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第34行： − − 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 <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a manifold. <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 function notation. 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. 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，通常使用术语随机（stochastic）过程和随机（random）过程，当索引集是 <math>n</math>-维欧几里得空间 <math>\mathbb{R}^n</math>或者是流形时，则使用随机场。虽然<math>\{X(t)\}</math>被认为是对函数表示法的滥用，但<math>\{X(t)\}</math>还是被简单地称为<math>X</math> 或 <math>X(t)</math>。例如，<math>X(t)</math> 或 <math>X_t</math> 用于指代带有索引 <math>t</math> 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"/> − − 当解释为时间时，如果随机过程的指标集有有限个或可数个元素，例如有限的一组数、一组整数或自然数，那么随机过程被称为“[[离散时间]]”<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> 。如果索引集是实数轴上的某个区间，则时间被称为“'[[连续时间]]”。这两类随机过程分别被称为“离散时间”和“[[连续时间随机过程]]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>。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，特别当索引集不可数时。<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> 如果索引集是整数或整数的子集，则随机过程也可以称为“随机序列”。<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"/> − − 如果状态空间是整数或自然数，则随机过程称为“离散”或“整值随机过程”。如果状态空间是实数轴，则随机过程被称为“实值随机过程”或“具有连续状态空间的过程”。如果状态空间是<math>n</math>-维欧几里德空间，则随机过程称为<math>n</math>-“维向量过程”或<math>n</math>—“向量过程”。<ref name=“florescu214page294”/><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. − − ''' 随机游走Random walks'''是一种随机过程，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，因此招致了批评。还有其他各种类型的''' 随机游走'''，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词源学=== − − 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. − − 一个经典的''' 随机游走'''的例子被称为''' 简单随机游走SimpleRandom walk'''，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Bernoulli process'''，其中每个 伯努利Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游走发生在整数上，它的值随概率<math>p</math>的增加而增加1，或随概率<math>1-p</math>的减少而减少1，所以这种''' 随机游走'''的索引集是自然数，而它的状态空间是整数。如果 <math>p=0.5</math>，这种随机漫步称为''' 对称随机游走Symmetric Random walk'''。 − − 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> − − 在[[英语语言|英语]]中，“随机”一词最初用作形容词，其定义是“与推测有关”，源于一个[[希腊语|希腊语]]词，意思是“瞄准一个标记，猜测”，而[[牛津英语词典]]将1662年作为最早出现的年份。<ref name=“Oxfordstraphic”>{Cite OED | random}</ref>在他关于概率“Ars conquectandi”的著作中，最初于1713年以拉丁文出版，[[Jakob Bernoulli]]使用了“Ars conquectandi istice”这个短语，这本书已经被翻译成“猜想或随机的艺术”。<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>这一短语是[[拉迪斯劳斯·博特基维茨]]]在关于伯努利问题中使用，<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>他在1917年用德语写下了“随机”一词。术语“随机过程”最早出现在1934年[[Joseph Doob]]的一篇论文中。<ref name=“oxfordstractical”/>对于这个术语和一个具体的数学定义，Doob引用了另一篇1934年的论文，其中[[Aleksandr Khinchin]]在德语中使用了术语“随机过程”，<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>尽管德语这个词在早些时候就被使用过，例如，安德烈·科尔莫戈洛夫（Andrei Kolmogorov）在1931年就使用过。<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> − − 根据《牛津英语词典》，英语中“random”（随机）一词的最早出现时间可追溯到16世纪，而早期有记载的用法则始于14世纪，意思是“急躁、速度快、力量大或暴力（骑马、跑步、击打等）”。这个词本身来自法语中间的一个词，意思是“速度，匆忙”，它可能是从法语动词“奔跑”或“飞奔”衍生而来。术语“随机过程”的首次书面出现是在“随机过程”之前出现的，牛津英语词典也将其作为同义词出现，并被[[Francis Edgeworth]]于1888年发表的一篇文章中使用。<ref name=“OxfordRandom”>{Cite OED | random}</ref> − − ===Terminology术语=== − − 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Wiener process'''是一个具有平稳和独立增量的随机过程，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. − − 随机过程的定义是不同的，<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> 但是随机过程传统上被定义为一组随机变量的集合<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/>。术语“随机random过程”和“随机stochastic过程”被视为同义词，可以互换使用，而无需精确指定索引集。<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>。两个“集合”<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/>，或“家庭”使用<ref name="Parzen1999"/><ref name="Ito2006page13">术语“参数集”<ref name="Lamperti1977page1"/> 或“参数空间”<ref name="AdlerTaylor2009page7"/> ，而不是“索引集”。 − − Realizations of Wiener processes (or Brownian motion processes) with drift () and without drift (). − − 带漂移()和无漂移()的 ''' 维纳过程'''(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"/> − − 术语''' “随机函数”Random function'''也用于指随机或随机过程，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Loeve1978”>{cite book | author=M.Loève|title=Probability Theory II | url=https://books.google.com/books？id=1y229ybulic | year=1978 | publisher=Springer Science&Business Media | isbn=978-0-387-90262-3 | page=163}</ref><ref name=“Brémaud2014page133”>{cite book |作者=Pierre Brémaud | title=Fourier Analysis and randocial Processes |网址=https://books.google.com/books？id=dP2JBAAAQBAJ&pg=PA1 | year=2014 | publisher=Springer | isbn=978-3-319-09590-5 | page=133}</ref>尽管有时它只在随机过程取实值时使用。<ref name=“Lamperti1977page1”/><ref name=“Ito2006page13”/>当索引集是数学空间而不是实线时，也使用这个术语，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“gusakkush2010page1”>{harvxt | Gusak | Kukush | Kulik | Mishura | 2010}，p.1</ref>，而术语“随机过程”和“随机过程”通常在指数集被解释为时间时使用，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“GusakKukush2010page1”/><ref name=“Bass2011page1”>{引用图书|作者=Richard F.Bass | title=随机过程| url=https://books.google.com/books？id=Ll0T7PIkcKMC | year=2011 | publisher=Cambridge University Press | isbn=978-1-139-50147-7 | page=1}</ref>和其他术语，例如当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或[[流形]].<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/><ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/>name=“adlertaylor2009第7页”/> − − 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 <math>n</math>-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 <math> \mu</math>, which is a real number, then the resulting stochastic process is said to have drift <math> \mu</math>. − − 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，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<math>n</math>维的''' 欧氏空间Euclidean space'''。如果增量的平均值为零，那么由此产生的维纳Wiener或布朗Brownian运动过程称为具有零漂移。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<math> \mu</math>，即一个实数，那么得到的随机过程就具有<math> \mu</math>漂移。 − − ===Notation符号=== − − 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"/> − − 随机过程可以用<math>\{X（t）\{t}</math>，<math>\{X（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>或简单地称为<math>X</math>或<math>X（t）</math>，尽管<math>X（t）</math>被视为[[符号滥用#函数表示法|函数表示法滥用]]。<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> 例如， <math>X(t)</math> 或 <math>X_t</math>引用具有索引<math>t</math>的随机变量，而不是整个随机过程。<ref name="Lamperti1977page3"/>如果索引集是<math>T=[0,\infty)</math>，然后，我们可以写，例如，<math>(X_t , t \geq 0)</math>来表示随机过程。<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，''' 维纳过程Wiener process'''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走的连续形式。这个过程作为其他随机过程的数学极限出现，例如某些随机游走的重新标度，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示例== − − 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. − − ''' 维纳过程Wiener process'''是马尔可夫过程、 列维Lévy 过程和 高斯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Black–Scholes–Merton model'''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伯努利过程'''=== − − {{Main|Bernoulli process}} − − {{Main |伯努利过程}} − − 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> − − 最简单的随机过程之一是[[伯努利过程]]，<ref name=“Florescu2014page293”/>它是[[独立且相同分布]]（iid）随机变量的序列，其中每个随机变量取1或0，比如概率<math>p</math>的值为1，概率<math>1-p</math>为零。