Two stochastic processes <math>X</math> and <math>Y</math> defined on the same probability space <math>(\Omega,\mathcal{F},P)</math> with the same index set <math>T</math> and set space <math>S</math> are said be '''indistinguishable''' if the following
After Cardano, Jakob Bernoulli wrote Ars Conjectandi, which is considered a significant event in the history of probability theory. But despite some renowned mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev, most of the mathematical community did not consider probability theory to be part of mathematics until the 20th century.
在 Cardano 之后，Jakob Bernoulli 写了 Ars Conjectandi，这被认为是概率论历史上的一个重大事件。但是，尽管一些著名的数学家为概率论做出了贡献，比如皮埃尔-西蒙·拉普拉斯，亚伯拉罕·棣莫弗，Carl Gauss，Siméon Poisson 和巴夫尼提·列波维奇·切比雪夫，大多数数学界直到20世纪才认为概率论是数学的一部分。
<center><math>
P(X_t=Y_t \text{ for all } t\in T )=1 ,
</math></center>
In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius, most of the work had little or no randomness.
在物理科学领域，科学家们在19世纪发展了统计力学学科，在这个学科中，物理系统，例如装满气体的容器，可以被看作或者从数学上被当作许多运动粒子的集合。尽管有一些科学家，比如鲁道夫 · 克劳修斯，试图将随机性纳入统计物理学，但大多数工作几乎没有或根本没有随机性。
holds.<ref name="FrizVictoir2010page571"/><ref name="RogersWilliams2000page130"/> If two <math>X</math> and <math>Y</math> are modifications of each other and are almost surely continuous, then <math>X</math> and <math>Y</math> are indistinguishable.<ref name="JeanblancYor2009page11">{{cite book|author1=Monique Jeanblanc|author1-link= Monique Jeanblanc |author2=Marc Yor|author2-link=Marc Yor|author3=Marc Chesney|title=Mathematical Methods for Financial Markets|url=https://books.google.com/books?id=ZhbROxoQ-ZMC|year=2009|publisher=Springer Science & Business Media|isbn=978-1-85233-376-8|page=11}}</ref>
This changed in 1859 when James Clerk Maxwell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities. The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement.
这种情况在1859年发生了改变，当时詹姆斯·克拉克·麦克斯韦对这个领域做出了重大贡献，更具体地说，他提出了假设气体粒子以随机速度向随机方向运动的工作，这对分子运动论研究有重大贡献。分子运动论和统计物理学在19世纪下半叶继续发展，主要由克劳修斯、路德维希·玻尔兹曼和约西亚吉布斯完成的工作，后来对阿尔伯特爱因斯坦的布朗运动的数学模型产生了影响。
====Separability====
'''Separability''' is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a [[separable space]],{{efn|The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic sense), in addition to other conditions.<ref name="Skorokhod2005page93"/>}} which means that the index set has a dense countable subset.<ref name="Adler2010page14"/><ref name="Ito2006page32">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|pages=32–33}}</ref>
At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms.}} and Andrei Kolmogorov. In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.
1900年在巴黎的国际数学家大会，David Hilbert 展示了一系列数学问题，其中他的第六个问题要求对物理学和涉及公理的概率进行数学处理和安德烈 · 科尔莫戈罗夫。在20世纪30年代早期，钦钦和科尔莫戈罗夫设立了概率研讨会，参加研讨会的研究人员有 Eugene Slutsky 和 Nikolai Smirnov，钦钦给出了第一个数学定义，随机过程是一组由实数线索引的随机变量。
More precisely, a real-valued continuous-time stochastic process <math>X</math> with a probability space <math>(\Omega,{\cal F},P)</math> is separable if its index set <math>T</math> has a dense countable subset <math>U\subset T</math> and there is a set <math>\Omega_0 \subset \Omega</math> of probability zero, so <math>P(\Omega_0)=0</math>, such that for every open set <math>G\subset T</math> and every closed set <math>F\subset \textstyle R =(-\infty,\infty) </math>, the two events <math>\{ X_t \in F \text{ for all } t \in G\cap U\}</math> and <math>\{ X_t \in F \text{ for all } t \in G\}</math> differ from each other at most on a subset of <math>\Omega_0</math>.<ref name="GikhmanSkorokhod1969page150">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=150}}</ref><ref name="Todorovic2012page19">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|pages=19–20}}</ref><ref name="Molchanov2005page340">{{cite book|author=Ilya Molchanov|title=Theory of Random Sets|url=https://books.google.com/books?id=kWEwk1UL42AC|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-892-3|page=340}}</ref>
The definition of separability{{efn|The definition of separability for a continuous-time real-valued stochastic process can be stated in other ways.<ref name="Billingsley2008page526">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=526–527}}</ref><ref name="Borovkov2013page535">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=535}}</ref>}} can also be stated for other index sets and state spaces,<ref name="GusakKukush2010page22">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 22</ref> such as in the case of random fields, where the index set as well as the state space can be <math>n</math>-dimensional Euclidean space.<ref name="AdlerTaylor2009page7"/><ref name="Adler2010page14"/>
In 1933 Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,{{efn|Later translated into English and published in 1950 as Foundations of the Theory of Probability
1933年 Andrei Kolmogorov 出版了一本关于概率论基础的书，名为 grundbigriffe der Wahrscheinlichkeitsrechnung，1950年被翻译成英文并出版为概率论基础
The concept of separability of a stochastic process was introduced by [[Joseph Doob]],<ref name="Ito2006page32"/>. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.<ref name="Billingsley2008page526"/> Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.<ref name="Doob1990page56">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=56}}</ref> A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.<ref name="Ito2006page32"/><ref name="Todorovic2012page19"/><ref name="Khoshnevisan2006page155">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|page=155}}</ref> Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.<ref name="Skorokhod2005page93"/>
Mathematician [[Joseph Doob did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales. Starting in the 1940s, Kiyosi Itô published papers developing the field of stochastic calculus, which involves stochastic integrals and stochastic differential equations based on the Wiener or Brownian motion process.