这个过程可以与反复翻动硬币有关，其中获得头部的概率为<math>p</math>，其值为1，而尾部的值为零=https://books.google.com/books？id=z5sebqaaqbaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | page=301}</ref>换句话说，伯努利过程是一个[[独立且同分布随机变量| iid]]伯努利随机变量的序列，<ref name=“Bertsekatsitsiklis2002page273”>{cite book | author1=Dimitri P.Bertsekas | author2=John N.Tsitsiklis | title=概率简介| url=https://books.google.com/books？id=bcHaAAAAMAAJ | year=2002 | publisher=Athena Scientific | isbn=978-1-886529-40-3 | page=273}</ref>每一次抛硬币都是[[Bernoulli审判]]的一个例子。<ref name=“Ibe2013page11”>{cite book | author=Oliver C.Ibe | title=Elements of Random Walk and Diffusion Processes |网址=https://books.google.com/books？id=duqaaaaqbaj&pg=PT10 |年份=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 <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. 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Poisson process'''是一个具有不同形式和定义的随机过程。它可以被定义为一个计数过程，这是一个随机过程，代表到某个时间，点或事件的随机数。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Homogeneous Poisson process'''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 < math > t </math > 的非负可积函数，则得到的过程称为''' 非齐次或非齐次泊松过程Inhomogeneous or nonhomogeneous Poisson process'''，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。 − − − − ===''' 随机游走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 <math>n</math>-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，它可以定义在维欧氏空间或其他数学空间上，在这些空间中，它通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}} − − {{Main |随机游走}} − − [[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> − − [[Random walks]]是随机过程，通常定义为欧几里德空间中[[iid]]随机变量或随机向量的和，因此它们是离散时间变化的过程=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 |网址=https://books.google.com/books？id=UBQdwAZDeOEC | year=2010 | publisher=Cambridge University Press | isbn=978-1-139-48876-1 | page=1}</ref><ref name=“Kallenberg 2002page136”>{cite book |作者=Olav Kallenberg | title=Foundations of Modern Probability |网址=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=概率与随机过程| url=https://books.google.com/books？id=z5sebqaaqbaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | page=383}}</ref><ref name=“Durrett2010page277”>{引用图书|作者=Rick Durrett | title=Probability:理论和示例| url=https://books.google.com/books？id=evbGTPhuvSoC | year=2010 | publisher=Cambridge University Press | isbn=978-1-139-49113-6 | page=277}</ref>但是有些人也使用这个术语来指代连续时间变化的过程，<ref name=“Weiss2006page1”>{cite book | last1=Weiss | first1=George H.| title=Statistical Sciences | chapter=Random Walks | year=2006 | doi=10.1002/0471667196.ess2180.pub2 | page=1 | isbn=978-0471667193}}</ref>尤其是金融中使用的维纳过程，这导致了一些混乱，导致其受到批评。<ref name=“Spanos1999page454”>{cite book | author=Aris Spanos | title=概率论和统计推断：观测数据的计量经济学建模|网址=https://books.google.com/books？id=G0|HxBubGAwC | year=1999 | publisher=Cambridge University Press | isbn=978-0-521-42408-0 | page=454}}</ref>还有其他各种类型的随机游动，它们的状态空间可以是其他数学对象，例如格和群，一般来说，它们都是高度研究的，在不同的学科中有许多应用。<ref name=“Weiss2006page1”/><ref name=“Klebaner2005page81”>{cite book | author=Fima C.Klebaner | title=随机微积分及其应用简介=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> − − ''' 随机游走Random walk'''的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，例如其值随概率<math>p</math>增加1，，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<ref name=“Gut2012page88”>{cite book | author=Allan Gut | title=Probability:a Graduate Course=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=“grimmetttstirzaker2001page71”>{引用图书| author1=Geoffrey Grimmett | author2=David Stirzaker | title=概率和随机过程| url=https://books.google.com/books？id=G3ig-0M4wSIC |年份=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 <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a sample space, <math>\mathcal{F}</math> is a <math>\sigma</math>-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>. − − 随机过程被定义为一系列随机变量的集合，这些随机变量定义在一个普通的概率空间上(Omega，mathcal { f } ，p) </math > ，其中 < math > Omega </math > 是一个样本空间，< math > mathcal { f } </math > 是 < math > sigma </math >-algebra，而 < math > p </math > 是一个机率量测;以及随机变量，用一些集合作为指标，它们都在同一个数学空间中取值，这些值必须是可以测量的。 − − − − <center><math> − − < 中心 > < 数学 > − − ===''' Wiener process维纳过程'''=== − − \{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> − − ''' Wiener process维纳过程'''是一个随机过程，具有平稳的[[独立的增量]]并且基于增量的大小是[[正态分布的].<ref name=“RogersWilliams2000page1”>{cite book | author1=L.C.G.Rogers | author2=David Williams | title=扩散、马尔可夫过程和鞅：第1卷，基金会网址=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=随机微积分及其应用简介|网址=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=56}</ref>维纳过程是以[[Norbert Wiener]]命名的，他证明了它的数学存在性，但是这个过程也被称为布朗运动过程或仅仅是布朗运动，因为它是液体中[[布朗运动]]的模型科学|卷=5 |议题=1 |年份=1968 |页数=1-2 | issn=0003-9519 | doi=10.1007/BF00328110}}</ref><ref name=“applebauma2004page1338”{{{引用杂志| last1=Applebaum | first1=David | title=Lévy过程：从概率到金融和量子群的概率到金融和量子群| journal=Na从概率到金融和量子群| journal=通知AMS | volume=51 | volume=11；年份=2004 |页数=1338}</ref><refname=“Applebaum2004page1338”/><ref name=“GikhmanSkorokhod1969page21”>{cite book | author1=Iosif Ilyich Gikhman | author2=Anatoly Vladimirovich skorokod | title=随机过程理论简介| url=https://books.google.com/books？id=yJyLzG7N7r8C&pg=PR2 |年份=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=21}</ref> − − 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>. 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>. − − 从历史上看，在自然科学的许多问题中，t 中的一个点<math>t\in T</math>代表时间，所以<math>X(t)</math>是一个随机变量，代表时间<math>t</math>观察到的值。一个随机过程也可以写成<math> \{X(t,\omega):t\in T \}</math> 来反映它实际上是一个双变量的函数，<math>t\in T</math> 且<math>\omega\in \Omega</math>。 − − − − [[File:DriftedWienerProcess1D.svg|thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).]] − − [[文件：floadWienerProcess1d.svg|拇指|左|实现维纳Wiener过程（或布朗运动过程），具有漂移（{color |蓝色}且不漂移（{color |红色}红色}）。]] − − 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 <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued functions of <math>t\in T</math> that map from the set <math>T</math> into the space <math>S</math>. 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. 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. 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一个 <math>S^T</math> 值随机变量，其中 <math>S^T</math> 是 <math>t\in T</math> 中所有可能的 <math>S</math>值函数的空间，这些函数从集合 <math>T</math> 映射到空间 <math>S</math> 。随机过程。这个集合通常是实数线的一些子集，比如使集合 <math>T</math> 时间有意义的自然数集或者区间。比如笛卡尔平面 <math>R^2</math> 或 <math>n</math> 维欧氏空间，其中的一个元素 <math>t\in T</math>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> − − ''' Wiener process维纳过程'''在概率论中起着中心作用，通常被认为是最重要和研究的随机过程，并与其他随机过程联系在一起微积分与金融应用|网址=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=29}</ref><ref name=“florescu214page471”>{cite book |作者=Ionut Florescu | title=概率与随机过程|网址=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=随机过程的第一门课程| url=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | pages=21，22}</ref><ref name=“karatzarshreeve2014pageviii”{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| 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=连续鞅和布朗运动| url=https://books.google.com/books？id=oybncaaqbaj | year=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-06400-9 | page=IX | author1 link=Daniel Revuz}</ref>其索引集和状态空间分别是非负数和实数，因此它既有连续索引集又有状态空间=https://books.google.com/books？id=am1IDQAAQBAJ | year=2006 | publisher=World Scientific Publishing Co Inc | isbn=978-981-310-165-4 | page=186}</ref>但是过程可以定义得更广泛，这样它的状态空间可以是维欧几里德空间。<ref name=“klebaner205page81”/><ref name=“KarlinTaylor2012page21”/><ref>{cite book | author1=Donald L。Snyder | author2=Michael I.Miller | title=时空中的随机点过程| 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>如果任何增量的[[平均值]]为零，则所得到的维纳或布朗运动过程称为零漂移。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<math>\mu</math>，即实数，由此产生的随机过程被称为漂移<math>\mu</math><ref name=“Steele2012page118”>{cite book | author=J.Michael Steele | title=随机微积分和金融应用程序| url=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=118}</ref><ref name=“MörtersPeres2010page1”/><ref name=“Karatzasshreeve2014page78”>{cite book | author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机演算| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 |年份=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=78}</ref> − − The mathematical space <math>S</math> of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, <math>n</math>-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 <math>S</math>称为状态空间。这个数学空间可以使用整数、实数线、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> − − [[几乎可以肯定]]，''' Wiener process维纳过程'''的样本路径处处连续，但[[无处可微函数|无处可微]]。它可以看作是简单随机游走的一个连续版本。<ref name=“Applebaum2004page1337”>{cite journal | last1=Applebaum | first1=David | title=Lévy过程：从概率到金融和量子群| journal=AMS的通知| volume=51 | issue=11 | year=2004|page=1337}</ref name=“MörtersPeres2010page1”>{citebook | author1=Peter Mörters | author2=Yuval Peres | title=布朗运动|网址=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>当其他随机过程（如某些随机游动重新缩放）的数学极限时，该过程出现，<ref name=“KaratzasShreve2014page61”>{cite book | author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| 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 |作者=Steven E.Shreve | title=金融随机微积分II：连续时间模型| url=https://books.google.com/books？id=O8kD1NwQBsQC | year=2004 | publisher=Springer Science&Business Media | isbn=978-0-387-40101-0 | page=93}</ref>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=https://books.google.com/books？id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | pages=225260}}</ref><ref name=“karatzarshreve2014page70”{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机演算| 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=布朗运动| url=https://books.google.com/books？id=e-TbA-dSrzYC |年=2010 | publisher=剑桥大学出版社| 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> − − ''' Wiener process维纳过程'''是一些重要的随机过程家族的成员，包括马尔可夫过程，Lévy过程和高斯过程。<ref name=“RogersWilliams2000page1”/><ref name=“Applebaum2004page1337”/>该过程也有许多应用，是随机微积分中使用的主要随机过程。<ref name=“Klebaner2005”>{cite book | author=Fima C.Klebaner | title=随机微积分简介应用程序| url=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7}</ref><ref name=“KaratzasShreve2014page”>{引用图书| author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2}</ref>它在数量金融中起着核心作用，{{本刊|从概率到金融金融和量子集团的过程〈124; journal从概率到金融和量子群的群| journal=journal=金融金融和量子群群| journal=noticof the AMS | volume=51 | issue=11 |年=2004年2004年| page=1341}}</ref><ref name=“KarlinTaylor2012page340 340{引用书〈author1=author1=Samuel Karlin | author1=Samuel Karlin；author2=Howard2=Howarde.Taylor | Howarde.Taylor标题=第一门课程随机过程| url=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=340}</ref>在Black-Scholes-Merton模型中使用它。<ref name=“Klebaner2005page124”>{cite book | author=Fima C.Klebaner | title=Introduction to Rastic Calculation with Applications |网址=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=124}</ref>该过程也被用于不同的领域，包括大多数自然科学以及社会科学的一些分支，作为各种随机现象的数学模型=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 |作者=Ubbo F.Wiersema | title=布朗运动演算| url=https://books.google.com/books？id=0h-n0WWuD9cC |年=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 <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, 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 < math > { x (t，Omega) : t in t } </math > 是随机过程，那么对于任意点 < math > Omega </math > ，映射就是 − − − − <center><math> − + − ===Poisson process泊松过程=== − X(\cdot,\omega): T \rightarrow S,+ − X (cdot，omega) : t, − {{Main|Poisson process}}+ − + − [数学中心] + − 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>. 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>. or path.+ + − 被称为一个样本函数，一个实现，或者，特别是当 < math > t </math > 被解释为时间，一个随机过程 < math > { x (t，omega) : t in t } </math > 的样本路径。这意味着对于 Omega </math > 中的一个固定的 < math > Omega，存在一个示例函数，它将索引集 < math > t </math > 映射到状态空间 < math > s </math > 。或者路径。 − 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"/>+ − {124jms{124tik=随机过程的定义是不同的=https://books.google.com/books？id=eBeNngEACAAJ | year=2003 | publisher=Wiley | isbn=978-0-471-49881-0 | pages=1，2}</ref><ref name=“daleyviere-Jones 2006chap2”>{cite book | author1=D.J.Daley | author2=D.Vere Jones | title=点过程理论导论：第一卷：基本理论与方法|网址=https://books.google.com/books？id=6Sv4BwAAQBAJ | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21564-8 | pages=19–36}</ref>它可以定义为一个计数过程，它是一个随机过程，表示某个时间点或事件的随机数量。在从零到某个给定时间区间内的过程点的数目是一个泊松随机变量，它取决于该时间和某个参数。该过程以自然数为状态空间，非负数为索引集。此过程也称为泊松计数过程，因为它可以被解释为计数过程的一个示例。<ref name=“tijms2303page1”/> − 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"/>+ + − 如果一个泊松过程是用一个正常数定义的，那么这个过程称为齐次泊松过程=https://books.google.com/books？id=PqUmjp7k1kEC | year=2011 | publisher=academical Press | isbn=978-0-12-381416-6 | page=241}</ref>齐次泊松过程是随机过程的一个重要类，如马尔可夫过程和Lévy过程 − 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.+ + + − 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 < math > { x (t) : t in t } </math > 是一个状态空间 < math > s </math > 并且索引设置为 < math > t = [0，infty ] </math > ，那么对于[0，infty ] </math > 中的任意两个非负数 t _ 1和[0，infty ] </math > </math > t _ 2在[0，infty ] </math </math > 这样 < t _ 1 leq t _ 2 </math > ,差值 < math > x _ { t _ 2}-x _ { t _ 1} </math > 是一个 < math > s </math >-valued 随机变量，称为递增量。 + − 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>+ − − 齐次泊松过程可以用不同的方法定义和推广。它的指标集可以定义为实线，这个随机过程也被称为平稳泊松过程=https://books.google.com/books？id=VEiM OtwDHkC | year=1992 | publisher=Clarendon Press | isbn=978-0-19-159124-2 | page=38}</ref><ref name=“daleyviere-Jones 2006page19”>{引用图书| author1=D.J.Daley | author2=D.Vere Jones | title=点过程理论导论：第一卷：基本理论与方法| url=https://books.google.com/books？id=6Sv4BwAAQBAJ | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21564-8 | page=19}</ref>如果泊松过程的参数常数被某个非负可积函数的<math>t</math>代替，则得到的过程称为非齐次或非齐次Poisson过程，其中过程点的平均密度不再是常数=https://books.google.com/books？id=VEiM OtwDHkC | year=1992 | publisher=Clarendon Press | isbn=978-0-19-159124-2 | page=22}</ref>作为排队论中的一个基本过程，泊松过程是数学模型的一个重要过程，在这里，它找到了在特定时间窗口中随机发生的事件模型的应用程序。<ref name=“KarlinTaylor2012page118”>{cite book | author1=Samuel Karlin | author2=Howard E.Taylor | title=A First Course in randocial Processes |网址=https://books.google.com/books？