数学家[约瑟夫 · 杜布在随机过程理论方面做了早期的工作，作出了基本的贡献，尤其是在鞅理论方面。从20世纪40年代开始，Kiyosi itô 发表了论文，拓展了随机分析的研究领域，包括随机积分和基于 Wiener 或 Brownian 运动过程的随机微分方程。
====Independence====
Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theory, with early ideas by Shizuo Kakutani and then later work by Joseph Doob.
同样从20世纪40年代开始，随机过程，特别是鞅，和势场理论的数学领域之间建立了联系，早期的思想由 Shizuo Kakutani 提出，后来由 Joseph Doob 提出。
Two stochastic processes <math>X</math> and <math>Y</math> defined on the same probability space <math>(\Omega,\mathcal{F},P)</math> with the same index set <math>T</math> are said be '''independent''' if for all <math>n \in \mathbb{N}</math> and for every choice of epochs <math>t_1,\ldots,t_n \in T</math>, the random vectors <math>\left( X(t_1),\ldots,X(t_n) \right)</math> and <math>\left( Y(t_1),\ldots,Y(t_n) \right)</math> are independent.<ref name=Lapidoth>Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.</ref>{{rp|p. 515}}
In 1953 Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-André Meyer. Earlier work had been carried out by Sergei Bernstein, Paul Lévy and Jean Ville, the latter adopting the term martingale for the stochastic process. Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes. which would later result in Varadhan winning the 2007 Abel Prize. In the 1990s and 2000s the theories of Schramm–Loewner evolution and rough paths were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin Werner in 2008 and to Martin Hairer in 2014.
1953年杜布出版了《随机过程》一书，该书对随机过程理论产生了重大影响，并强调了概率测度理论的重要性。Doob 还主要发展了鞅理论，后来保罗-安德烈 · 迈耶做出了重大贡献。早期的工作是由 Sergei Bernstein，Paul Lévy 和 Jean Ville 完成的，Jean Ville 采用了鞅这个术语来称呼随机过程。从鞅理论开始，解决各种概率问题的方法变得流行起来。研究马尔可夫过程的技术和理论得到了发展，并应用于鞅。相反，从鞅理论中建立了处理马尔可夫过程的方法。后来 Varadhan 赢得了2007年的阿贝尔奖。20世纪90年代和21世纪初，Schramm-Loewner 进化理论和粗糙路径理论被引入并发展起来，用于研究21概率论的随机过程和其他数学对象，结果分别在2008年和2014年分别授予 Wendelin Werner 和 Martin Hairer 菲尔兹奖。
====Uncorrelatedness====
Two stochastic processes <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are called '''uncorrelated''' if their cross-covariance <math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right]</math> is zero for all times.<ref name=KunIlPark>Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3</ref>{{rp|p. 142}} Formally:
The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.
随机过程理论仍然是研究的焦点，每年都有关于随机过程的国际会议。
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. Bernoulli's work, including the Bernoulli process, were published in his book Ars Conjectandi in 1713.
伯努利过程可以作为一个数学模型来抛出一个有偏见的硬币，它可能是第一个被研究的随机过程。伯努利的著作，包括《伯努利过程，于1713年在他的书《猜测》中出版。
====Independence implies uncorrelatedness====
If two stochastic processes <math>X</math> and <math>Y</math> are independent, then they are also uncorrelated.<ref name=KunIlPark/>{{rp|p. 151}}
In 1905 Karl Pearson coined the term random walk while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. and is an example of a random walk with absorbing barriers. Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods, and then more detailed solutions were presented by Jakob Bernoulli and Abraham de Moivre.
1905年，卡尔 · 皮尔森在提出一个描述平面上随机漫步的问题时，创造了随机漫步这个术语，这个问题的动机是生物学中的一个应用，但是这种涉及随机漫步的问题已经在其他领域得到了研究。几个世纪前研究过的某些赌博问题可以被认为是涉及随机漫步的问题。这是一个带有吸收屏障的随机漫步的例子。和 Huyens 都给出了这个问题的数值解，但没有详细介绍他们的方法，然后 Jakob Bernoulli 和亚伯拉罕·棣莫弗提供了更详细的解。
====Orthogonality====
Two stochastic processes <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are called '''orthogonal''' if their cross-correlation <math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E}[X(t_1) \overline{Y(t_2)}]</math> is zero for all times.<ref name=KunIlPark/>{{rp|p. 142}} Formally:
For random walks in n-dimensional integer lattices, George Pólya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.