id=dSDxjX9nmmMC |年份=2012 |出版商=学术出版社| isbn=978-0-08-057041-9 |页数=118，119}}</ref><ref name=“Kleinrock1976page61”>{cite book | author=Leonard Kleinrock | title=排队系统：理论|网址=https://archive.org/details/queueingsystems00klei|url access=registration |年份=1976 | publisher=Wiley | isbn=978-0-471-49110-1 |页=[https://archive.org/details/queueingsystems00klei/page/6161]}}</ref> − − For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives − 对于 < math > s ^ t </math > 的可测子集 < math > b </math > ，< math > x </math > 的前映像给出了 + − <center><math>+ − + − 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>+ − 在实线上定义的泊松过程可以解释为一个随机过程，<ref name=“Applebaum2004page1337”/><ref name=“Rosenblatt1962page94”>{cite book | author=Murray Rosenblatt | title=Random Processes |网址=https://archive.org/details/randomprocess00rose\u 0|url access=注册|年份=1962 | publisher=牛津大学出版社|页=[归档文件https://randomesu/0094/94]}}</ref>等随机变量对象。<ref name=“Haenggi2013page10and18”>{cite book | author=Martin Haenggi | title=无线网络随机几何| 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=随机几何及其应用| url=https://books.google.com/books？id=825NfM6Nc EC | year=2013 | publisher=John Wiley&Sons | isbn=978-1-118-65825-3 | pages=41108}</ref>但是它可以定义在<math>n</math>维欧几里德空间或其他数学空间上，<ref name=“kingmann1992page11”>{cite book | author=J.F.C.Kingman | title=Poisson Processess | url=https://books.google.com/books？id=VEiM OtwDHkC | year=1992 | publisher=Clarendon Press | isbn=978-0-19-159124-2 | page=11}</ref>其中它通常被解释为随机集或随机计数度量，而不是随机过程。<ref name=“Haenggi2013page10and18”/><ref name=“ChiuStoyan2013page41and108”/>在此设置中，是泊松过程，也称为泊松点过程，是概率论中最重要的研究对象之一，无论是应用还是理论原因=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>但有人指出，Poisson过程并没有得到应有的重视，部分原因是它经常被认为只是在实线上，而不是在其他数学空间中。<ref name=“Streit2010page1”/><refname=“kingmann1992pagev”>{cite book | author=J.F.C.Kingman | title=Poisson进程| 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]+ − + − so the law of a <math>X</math> can be written as: − 所以 <math>X</math>的定律可以写成:+ + − ===Stochastic process随机过程=== − 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"/>+ − 随机过程被定义为在一个公共[[概率空间]]<math>(\Omega, \mathcal{F}, P)</math>上定义的随机变量集合，其中<math>\Omega</math> 是[[样本空间]]，<math>\mathcal{F}</math>是一个<math>\sigma</math>-[[sigma代数|代数]]，<math>P</math>是[[概率测度]]；而随机变量，由某个集合<math>T</math>索引，所有值都取同一个数学空间<math>S</math>，对于某些<math>\sigma</math>-代数<math>\sigma</math><ref name=“Lamperti1977page1”/> − For a stochastic process <math>X</math> with law <math>\mu</math>, its finite-dimensional distributions are defined as:+ + − 对于一个随机过程，其有限维分布被定义为:+ − 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> − − \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} , 第312行：
第121行： − </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>. − − 其中 <math>n\geq 1</math> 是一个计数数字，每个集 <math>t_i</math> 是指数集 <math>T</math> 的非空有限子集，因此每个 <math>t_i\subset T</math> ，这意味着 <math>t_1,\dots,t_n</math>是指数集 <math>T</math> 的任何有限子集。 − − − − 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> − − 历史上，在许多自然科学问题中，一个点<math>t\in T</math> 具有时间的意义，因此，<math>X(t)</math>表示是一个在时间<math>t</math>的随机变量。<ref name=“Borovkov2013page528”>{cite book | 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>随机过程也可以写成<math>\{X（t，omega）：t\ in t\}</math>来反映它实际上是两个变量的函数，<math>t\in t</math>和<math>\omega\in\omega</math><ref name=“Lamperti1977page1”/><ref name=“LindgrenRootzen2013page11”>{cite book | author1=Georg Lindgren | author2=Holger Rootzen | author3=Maria Sandsten | title=科学家和工程师的平稳随机过程| url=https://books.google.com/books？id=fyjfaqbaj&pg=PR1 | year=2013 | publisher=CRC出版社| isbn=978-1-4665-8618-5 | pages=11}</ref> − − 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: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time. − − 对于任何可测量的子集 < math > c </math > n </math >-fold 笛卡尔幂 < math > s ^ n = s 乘以 s </math > ，一个随机过程 < math > x </math > 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，其中指数集不被解释为时间。 − − − − 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"/> − 还有其他方法可以考虑随机过程，上面的定义被认为是传统的=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=S|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>例如，一个随机过程可以解释或定义为一个<math>S^T</math>值的随机变量，其中<math>S^T</math>是所有可能的<math>S</math>-值[[函数（数学）|函数]]的空间T</math>从集合<math>T</math>到空间<math>S</math><ref name=“Kallenbergg2002page24”/><ref name=“RogersWilliams2000page121”/> − 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. 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 < math > t </math > 可以被解释为时间时，如果一个随机过程的有限维分布在时间平移下是不变的，那么它就是静止的。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。 − + − − 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 <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>. − − 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。更正式的说法是，如果一个随机过程有一个总序的索引集，那么在一个总序的索引集上，对一个概率空间的索引集进行一次过滤，这样的索引集就是一个总序的索引集，这样的索引集的总序就是数学的。 − − − − === State space 状态空间=== − − 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> − − 随机过程的[[数学空间]]<math>S</math>称为其“状态空间”。这个数学空间可以用[[integer]]s、[[real line]]s、<math>n</math>-dimensional[[Euclidean space]]s、复杂平面或更抽象的数学空间来定义。状态空间是用反映随机过程可以采用的不同值的元素来定义的进程| url=https://books.google.com/books？id=Z5xEBQAAQBAJ&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | pages=294295}</ref><ref name=“Brémaud2014page120”>{cite book |作者=Pierre Brémaud | title=Fourier Analysis and random Processes |网址=https://books.google.com/books？id=dP2JBAAAQBAJ&pg=PA1 |年=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 <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 − − 一个随机过程的修正是另一个随机过程，它与原始随机过程密切相关。更确切地说，一个随机过程<math>X</math>具有相同的索引集<math>T</math>、集空间<math>和概率空间<math>（\Omega，{\cal F}，P）</math>作为另一个随机过程<math>Y</math>的随机过程被称为<math>Y</math>的修改，如果T</math>中的所有<math>T\ − + − + − < 中心 > < 数学 >+ − − − − P(X_t=Y_t)=1 , − P (x _ t = y _ t) = 1, − − </math></center> − − [数学中心] − − 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。 − − − 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 <math>n</math>-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces. − 代替修正，术语版本也被使用，然而当两个随机过程具有相同的有限维分布一些作者使用术语版本，但他们可能被定义在不同的概率空间，因此在后一种意义上，两个相互修改的过程也是彼此的版本，但不是相反。该定理还可以推广到随机域，使指标集是<math>n</math>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。 − − 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，因此指数集是 < math > n </math > 维欧氏空间，以及以度量空间为状态空间的随机过程。 第401行：
第155行： − + − − 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 − − 两个随机过程 < math > x </math > 和 < math > y </math > 定义在同一个概率空间 < math > (Omega，cal { f } ，p) </math > 具有相同的指数集 < math > t </math > 和集合空间 < math > s </math > 如果下列情况，这两个随机过程是无法区分的 第411行：
第161行： − <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。+ − + 第440行：
第171行： − <中心><数学> − − 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。 − − 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>. − − 更确切地说，一个带有随机过程的实值连续时间子集 x < math > (Omega，{ cal f } ，p) </math > 是可分的，如果它的指数集 < math > t </math > 有一个稠密的可数子集 < math > u t </math > 并且存在一个集合 < math > Omega 0子集 ω </math > 概率为0,所以 < math > p (Omega _ 0) = 0 </math > ，对于每个开集 < math > g 子集 t </math > 和每个闭集 < math > f 子集文本样式 r = (- infty，infty) </math > ,两个事件 < math > > { x _ t in f text { for all } t in g cap u } </math > 和 < math > { x _ t in f text { for all } t in g } </math > 在 < math > 的子集上最多不同。 − − 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 <math>n</math>-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，索引集和状态空间可以是 < math > n </math >-dimensional Euclidean 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。 − 第468行：