对于 n 维整数格中的随机游动，George Pólya 在1919年和1921年发表的著作中，研究了对称随机游动回到格中先前位置的概率。Pólya 证明了对称随机游动，它在格子中向任何方向前进的概率相等，将无限次地回到格子中的一个先前的位置，概率为1在一维和2维，但概率为0在三维或更高维。
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ orthogonal} \quad \iff \quad \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics. The work is now considered as an early discovery of the statistical method known as Kalman filtering, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time. in order to model price changes on the Paris Bourse, a stock exchange, without knowing the work of Thiele. and Bachelier's thesis is now considered pioneering in the field of financial mathematics.
维纳过程或布朗运动过程起源于不同的领域，包括统计学、金融学和物理学。这项工作现在被认为是卡尔曼滤波统计方法的早期发现，但是这项工作在很大程度上被忽视了。人们认为，蒂勒论文中的观点太过先进，当时更广泛的数学和统计学界无法理解。为了模拟巴黎证券交易所的价格变化，不知道蒂勒的工作。巴切利耶的论文现在被认为是金融数学领域的先驱。
====Skorokhod space====
{{Main|Skorokhod space}}
Einstein's work, as well as experimental results obtained by Jean Perrin, later inspired Norbert Wiener in the 1920s to use a type of measure theory, developed by Percy Daniell, and Fourier analysis to prove the existence of the Wiener process as a mathematical object. There are a number of claims for early uses or discoveries of the Poisson
爱因斯坦的工作，以及 Jean Perrin 获得的实验结果，后来激发了 Norbert Wiener 在20世纪20年代使用一种由 Percy Daniell 和傅立叶变换家族中的关系提出的测量理论来证明 Wiener 过程作为一个数学对象的存在。关于泊松鱼的早期用途和发现，有许多说法
A '''Skorokhod space''', also written as '''Skorohod space''', is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as <math>[0,1]</math> or <math>[0,\infty)</math>, and take values on the real line or on some metric space.<ref name="Whitt2006page78">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|pages=78–79}}</ref><ref name="GusakKukush2010page24">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 24</ref><ref name="Bogachev2007Vol2page53">{{cite book|author=Vladimir I. Bogachev|title=Measure Theory (Volume 2)|url=https://books.google.com/books?id=CoSIe7h5mTsC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-34514-5|page=53}}</ref> Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression ''continue à droite, limite à gauche'', due to the functions being right-continuous with left limits.<ref name="Whitt2006page78"/><ref name="Klebaner2005page4">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=4}}</ref> A Skorokhod function space, introduced by [[Anatoliy Skorokhod]],<ref name="Bogachev2007Vol2page53"/> is often denoted with the letter <math>D</math>,<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/> so the function space is also referred to as space <math>D</math>.<ref name="Whitt2006page78"/><ref name="Asmussen2003page420">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=420}}</ref><ref name="Billingsley2013page121">{{cite book|author=Patrick Billingsley|title=Convergence of Probability Measures|url=https://books.google.com/books?id=6ItqtwaWZZQC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-62596-5|page=121}}</ref> The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, <math>D[0,1]</math> denotes the space of càdlàg functions defined on the [[unit interval]] <math>[0,1]</math>.<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>
process.
过程。
Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.<ref name="Bogachev2007Vol2page53"/><ref name="Asmussen2003page420"/> Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.<ref name="Billingsley2013page121"/><ref name="BinghamKiesel2013page154">{{cite book|author1=Nicholas H. Bingham|author2=Rüdiger Kiesel|title=Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives|url=https://books.google.com/books?id=AOIlBQAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-3856-3|page=154}}</ref>
Another discovery occurred in Denmark in 1909 when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution. Markov was interested in studying an extension of independent random sequences. which had been commonly regarded as a requirement for such mathematical laws to hold. Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.
另一个发现发生在1909年的丹麦。在开发一个有限时间间隔内接听电话数量的数学模型时，Erlang 得出了这个泊松分佈。当时 Erlang 并不知道 Poisson 的早期工作，并且假设每个时间间隔内到达的号码电话是相互独立的。然后他发现了极限情况，这是有效地重铸泊松分佈作为一个二项分布的限制。马尔科夫对研究独立随机序列的推广很感兴趣。这被普遍认为是这样的数学定律的一个必要条件。从1928年开始，莫里斯 · 弗雷切特对马尔可夫链产生了兴趣，最终导致他在1938年发表了一篇关于马尔可夫链的详细研究。
====Regularity====
Andrei Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s. In addition to Lévy, Khinchin and Kolomogrov, early fundamental contributions to the theory of Lévy processes were made by Bruno de Finetti and Kiyosi Itô.