第185行： − − 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. Formally: − − 两个随机过程 < math > 左{ x _ t 右} </math > 和 < math > 左{ y _ t 右} </math > 如果它们的互协方差 < math > 操作者名{ k }{ mathbf { x }{ y }}(t _ 1，t _ 2) = 操作者名{ e }左(x (t _ 1)-mu _ x (t _ 1)右)左(y (t _ 2)-y _ 2)右] </math > 始终为零，则称为不相关过程。形式上: 第477行：
第190行： − <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>. − − 左{ x _ t 右} ，左{ y _ t 右}文本{ uncorrelated }四匹配四匹配操作器名{ k }{ mathbf { x }{ y }(t _ 1，t _ 2) = 0四匹配所有 t _ 1，t _ 2 </math > 。 第488行：
第198行： − − If two stochastic processes <math>X</math> and <math>Y</math> 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, <math>D[0,1]</math> denotes the space of càdlàg functions defined on the unit interval <math>[0,1]</math>. − − 如果两个随机过程<math>X</math>和<math>Y</math>是独立的，那么它们也是不相关的。这类函数被称为cádla g或cadlag函数，基于法语表达式continue a droite，limiteégauche的首字母缩略词，因为这些函数是右连左限的。由Anatoliy Skorokod引入的一个Skorokod函数空间，该函数空间的表示法还可以包括定义所有cédlág函数的区间，因此，例如，<math>D[0,1]</math>表示定义在单位区间<math>[0,1]</math>上的cádlág函数空间。 − − 如果两个随机过程 < math > x </math > 和 < math > y </math > 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函数的区间，因此，例如，< math > d [0,1] </math > 表示在单位区间 < math > [0,1] </math > 上定义的函数的空间。 − − − − 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。 第509行：
第207行： − + − − 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}} − {{Main |有限维分布}} 第528行：
第219行： − 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>+ − + − 其中<math>n\geq 1</math>是一个计数数，而每个集合<math>t_i</math>>是索引集<math>T</math>的一个非空有限子集，因此每个<math>t_i\subset T</math>，这意味着<math>t_1,\dots,t_n</math>是索引集<math>T</math>的任何子集的有限集合<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>id=W0ydAgAAQBAJ&pg=PA356 | year=2000 | publisher=剑桥大学出版社| 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。 第541行：
第227行： − − 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。 − − 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Markov chain'''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Markov chain'''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)。 第559行：
第237行： − 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Markov processes'''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Markov chain MonteCarlo'''，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。 − − 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. − − 马尔可夫性的概念最初是用于连续和离散时间的随机过程，但是这个性质已经适用于其他指标集，如 <math>n</math>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}} − {{Main |稳定过程}} 第579行：
第247行： − − 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Martingale'''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Martingale'''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Martingale'''通常被定义为实值的，但是它们也可以取复值，甚至是更一般的值。 第589行：
第253行： − − − 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Symmetric random walk'''和 维纳Wiener 过程(带零漂移)都是''' 鞅Martingale'''的例子。在这方面，离散''' 鞅Martingale'''推广了独立随机变量部分和的概念。 − 第600行：
第258行： − − 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Martingale'''，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Martingale'''的情形。 − 第611行：
第264行： − 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.+ − − 数学上的鞅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。 − − − − 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 <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 − − ''' 鞅Martingales'''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，如果对于 <math>n</math> 非负数，<math>0\leq t_1\leq \dots \leq t_n</math> ，相应的 <math>n-1</math> 递增值是一个列维 Lévy 过程 第625行：
第270行： − <center><math> − − − − − 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修正==== − − </math></center> + 第643行：
第277行： − − 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。 第656行：
第286行： − 注意。两个相互修正的随机过程具有相同的有限维法则<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>它们被称为“随机等价”或“等价物”<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, <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. 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 <math>n</math>-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. − − 点过程是一个点的集合，这些点随机地分布在一些数学空间上，比如实数直线、 n 维欧氏空间或者更多的抽象空间。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。}它被定义在其上，例如实线或者<math>n</math> 维欧氏空间。在点过程理论中研究了更新和计数过程等其他随机过程。 − 第674行：
第298行： − 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。+ 第738行：
第361行： − 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。+ 第754行：
第377行： − 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。+ 第1,063行：
第686行： − 1953年，杜布出版了《随机过程》一书，这本书对随机过程理论产生了重大影响，并强调了测度理论在概率论中的重要性。<ref name=“Meyer2009”/>+
无编辑摘要
===分类===
===分类===
随机过程可以用不同的方法进行分类，例如，根据其状态空间、索引集或随机变量之间的相关性。一种常见的分类方法是通过索引集和状态空间的[[基数]]进行分类。<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>
随机过程可以用不同的方法进行分类，例如，根据其状态空间、索引集或随机变量之间的相关性。一种常见的分类方法是通过索引集和状态空间的[[基数]]进行分类。<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>
< 中心 > < 数学 >
当解释为时间时，如果随机过程的指标集有有限个或可数个元素，例如有限的一组数、一组整数或自然数，那么随机过程被称为离散时间<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> 。如果索引集是实数轴上的某个区间，则时间被称为连续时间。这两类随机过程分别被称为'''离散时间随机过程'''和'''连续时间随机过程'''<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>。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，特别当索引集不可数时。<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> 如果索引集是整数或整数的子集，则随机过程也可以称为'''随机序列 random sequence'''。<ref name=“Borovkov2013page527”/>
如果状态空间是整数或自然数，则随机过程称为“离散随机过程”或“整值随机过程”。如果状态空间是实数轴，则随机过程被称为“实值随机过程”或“具有连续状态空间的过程”。如果状态空间是<math>n</math>-维欧几里德空间，则随机过程称为<math>n</math>-“维向量过程”或<math>n</math>—“向量过程”。<ref name=“florescu214page294”/><ref name=“KarlinTaylor2012page26”/>
===词源学===
</math></center>
在英语中，“随机”一词最初用作形容词，其定义是“与推测有关”，源于一个希腊语词，意思是“瞄准一个标记，猜测”，而牛津英语词典将1662年作为最早出现的年份。<ref name=“Oxfordstraphic”>{Cite OED | random}</ref>在他关于概率“Ars conquectandi”的著作中，最初于1713年以拉丁文出版，[[雅各布·伯努利 Jakob Bernoulli]]使用了“Ars conquectandi istice”这个短语，这本书已经被翻译成“猜想或随机的艺术”。<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>这一短语是[adislaus Bortkiewicz在关于伯努利问题中使用，<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>他在1917年用德语写下了“随机”一词。术语“随机过程”最早出现在1934年Joseph Doob的一篇论文中。<ref name=“oxfordstractical”/>对于这个术语和一个具体的数学定义，Doob引用了另一篇1934年的论文，其中Aleksandr Khinchin在德语中使用了术语“随机过程”，<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>尽管德语这个词在早些时候就被使用过，例如，Andrei Kolmogorov在1931年就使用过。<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>
根据《牛津英语词典》，英语中“random”（随机）一词的最早出现时间可追溯到16世纪，而早期有记载的用法则始于14世纪，意思是“急躁、速度快、力量大或暴力（骑马、跑步、击打等）”。这个词本身来自法语中间的一个词，意思是“速度，匆忙”，它可能是从法语动词“奔跑”或“飞奔”衍生而来。术语“随机过程”的首次书面出现是在“随机过程”之前出现的，牛津英语词典也将其作为同义词出现，并被Francis Edgeworth于1888年发表的一篇文章中使用。<ref name=“OxfordRandom”>{Cite OED | random}</ref>
===术语===
随机过程的定义是不同的，<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> 但是随机过程传统上被定义为一组随机变量的集合<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/>。术语“随机（random）过程”和“随机（stochastic）过程”被视为同义词，可以互换使用，而无需精确指定索引集。<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>。两个“集合”<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/>，或“家庭”使用<ref name="Parzen1999"/><ref name="Ito2006page13">术语“参数集”<ref name="Lamperti1977page1"/> 或“参数空间”<ref name="AdlerTaylor2009page7"/> ，而不是“索引集”。