安德烈 · 科尔莫戈罗夫在1931年的一篇论文中发展了早期连续时间马尔可夫过程理论的很大一部分。他介绍并研究了一组特殊的马尔可夫过程，称为扩散过程，在这组过程中他推导出了一组描述这些过程的微分方程。在研究布朗运动时，Sydney Chapman 在1928年的一篇论文中，独立于 Kolmogorov 的工作，用一种比 Kolmogorov 更不严密的数学方法，推导出了一个方程，现在称为 Chapman-Kolmogorov 方程。这些微分方程现在被称为 Kolmogorov 方程或 Kolmogorov-Chapman 方程。其他对马尔可夫过程的基础做出了重大贡献的数学家包括威廉 · 费勒，从20世纪30年代开始，然后是尤金 · 戴金，从20世纪50年代开始。除了 Lévy，Khinchin 和 Kolomogrov，早期对 Lévy 过程理论的根本性贡献是由德福内梯和 Kiyosi itô。
In the context of mathematical construction of stochastic processes, the term '''regularity''' is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.<ref name="Borovkov2013page532">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=532}}</ref><ref name="Khoshnevisan2006page148to165">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=148–165}}</ref> For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.<ref name="Todorovic2012page22">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|page=22}}</ref><ref name="Whitt2006page79">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|page=79}}</ref>
Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem to prove a corresponding stochastic process exists. says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions. This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.
另一种方法是定义一组具有特定有限维分布的随机变量，然后用 Kolmogorov 的存在性定理证明相应的随机过程存在。他说，如果任何有限维分布满足两个条件，也就是所谓的一致性条件，那么就存在这些有限维分布的随机过程。这意味着随机过程的分布并不一定唯一地指定随机过程的样本函数的属性。
==Further examples==
Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined. Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set. Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied. for a continuous-time stochastic process with any metric space as its state space. For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption, but such a stochastic process based on this approach will be automatically separable.
另一个问题是，连续时间过程的泛函依赖于指数集中无法计算的点数，因此某些事件的概率可能无法很好地定义。可分性保证了无穷维分布决定样本函数的性质，它要求样本函数本质上是由指数集中的稠密可数点集上的值决定的。此外，如果随机过程是可分的，那么指数集上不可数个点的泛函是可测的，并且可以研究它们的概率。对于任意度量空间作为状态空间的连续时间随机过程。为了构造这样一个随机过程，我们假设随机过程的样本函数属于某个适当的函数空间，这个空间通常是由所有右连续函数和左极限组成的 Skorokhod 空间。这种方法现在比可分离性假设更常用，但是基于这种方法的随机过程可自动分离。
===Markov processes and chains===
{{Main|Markov process}}
Although less used, the separability assumption is considered more general because every stochastic process has a separable version. For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as n-dimensional Euclidean space.
尽管很少使用，但是可分性假设被认为是更一般的，因为每个随机过程都有一个可分离的版本。例如，在构造和研究随机场时假设可分性，其中随机变量的集合现在由实线以外的集合索引，如 n 维欧氏空间。
Markov processes are stochastic processes, traditionally in [[Discrete time and continuous time|discrete or continuous time]], that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.<ref name="Serfozo2009page2">{{cite book|author=Richard Serfozo|title=Basics of Applied Stochastic Processes|url=https://books.google.com/books?id=JBBRiuxTN0QC|year=2009|publisher=Springer Science & Business Media|isbn=978-3-540-89332-5|page=2}}</ref><ref name="Rozanov2012page58">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=58}}</ref>
The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes<ref name="Ross1996page235and358">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|pages=235, 358}}</ref> in continuous time, while [[random walk]]s on the integers and the [[gambler's ruin]] problem are examples of Markov processes in discrete time.<ref name="Florescu2014page373">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=373, 374}}</ref><ref name="KarlinTaylor2012page49">{{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|page=49}}</ref>
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A Markov chain is a type of Markov process that has either discrete [[state space]] or discrete index set (often representing time), but the precise definition of a Markov chain varies.<ref name="Asmussen2003page7">{{cite book|url=https://books.google.com/books?id=BeYaTxesKy0C|title=Applied Probability and Queues|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=7|author=Søren Asmussen}}</ref> For example, it is common to define a Markov chain as a Markov process in either [[Continuous and discrete variables|discrete or continuous time]] with a countable state space (thus regardless of the nature of time),<ref name="Parzen1999page188">{{cite book|url=https://books.google.com/books?id=0mB2CQAAQBAJ|title=Stochastic Processes|year=2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|page=188|author=Emanuel Parzen}}</ref><ref name="KarlinTaylor2012page29">{{cite book|url=https://books.google.com/books?id=dSDxjX9nmmMC|title=A First Course in Stochastic Processes|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=29, 30|author1=Samuel Karlin|author2=Howard E. Taylor}}</ref><ref name="Lamperti1977chap6">{{cite book|url=https://books.google.com/books?id=Pd4cvgAACAAJ|title=Stochastic processes: a survey of the mathematical theory|publisher=Springer-Verlag|year=1977|isbn=978-3-540-90275-1|pages=106–121|author=John Lamperti}}</ref><ref name="Ross1996page174and231">{{cite book|url=https://books.google.com/books?id=ImUPAQAAMAAJ|title=Stochastic processes|publisher=Wiley|year=1996|isbn=978-0-471-12062-9|pages=174, 231|author=Sheldon M. Ross}}</ref> but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).<ref name="Asmussen2003page7" /> It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like [[Joseph Doob]] and [[Kai Lai Chung]].<ref name="MeynTweedie2009">{{cite book|author1=Sean Meyn|author2=Richard L. Tweedie|title=Markov Chains and Stochastic Stability|url=https://books.google.com/books?id=Md7RnYEPkJwC|year=2009|publisher=Cambridge University Press|isbn=978-0-521-73182-9|page=19}}</ref>
Markov processes form an important class of stochastic processes and have applications in many areas.<ref name="LatoucheRamaswami1999"/><ref name="KarlinTaylor2012page47">{{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|page=47}}</ref> For example, they are the basis for a general stochastic simulation method known as [[Markov chain Monte Carlo]], which is used for simulating random objects with specific probability distributions, and has found application in [[Bayesian statistics]].<ref name="RubinsteinKroese2011page225">{{cite book|author1=Reuven Y. Rubinstein|author2=Dirk P. Kroese|title=Simulation and the Monte Carlo Method|url=https://books.google.com/books?id=yWcvT80gQK4C|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-21052-9|page=225}}</ref><ref name="GamermanLopes2006">{{cite book|author1=Dani Gamerman|author2=Hedibert F. Lopes|title=Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition|url=https://books.google.com/books?id=yPvECi_L3bwC|year=2006|publisher=CRC Press|isbn=978-1-58488-587-0}}</ref>
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math>n</math>-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.<ref name="Rozanov2012page61">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=61}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=27}}</ref><ref name="Bremaud2013page253">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=253}}</ref>
===Martingale===
{{Main|Martingale (probability theory)}}
A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,<ref name="Klebaner2005page65">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=65}}</ref><ref name="KaratzasShreve2014page11">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=11}}</ref><ref name="Williams1991page93">{{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|pages=93, 94}}</ref> but they can also be complex-valued<ref name="Doob1990page292">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref> or even more general.<ref name="Pisier2016">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref>
A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.<ref name="Klebaner2005page65"/><ref name="KaratzasShreve2014page11"/> For a [[sequence]] of [[independent and identically distributed]] random variables <math>X_1, X_2, X_3, \dots</math> with zero mean, the stochastic process formed from the successive partial sums <math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> is a discrete-time martingale.<ref name="Steele2012page12">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|pages=12, 13}}</ref> In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.<ref name="HallHeyde2014page2">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</ref>
Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the ''compensated Poisson process''.<ref name="KaratzasShreve2014page11"/> Martingales can also be built from other martingales.<ref name="Steele2012page12"/> For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.<ref name="Klebaner2005page65"/><ref name="Steele2012page115">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=115}}</ref>
Martingales mathematically formalize the idea of a fair game,<ref name="Ross1996page295">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=295}}</ref> and they were originally developed to show that it is not possible to win a fair game.<ref name="Steele2012page11"/> But now they are used in many areas of probability, which is one of the main reasons for studying them.<ref name="Williams1991page93"/><ref name="Steele2012page11">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=11}}</ref><ref name="Kallenberg2002page96">{{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=96}}</ref> Many problems in probability have been solved by finding a martingale in the problem and studying it.<ref name="Steele2012page371">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=371}}</ref> Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to [[martingale convergence theorem]]s.<ref name="HallHeyde2014page2"/><ref name="Steele2012page22">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=22}}</ref><ref name="GrimmettStirzaker2001page336">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=336}}</ref>
Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref> They have found applications in areas in probability theory such as queueing theory and Palm calculus<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref> and other fields such as economics<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/>
===Lévy process===
{{Main|Lévy process}}
Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref> These processes have many applications in fields such as finance, fluid mechanics, physics and biology.<ref name="Applebaum2004page1336">{{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}}</ref><ref name="ApplebaumBook2004page69">{{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|page=69}}</ref> The main defining characteristics of these processes are their stationarity and independence properties, so they were known as ''processes with stationary and independent increments''. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments
<center><math>
X_{t_2}-X_{t_1}, \dots , X_{t_{n-1}}-X_{t_n},
</math></center>
are all independent of each other, and the distribution of each increment only depends on the difference in time.<ref name="Applebaum2004page1337"/>
<!-- Tips for referencing:
< ! -- 参考提示:
A Lévy process can be defined such that its state space is some abstract mathematical space, such as a [[Banach space]], but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so <math> I= [0,\infty) </math>, which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and [[subordinator (mathematics)|subordinators]] are all Lévy processes.<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII"/>
For books, use:
对于书籍，使用:
===Random field===
{{Main|Random field}}