术语“随机函数”也用于指随机或随机过程，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Loeve1978”>{cite book | author=M.Loève|title=Probability Theory II | url=https://books.google.com/books？id=1y229ybulic | year=1978 | publisher=Springer Science&Business Media | isbn=978-0-387-90262-3 | page=163}</ref><ref name=“Brémaud2014page133”>{cite book |作者=Pierre Brémaud | title=Fourier Analysis and randocial Processes |网址=https://books.google.com/books？id=dP2JBAAAQBAJ&pg=PA1 | year=2014 | publisher=Springer | isbn=978-3-319-09590-5 | page=133}</ref>尽管有时它只在随机过程取实值时使用。<ref name=“Lamperti1977page1”/><ref name=“Ito2006page13”/>当索引集是数学空间而不是实线时，也使用这个术语，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“gusakkush2010page1”>{harvxt | Gusak | Kukush | Kulik | Mishura | 2010}，p.1</ref>，而术语“随机过程”和“随机过程”通常在指数集被解释为时间时使用，<ref name=“GikhmanSkorokhod1969page1”/><ref name=“GusakKukush2010page1”/><ref name=“Bass2011page1”>{引用图书|作者=Richard F.Bass | title=随机过程| url=https://books.google.com/books？id=Ll0T7PIkcKMC | year=2011 | publisher=Cambridge University Press | isbn=978-1-139-50147-7 | page=1}</ref>和其他术语，例如当索引集是<math>n</math>-维欧几里德空间<math>\mathbb{R}^n</math>或流形.<ref name=“GikhmanSkorokhod1969page1”/><ref name=“Lamperti1977page1”/><ref name=“GikhmanSkorokhod1969page1”/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>
===符号===
随机过程可以用<math>\{X（t）\{t}</math>，<math>\{X（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>或简单地称为<math>X</math>或<math>X（t）</math>，尽管<math>X（t）</math>被视为函数表示法滥用。<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> 例如， <math>X(t)</math> 或 <math>X_t</math>引用具有索引<math>t</math>的随机变量，而不是整个随机过程。<ref name="Lamperti1977page3"/>如果索引集是<math>T=[0,\infty)</math>，然后，我们可以写，例如，<math>(X_t , t \geq 0)</math>来表示随机过程。<ref name=“ChaumontYor2012”/>
==示例==
===伯努利过程 Bernoulli process===
最简单的随机过程之一是伯努利过程，<ref name=“Florescu2014page293”/>它是独立且相同分布随机变量的序列，其中每个随机变量取1或0，比如概率<math>p</math>的值为1，概率<math>1-p</math>为零。这个过程可以与反复翻动硬币有关，其中获得头部的概率为<math>p</math>，其值为1，而尾部的值为零=https://books.google.com/books？id=z5sebqaaqbaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | page=301}</ref>换句话说，伯努利过程是一个独立且同分布随机变量伯努利随机变量的序列，<ref name=“Bertsekatsitsiklis2002page273”>{cite book | author1=Dimitri P.Bertsekas | author2=John N.Tsitsiklis | title=概率简介| url=https://books.google.com/books？id=bcHaAAAAMAAJ | year=2002 | publisher=Athena Scientific | isbn=978-1-886529-40-3 | page=273}</ref>每一次抛硬币都是[[伯努利试验]]的一个例子。<ref name=“Ibe2013page11”>{cite book | author=Oliver C.Ibe | title=Elements of Random Walk and Diffusion Processes |year=2013 | publisher=John Wiley&Sons | isbn=978-1-118-61793-9 | page=11}</ref>
===随机游走 Random walk===
随机游走是随机过程，通常定义为欧几里德空间中独立同分布的随机变量随机变量或随机向量的和，因此它们是离散时间变化的过程。<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>但是有些人也使用这个术语来指代连续时间变化的过程，<ref name=“Weiss2006page1”>{cite book | last1=Weiss | first1=George H.| title=Statistical Sciences | chapter=Random Walks | year=2006 | doi=10.1002/0471667196.ess2180.pub2 | page=1 | isbn=978-0471667193}}</ref>尤其是金融中使用的维纳过程，这导致了一些混乱，导致其受到批评。<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>还有其他各种类型的随机游动，它们的状态空间可以是其他数学对象，例如格和群，一般来说，它们都是高度研究的，在不同的学科中有许多应用。<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>
随机游走的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，例如其值随概率<math>p</math>增加1，，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<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>
===维纳过程 Wiener process===
< 中心 > < 数学 >
维纳过程是一个随机过程，具有平稳的独立的增量并且基于增量的大小是正态分布的.<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>维纳过程是以Norbert Wiener命名的，他证明了它的数学存在性，但是这个过程也被称为布朗运动过程或仅仅是布朗运动，因为它是液体中[[布朗运动]]的模型。<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>
[[File:DriftedWienerProcess1D.svg|thumb|left|实现维纳Wiener过程（或布朗运动过程），具有漂移（<font color=blue>蓝色</font>}且不漂移（{<font color=red>红色</font>）。]]
''' Wiener process维纳过程'''在概率论中起着中心作用，通常被认为是最重要和研究的随机过程，并与其他随机过程联系在一起<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>其索引集和状态空间分别是非负数和实数，因此它既有连续索引集又有状态空间<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>但是过程可以定义得更广泛，这样它的状态空间可以是维欧几里德空间。<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>如果任何增量的[[平均值]]为零，则所得到的维纳或布朗运动过程称为零漂移。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<math>\mu</math>，即实数，由此产生的随机过程被称为'''漂移'''。<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>
几乎可以肯定，维纳过程的样本路径处处连续，但无处可微。它可以看作是简单随机游走的一个连续版本。<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>当其他随机过程（如某些随机游动重新缩放）的数学极限时，该过程出现，<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>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'''。<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>
维纳过程是一些重要的随机过程家族的成员，包括马尔可夫过程，Lévy过程和高斯过程。<ref name="RogersWilliams2000page1"/><ref name="Applebaum2004page1337"/>该过程也有许多应用，是随机微积分中使用的主要随机过程。<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>它在数量金融中起着核心作用，<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>在Black-Scholes-Merton模型中使用它。<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>该过程也被用于不同的领域，包括大多数自然科学以及社会科学的一些分支，作为各种随机现象的数学模型。<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>
===泊松过程 Poisson process===
==Definitions定义==
泊松过程是一个随机过程，有不同的形式和定义。<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>它可以定义为一个计数过程，它是一个随机过程，表示某个时间点或事件的随机数量。在从零到某个给定时间区间内的过程点的数目是一个泊松随机变量，它取决于该时间和某个参数。该过程以自然数为状态空间，非负数为索引集。此过程也称为泊松计数过程，因为它可以被解释为计数过程的一个示例。<ref name=“tijms2303page1”/>
如果一个泊松过程是用一个正常数定义的，那么这个过程称为齐次泊松过程。<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>齐次泊松过程是随机过程的一个重要类，如马尔可夫过程和Lévy过程。<ref name="Applebaum2004page1337"/>
齐次泊松过程可以用不同的方法定义和推广。它的指标集可以定义为实线，这个随机过程也被称为平稳泊松过程<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>如果泊松过程的参数常数被某个非负可积函数的<math>t</math>代替，则得到的过程称为非齐次或非齐次Poisson过程，其中过程点的平均密度不再是常数。<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>作为排队论中的一个基本过程，泊松过程是数学模型的一个重要过程，在这里，它找到了在特定时间窗口中随机发生的事件模型的应用程序。<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>
在实线上定义的泊松过程可以解释为一个随机过程，<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>等随机变量对象。<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>但是它可以定义在<math>n</math>维欧几里德空间或其他数学空间上，<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>其中它通常被解释为随机集或随机计数度量，而不是随机过程。<ref name="Haenggi2013page10and18"/><ref name="ChiuStoyan2013page41and108"/>在此设置中，是泊松过程，也称为泊松点过程，是概率论中最重要的研究对象之一，无论是应用还是理论原因。<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>但有人指出，Poisson过程并没有得到应有的重视，部分原因是它经常被认为只是在实线上，而不是在其他数学空间中。<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>
==定义==
===随机过程 Stochastic process===
随机过程被定义为在一个公共概率空间<math>(\Omega, \mathcal{F}, P)</math>上定义的随机变量集合，其中<math>\Omega</math> 是[[样本空间]]，<math>\mathcal{F}</math>是一个<math>\sigma</math>-代数，<math>P</math>是概率测度；而随机变量，由某个集合<math>T</math>索引，所有值都取同一个数学空间<math>S</math>，对于某些<math>\sigma</math>-代数<math>\sigma</math><ref name=“Lamperti1977page1”/>
换言之，对于给定的概率空间<math>（\Omega，\mathcal{F}，P）</math>和可测空间<math>（S，Sigma）</math>，随机过程是一个值为<math>S</math>的随机变量的集合，可以写成：<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>
换言之，对于给定的概率空间<math>（\Omega，\mathcal{F}，P）</math>和可测空间<math>（S，Sigma）</math>，随机过程是一个值为<math>S</math>的随机变量的集合，可以写成：<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>
<center><math>
\{X(t):t\in T \}.