A random field is a collection of random variables indexed by a <math>n</math>-dimensional Euclidean space or some manifold. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line.<ref name="AdlerTaylor2009page7"/> But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions.<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="KoralovSinai2007page171">{{cite book|author1=Leonid Koralov|author2=Yakov G. Sinai|title=Theory of Probability and Random Processes|url=https://books.google.com/books?id=tlWOphOFRgwC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-68829-7|page=171}}</ref> If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.<ref name="ApplebaumBook2004page19">{{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|page=19}}</ref>
Also use a web tool for getting book citation details via Google Books:
还可以使用网络工具通过 Google Books 获取图书的引用细节:
===Point process===
{{Main|Point process}}
http://reftag.appspot.com/
Http://reftag.appspot.com/
A point process is a collection of points randomly located on some mathematical space such as the real line, <math>n</math>-dimensional Euclidean space, or more abstract spaces. Sometimes the term ''point process'' is not preferred, as historically the word ''process'' denoted an evolution of some system in time, so a point process is also called a '''random point field'''.<ref name="ChiuStoyan2013page109">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=109}}</ref> There are different interpretations of a point process, such a random counting measure or a random set.<ref name="ChiuStoyan2013page108">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=108}}</ref><ref name="Haenggi2013page10">{{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5|page=10}}</ref> Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process,<ref name="DaleyVere-Jones2006page194">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=194}}</ref><ref name="CoxIsham1980page3">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8|page=3}}</ref> though it has been remarked that the difference between point processes and stochastic processes is not clear.<ref name="CoxIsham1980page3"/>
or article citation details via DOI numbers:
或文章引用细节通过 DOI 编号:
Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space{{efn|In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line,<ref name="Kingman1992page8">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=8}}</ref><ref name="MollerWaagepetersen2003page7">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|year=2003|publisher=CRC Press|isbn=978-0-203-49693-0|page=7}}</ref> which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or <math>n</math>-dimensional Euclidean space.<ref name="KarlinTaylor2012page31">{{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|page=31}}</ref><ref name="Schmidt2014page99">{{cite book|author=Volker Schmidt|title=Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms|url=https://books.google.com/books?id=brsUBQAAQBAJ&pg=PR5|date= 2014|publisher=Springer|isbn=978-3-319-10064-7|page=99}}</ref> Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.<ref name="DaleyVere-Jones200">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8}}</ref><ref name="CoxIsham1980">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8}}</ref>
http://reftag.appspot.com/doiweb.py
Http://reftag.appspot.com/doiweb.py
==History==
For other sources, see: WP:CITET
有关其他来源，请参阅: WP: CITET
===Early probability theory===
Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,<ref name=":1">{{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=US|pages=1–2}}</ref><ref name="David1955">{{cite journal|last1=David|first1=F. N.|title=Studies in the History of Probability and Statistics I. Dicing and Gaming (A Note on the History of Probability)|journal=Biometrika|volume=42|issue=1/2|pages=1–15|year=1955|issn=0006-3444|doi=10.2307/2333419|jstor=2333419}}</ref> but very little analysis on them was done in terms of probability.<ref name=":1" /><ref name="Maistrov2014page1">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=1}}</ref> The year 1654 is often considered the birth of probability theory when French mathematicians [[Pierre Fermat]] and [[Blaise Pascal]] had a written correspondence on probability, motivated by a [[Problem of points|gambling problem]].<ref name=":1" /><ref name="Seneta2006page1">{{cite book|last1=Seneta|first1=E.|title=Encyclopedia of Statistical Sciences|chapter=Probability, History of|year=2006|doi=10.1002/0471667196.ess2065.pub2|page=1|isbn=978-0471667193}}</ref><ref name="Tabak2014page24to26">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|pages=24–26}}</ref> But there was earlier mathematical work done on the probability of gambling games such as ''Liber de Ludo Aleae'' by [[Gerolamo Cardano]], written in the 16th century but posthumously published later in 1663.<ref name=":1" /><ref name="Bellhouse2005">{{cite journal|last1=Bellhouse|first1=David|title=Decoding Cardano's Liber de Ludo Aleae|journal=Historia Mathematica|volume=32|issue=2|year=2005|pages=180–202|issn=0315-0860|doi=10.1016/j.hm.2004.04.001|doi-access=free}}</ref>
-->
-->
After Cardano, [[Jakob Bernoulli]]{{efn|Also known as James or Jacques Bernoulli.<ref name="Hald2005page221">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=221}}</ref>}} wrote [[Ars Conjectandi]], which is considered a significant event in the history of probability theory.<ref name=":1" /> Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability.<ref name=":1" /><ref name="Maistrov2014page56">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=56}}</ref><ref name="Tabak2014page37">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|page=37}}</ref> But despite some renowned mathematicians contributing to probability theory, such as [[Pierre-Simon Laplace]], [[Abraham de Moivre]], [[Carl Gauss]], [[Siméon Poisson]] and [[Pafnuty Chebyshev]],<ref name="Chung1998">{{cite journal|last1=Chung|first1=Kai Lai|title=Probability and Doob|journal=The American Mathematical Monthly|volume=105|issue=1|pages=28–35|year=1998|issn=0002-9890|doi=10.2307/2589523|jstor=2589523}}</ref><ref name="Bingham2000">{{cite journal|last1=Bingham|first1=N.