\{X(t):t\in T \}.
历史上，在许多自然科学问题中，一个点<math>t\in T</math> 具有时间的意义，因此，<math>X(t)</math>表示是一个在时间<math>t</math>的随机变量。<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>随机过程也可以写成<math>\{X（t，omega）：t\ in t\}</math>来反映它实际上是两个变量的函数，<math>t\in t</math>和<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>
还有其他方法可以考虑随机过程，上面的定义被认为是传统的。<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>例如，一个随机过程可以解释或定义为一个<math>S^T</math>值的随机变量，其中<math>S^T</math>是所有可能的<math>S</math>-值函数的空间T</math>从集合<math>T</math>到空间<math>S</math>。<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/>
===Index set索引集===
===索引集 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>
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>
集合<math>T</math>称为“索引集”<ref name=“Parzen1999”/><ref name=“Florescu2014page294”/>或“‘参数集’”<ref name=“Lamperti1977page1”/><ref name=“Skorokhod2005page93”>{cite book | author=Valeriy skorokord | title=概率论的基本原理和应用=https://books.google.com/books？随机过程的id=dQkYMjRK3fYC | year=2005 | publisher=Springer Science&Business Media | isbn=978-3-540-26312-8 | pages=93，94}}</ref>。通常，这个集合是[[实线]]的一个子集，例如[[自然数]]或一个区间，使集合<math>T</math>能够解释时间。<ref name=“doob1953stochasticP46to47”/>除了这些集合，索引集<math>T</math>可以是其他线性有序集或更一般的数学集，<ref name=“doob1953stochasticP46to47”/><ref name=“Billingsley2008page482”>{cite book | author=Patrick Billingsley | title=Probability and Measure |网址=https://books.google.com/books？id=qyxqoxyeic | year=2008 | publisher=Wiley India Pvt.Limited | isbn=978-81-265-1771-8 | page=482}}</ref>例如笛卡尔平面<math>R^2</math>或<math>n</math>维欧几里得空间，其中t中的元素可以表示空间中的一个点=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=27}</ref>{cite book | author1=Donald L.Snyder | author2=Michael I.Miller | title=时空中的随机点过程| 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>但一般情况下，当索引集有序时，随机过程可以得到更多的结果和定理。<ref name=“skorokod2005page104”>{cite book | author=Valeriy skorokorokod | title=概率的基本原理和应用理论|网址=https://books.google.com/books？id=dQkYMjRK3fYC |年=2005 | publisher=Springer Science&Business Media | isbn=978-3-540-26312-8 | page=104}</ref>
集合<math>T</math>称为“索引集”<ref name=“Parzen1999”/><ref name=“Florescu2014page294”/>或“‘参数集’”<ref name=“Lamperti1977page1”/><ref name=“Skorokhod2005page93”>{cite book | author=Valeriy skorokord | title=概率论的基本原理和应用=https://books.google.com/books？随机过程的id=dQkYMjRK3fYC | year=2005 | publisher=Springer Science&Business Media | isbn=978-3-540-26312-8 | pages=93，94}}</ref>。通常，这个集合是[[实线]]的一个子集，例如[[自然数]]或一个区间，使集合<math>T</math>能够解释时间。<ref name=“doob1953stochasticP46to47”/>除了这些集合，索引集<math>T</math>可以是其他线性有序集或更一般的数学集，<ref name=“doob1953stochasticP46to47”/><ref name=“Billingsley2008page482”>{cite book | author=Patrick Billingsley | title=Probability and Measure |网址=https://books.google.com/books？id=qyxqoxyeic | year=2008 | publisher=Wiley India Pvt.Limited | isbn=978-81-265-1771-8 | page=482}}</ref>例如笛卡尔平面<math>R^2</math>或<math>n</math>维欧几里得空间，其中t中的元素可以表示空间中的一个点=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=27}</ref>{cite book | author1=Donald L.Snyder | author2=Michael I.Miller | title=时空中的随机点过程| 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>但一般情况下，当索引集有序时，随机过程可以得到更多的结果和定理。<ref name=“skorokod2005page104”>{cite book | author=Valeriy skorokorokod | title=概率的基本原理和应用理论|网址=https://books.google.com/books？id=dQkYMjRK3fYC |年=2005 | publisher=Springer Science&Business Media | isbn=978-3-540-26312-8 | page=104}</ref>
===状态空间 State space ===
<center><math>
随机过程的数学空间<math>S</math>称为其“状态空间”。这个数学空间可以用[[integer]]s、[[real line]]s、<math>n</math>-dimensional[[Euclidean space]]s、复杂平面或更抽象的数学空间来定义。状态空间是用反映随机过程可以采用的不同值的元素来定义的进程。<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>
===样本函数 Sample function===
===Sample function样本函数===
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]]
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]]
“样本函数”是随机过程的单个[[结果（概率）|结果]]，因此，它是由随机过程中每个随机变量的一个可能值构成的=https://books.google.com/books？id=z5sebqaaqbaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | page=296}</ref>更准确地说，如果<math>\{X（t，omega）：t\in t\}</math>是一个随机过程，那么对于任何点<math>\omega\in\omega</math>，则[[Map（mathematics）| mapping]]
“样本函数”是随机过程的单个[[结果（概率）|结果]]，因此，它是由随机过程中每个随机变量的一个可能值构成的=https://books.google.com/books？id=z5sebqaaqbaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | page=296}</ref>更准确地说，如果<math>\{X（t，omega）：t\in t\}</math>是一个随机过程，那么对于任何点<math>\omega\in\omega</math>，则[[Map（mathematics）| mapping]]
<center><math>
<center><math>
X(\cdot,\omega): T \rightarrow S,
X(\cdot,\omega): T \rightarrow S,
</math></center>
</math></center>
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>
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增量===
===增量 Increment===
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"/>
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"/>
随机过程的“增量”是同一随机过程的两个随机变量之间的差值。对于一个指数集可以解释为时间的随机过程，增量是随机过程在某个时间段内的变化量。例如，如果t\}</math>中的<math>\{X（t）：t\in t\}</math>是状态空间<math>S</math>且索引集<math>t=[0，infty）</math>中的任意两个非负数<math>t\u 1\和[0，\infty）</math>中的<math>t_2\使得<math>tˉ，差异<math>X{tu 2}-X{t_1}</math>是一个称为增量的<math>S</math>值随机变量。<ref name=“KarlinTaylor2012page27”/><ref name=“Applebaum2004page1337”/>当对增量感兴趣时，通常状态空间<math>S</math>是实线或自然数，但它可以是<math>n</math>维欧几里德空间或更抽象的空间，如[[Banach space]]s.<ref name=“Applebaum2004page1337”/>