|title=Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov|journal=Biometrika|volume=87|issue=1|year=2000|pages=145–156|issn=0006-3444|doi=10.1093/biomet/87.1.145}}</ref> most of the mathematical community{{efn|It has been remarked that a notable exception was the St Petersburg School in Russia, where mathematicians led by Chebyshev studied probability theory.<ref name="BenziBenzi2007">{{cite journal|last1=Benzi|first1=Margherita|last2=Benzi|first2=Michele|last3=Seneta|first3=Eugene|title=Francesco Paolo Cantelli. b. 20 December 1875 d. 21 July 1966|journal=International Statistical Review|volume=75|issue=2|year=2007|page=128|issn=0306-7734|doi=10.1111/j.1751-5823.2007.00009.x}}</ref>}} did not consider probability theory to be part of mathematics until the 20th century.<ref name="Chung1998"/><ref name="BenziBenzi2007"/><ref name="Doob1996">{{cite journal|last1=Doob|first1=Joseph L.|title=The Development of Rigor in Mathematical Probability (1900-1950)|journal=The American Mathematical Monthly|volume=103|issue=7|pages=586–595|year=1996|issn=0002-9890|doi=10.2307/2974673|jstor=2974673}}</ref><ref name="Cramer1976">{{cite journal|last1=Cramer|first1=Harald|title=Half a Century with Probability Theory: Some Personal Recollections|journal=The Annals of Probability|volume=4|issue=4|year=1976|pages=509–546|issn=0091-1798|doi=10.1214/aop/1176996025|doi-access=free}}</ref>
===Statistical mechanics===
In the physical sciences, scientists developed in the 19th century the discipline of [[statistical mechanics]], where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as [[Rudolf Clausius]], most of the work had little or no randomness.<ref name="Truesdell1975page22">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=22–23|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1967page150">{{cite journal|last1=Brush|first1=Stephen G.|title=Foundations of statistical mechanics 1845?1915|journal=Archive for History of Exact Sciences|volume=4|issue=3|year=1967|pages=150–151|issn=0003-9519|doi=10.1007/BF00412958}}</ref>
This changed in 1859 when [[James Clerk Maxwell]] contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities.<ref name="Truesdell1975page31">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=31–32|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1958">{{cite journal|last1=Brush|first1=S.G.|title=The development of the kinetic theory of gases IV. Maxwell|journal=Annals of Science|volume=14|issue=4|year=1958|pages=243–255|issn=0003-3790|doi=10.1080/00033795800200147}}</ref> The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, [[Ludwig Boltzmann]] and [[Josiah Gibbs]], which would later have an influence on [[Albert Einstein]]'s mathematical model for [[Brownian movement]].<ref name="Brush1968page15">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>
===Measure theory and probability theory===
At the [[International Congress of Mathematicians]] in [[Paris]] in 1900, [[David Hilbert]] presented a list of [[Hilbert's problems|mathematical problems]], where his sixth problem asked for a mathematical treatment of physics and probability involving [[axiom]]s.<ref name="Bingham2000"/> Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians, [[Henri Lebesgue]] and [[Émile Borel]]. In 1925 another French mathematician [[Paul Lévy (mathematician)|Paul Lévy]] published the first probability book that used ideas from measure theory.<ref name="Bingham2000"/>
In 1920s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as [[Sergei Bernstein]], [[Aleksandr Khinchin]],{{efn|The name Khinchin is also written in (or transliterated into) English as Khintchine.<ref name="Doob1934">{{cite journal|last1=Doob|first1=Joseph|title=Stochastic Processes and Statistics|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=20|issue=6|year=1934|pages=376–379|doi=10.1073/pnas.20.6.376|pmid=16587907|pmc=1076423|bibcode=1934PNAS...20..376D}}</ref>}} and [[Andrei Kolmogorov]].<ref name="Cramer1976"/> Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory.<ref name="KendallBatchelor1990page33">{{cite journal|last1=Kendall|first1=D. G.|last2=Batchelor|first2=G. K.|last3=Bingham|first3=N. H.|last4=Hayman|first4=W. K.|last5=Hyland|first5=J. M. E.|last6=Lorentz|first6=G. G.|last7=Moffatt|first7=H. K.|last8=Parry|first8=W.|last9=Razborov|first9=A. A.|last10=Robinson|first10=C. A.|last11=Whittle|first11=P.|title=Andrei Nikolaevich Kolmogorov (1903–1987)|journal=Bulletin of the London Mathematical Society|volume=22|issue=1|year=1990|page=33|issn=0024-6093|doi=10.1112/blms/22.1.31}}</ref> In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as [[Eugene Slutsky]] and [[Nikolai Smirnov (mathematician)|Nikolai Smirnov]],<ref name="Vere-Jones2006page1">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=1|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref> and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.<ref name="Doob1934"/><ref name="Vere-Jones2006page4">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=4|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>{{efn|Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.<ref name="Snell2005">{{cite journal|last1=Snell|first1=J. Laurie|title=Obituary: Joseph Leonard Doob|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|page=251|issn=0021-9002|doi=10.1239/jap/1110381384|doi-access=free}}</ref> }}
===Birth of modern probability theory===
In 1933 Andrei Kolmogorov published in German, his book on the foundations of probability theory titled ''Grundbegriffe der Wahrscheinlichkeitsrechnung'',{{efn|Later translated into English and published in 1950 as Foundations of the Theory of Probability<ref name="Bingham2000"/>}} where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.<ref name="Bingham2000"/><ref name="Cramer1976"/>
After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as [[Joseph Doob]], [[William Feller]], [[Maurice Fréchet]], [[Paul Lévy (mathematician)|Paul Lévy]], [[Wolfgang Doeblin]], and [[Harald Cramér]].<ref name="Bingham2000"/><ref name="Cramer1976"/>
Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".<ref name="Cramer1976"/> [[World War II]] greatly interrupted the development of probability theory, causing, for example, the migration of Feller from [[Sweden]] to the [[United States|United States of America]]<ref name="Cramer1976"/> and the death of Doeblin, considered now a pioneer in stochastic processes.<ref name="Lindvall1991">{{cite journal|last1=Lindvall|first1=Torgny|title=W. Doeblin, 1915-1940|journal=The Annals of Probability|volume=19|issue=3|year=1991|pages=929–934|issn=0091-1798|doi=10.1214/aop/1176990329|doi-access=free}}</ref>
[[File:Joseph Doob.jpg|thumb|right|Mathematician [[Joseph Doob]] did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales.<ref name="Getoor2009"/><ref name="Snell2005"/> His book ''Stochastic Processes'' is considered highly influential in the field of probability theory.<ref name="Bingham2005"/> ]]
===Stochastic processes after World War II===
After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas.<ref name="Cramer1976"/><ref name="Meyer2009">{{cite journal|last1=Meyer|first1=Paul-André|title=Stochastic Processes from 1950 to the Present|journal=Electronic Journal for History of Probability and Statistics|volume=5|issue=1|year=2009|pages=1–42}}</ref> Starting in the 1940s, [[Kiyosi Itô]] published papers developing the field of [[stochastic calculus]], which involves stochastic [[integrals]] and stochastic [[differential equations]] based on the Wiener or Brownian motion process.<ref name="Ito1998Prize">{{cite journal|title=Kiyosi Itô receives Kyoto Prize|journal=Notices of the AMS|volume=45|issue=8|year=1998|pages=981–982}}</ref>
Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of [[potential theory]], with early ideas by [[Shizuo Kakutani]] and then later work by Joseph Doob.<ref name="Meyer2009"/> Further work, considered pioneering, was done by [[Gilbert Hunt]] in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.<ref name="JarrowProtter2004"/><ref name="Bertoin1998pageVIIIandIX">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii and ix}}</ref><ref name="Steele2012page176">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=176}}</ref>
In 1953 Doob published his book ''Stochastic processes'', which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.<ref name="Meyer2009"/>
<ref name="Bingham2005">{{cite journal|last1=Bingham|first1=N. H.|title=Doob: a half-century on|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|pages=257–266|issn=0021-9002|doi=10.1239/jap/1110381385|doi-access=free}}</ref> Doob also chiefly developed the theory of martingales, with later substantial contributions by [[Paul-André Meyer]]. Earlier work had been carried out by [[Sergei Bernstein]], [[Paul Lévy (mathematician)|Paul Lévy]] and [[Jean Ville]], the latter adopting the term martingale for the stochastic process.<ref name="HallHeyde2014page1">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|pages=1, 2}}</ref><ref name="Dynkin1989">{{cite journal|last1=Dynkin|first1=E. B.|title=Kolmogorov and the Theory of Markov Processes|journal=The Annals of Probability|volume=17|issue=3|year=1989|pages=822–832|issn=0091-1798|doi=10.1214/aop/1176991248|doi-access=free}}</ref> Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.<ref name="Meyer2009"/>
Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations.<ref name="Meyer2009"/> The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s fundamental work was done by Alexander Wentzell in the Soviet Union and [[Monroe D. Donsker]] and [[Srinivasa Varadhan]] in the United States of America,<ref name="Ellis1995page98">{{cite journal|last1=Ellis|first1=Richard S.|title=An overview of the theory of large deviations and applications to statistical mechanics|journal=Scandinavian Actuarial Journal|volume=1995|issue=1|year=1995|page=98|issn=0346-1238|doi=10.1080/03461238.1995.10413952}}</ref> which would later result in Varadhan winning the 2007 Abel Prize.<ref name="RaussenSkau2008">{{cite journal|last1=Raussen|first1=Martin|last2=Skau|first2=Christian|title=Interview with Srinivasa Varadhan|journal=Notices of the AMS|volume=55|issue=2|year=2008|pages=238–246}}</ref> In the 1990s and 2000s the theories of [[Schramm–Loewner evolution]]<ref name="HenkelKarevski2012page113">{{cite book|author1=Malte Henkel|author2=Dragi Karevski|title=Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution|url=https://books.google.com/books?id=fnCQWd0GEZ8C&pg=PA113|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-27933-1|page=113}}</ref> and [[rough paths]]<ref name="FrizVictoir2010page571">{{cite book|author1=Peter K. Friz|author2=Nicolas B. Victoir|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|url=https://books.google.com/books?id=CVgwLatxfGsC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48721-4|page=571}}</ref> were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in [[Fields Medal]]s being awarded to [[Wendelin Werner]]<ref name="Werner2004Fields">{{cite journal|title=2006 Fields Medals Awarded|journal=Notices of the AMS|volume=53|issue=9|year=2015|pages=1041–1044}}</ref> in 2008 and to [[Martin Hairer]] in 2014.<ref name="Hairer2004Fields">{{cite journal|last1=Quastel|first1=Jeremy|title=The Work of the 2014 Fields Medalists|journal=Notices of the AMS|volume=62|issue=11|year=2015|pages=1341–1344}}</ref>
The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.<ref name="BlathImkeller2011"/><ref name="Applebaum2004page1336"/>
Category:Stochastic models
类别: 随机模型
Category:Statistical data types
类别: 统计数据类型
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<small>This page was moved from [[wikipedia:en:Stochastic Process]]. Its edit history can be viewed at [[随机过程/edithistory]]</small></noinclude>
<small>This page was moved from [[wikipedia:en:Stochastic process]]. Its edit history can be viewed at [[随机过程/edithistory]]</small></noinclude>
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