随机过程的“增量”是同一随机过程的两个随机变量之间的差值。对于一个指数集可以解释为时间的随机过程，增量是随机过程在某个时间段内的变化量。例如，如果t\}</math>中的<math>\{X（t）：t\in t\}</math>是状态空间<math>S</math>且索引集<math>t=[0，infty）</math>中的任意两个非负数<math>t\u 1\和[0，\infty）</math>中的<math>t_2\使得<math>tˉ，差异<math>X{tu 2}-X{t_1}</math>是一个称为增量的<math>S</math>值随机变量。<ref name=“KarlinTaylor2012page27”/><ref name=“Applebaum2004page1337”/>当对增量感兴趣时，通常状态空间<math>S</math>是实线或自然数，但它可以是<math>n</math>维欧几里德空间或更抽象的空间，如[[Banach space]]s.<ref name=“Applebaum2004page1337”/>
===进一步定义===
====Law定律====
====定律====
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]]:
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>
<center><math>
\mu=P\circ X^{-1},
\mu=P\circ X^{-1},
</math></center>
</math></center>
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>
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>
其中<math>P</math>是一个概率度量，符号<math>\circ</math>表示函数组合，<math>X^{-1}</math>是可测量函数的前映像，或者等价地，<math>S^T</math>值随机变量<math>X</math>，其中<math>S^T</math>是T</math>中所有可能的<math>S</math>值函数的空间，所以随机过程的规律就是一个概率测度=https://books.google.com/books？id=VQrpBwAAQBAJ |年=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4612-0387-2 |页=40–41}</ref>
其中<math>P</math>是一个概率度量，符号<math>\circ</math>表示函数组合，<math>X^{-1}</math>是可测量函数的前映像，或者等价地，<math>S^T</math>值随机变量<math>X</math>，其中<math>S^T</math>是T</math>中所有可能的<math>S</math>值函数的空间，所以随机过程的规律就是一个概率测度=https://books.google.com/books？id=VQrpBwAAQBAJ |年=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4612-0387-2 |页=40–41}</ref>
<center><math>
<center><math>
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
</math></center>
</math></center>
so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>
so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>
\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).
\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).
</math></center>
</math></center>
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>
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有限维概率分布====
====有限维概率分布 Finite-dimensional probability distributions====
For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distributions''' are defined as:
For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distributions''' are defined as:
</math></center>
</math></center>
这项措施<math>\mu_{t_1,..,t_n}</math>是随机向量的联合分布 <math>(X({t_1}),\dots, X({t_n}))</math>；它可以被视为法律的“投影”<math>\mu</math>到一个有限子集<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>
对于<math>n</math>级[[笛卡尔幂]]<math>S^n=S\times\dots \times S</math>的任何可测子集<math>C</math>，<math>X</math>的有限维分布可以写成：<ref name=“Lamperti1977page1”/>
对于<math>n</math>级[[笛卡尔幂]]<math>S^n=S\times\dots \times S</math>的任何可测子集<math>C</math>，<math>X</math>的有限维分布可以写成：<ref name=“Lamperti1977page1”/>
<center><math>
<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).
\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).
</math></center>
</math></center>
随机过程的有限维分布满足两个称为一致性条件的数学条件。<ref name=“Rosenthal2006page177”/>
随机过程的有限维分布满足两个称为一致性条件的数学条件。<ref name=“Rosenthal2006page177”/>
====Stationarity稳定性====
====Stationarity稳定性====
'''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
'''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
X_{t_1}, \dots X_{t_n},
X_{t_1}, \dots X_{t_n},
</math></center>
</math></center>
它们都有相同的[[概率分布]]。平稳随机过程的指标集通常被解释为时间，因此可以是整数或实线。<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>
它们都有相同的[[概率分布]]。平稳随机过程的指标集通常被解释为时间，因此可以是整数或实线。<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>
当指标集<math>T</math>可以解释为时间时，如果随机过程的有限维分布在时间平移下是不变的，则称其为平稳过程。这种随机过程可以用来描述处于稳态的物理系统，但是仍然会经历随机波动。<ref name=“Lamperti1977page6”/>平稳性背后的直觉是，随着时间的推移，平稳随机过程的分布保持不变。<ref name=“Doob1990page94”>{cite book | author=Joseph L.Doob | title=randours | url=图书https://books.com/？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=94–96}}</ref>只有当随机变量相同分布时，一系列随机变量才会形成平稳随机过程。<ref name=“Lamperti1977page6”/>
当指标集<math>T</math>可以解释为时间时，如果随机过程的有限维分布在时间平移下是不变的，则称其为平稳过程。这种随机过程可以用来描述处于稳态的物理系统，但是仍然会经历随机波动。<ref name=“Lamperti1977page6”/>平稳性背后的直觉是，随着时间的推移，平稳随机过程的分布保持不变。<ref name=“Doob1990page94”>{cite book | author=Joseph L.Doob | title=randours | url=图书https://books.com/？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=94–96}}</ref>只有当随机变量相同分布时，一系列随机变量才会形成平稳随机过程。<ref name=“Lamperti1977page6”/>
具有上述平稳性定义的随机过程有时被称为严格平稳的，但也有其他形式的平稳性。一个例子是当离散时间或连续时间随机过程<math>X</math>被称为广义平稳时，那么，对于t</math>中的所有<math>t\n，过程<math>X</math>有一个有限的第二时刻，两个随机变量的协方差只取决于t</math>中所有<math>t\t\的数<math>h</math>Florescu | title=概率和随机过程| url=图书https://books.com/？id=z5sebqaaqabaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | pages=298299}}</ref>[[Aleksandr Khinchin | Khinchin]]介绍了“广义平稳性”的相关概念，其他名称包括“协方差平稳性”或“广义平稳性”。<ref name=“Florescu2014page298”/><ref name=“GikhmanSkorokhod1969page8”>{cite book | author1=Iosif Ilyich Gikhman | author2=Anatoly Vladimirovich skorokod | title=随机过程理论导论| url=图书https://books.com/？id=yJyLzG7N7r8C&pg=PR2 |年份=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=8}</ref>
具有上述平稳性定义的随机过程有时被称为严格平稳的，但也有其他形式的平稳性。一个例子是当离散时间或连续时间随机过程<math>X</math>被称为广义平稳时，那么，对于t</math>中的所有<math>t\n，过程<math>X</math>有一个有限的第二时刻，两个随机变量的协方差只取决于t</math>中所有<math>t\t\的数<math>h</math>Florescu | title=概率和随机过程| url=图书https://books.com/？id=z5sebqaaqabaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | pages=298299}}</ref>[[Aleksandr Khinchin | Khinchin]]介绍了“广义平稳性”的相关概念，其他名称包括“协方差平稳性”或“广义平稳性”。<ref name=“Florescu2014page298”/><ref name=“GikhmanSkorokhod1969page8”>{cite book | author1=Iosif Ilyich Gikhman | author2=Anatoly Vladimirovich skorokod | title=随机过程理论导论| url=图书https://books.com/？id=yJyLzG7N7r8C&pg=PR2 |年份=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=8}</ref>
====过滤 Filtration====
====Filtration过滤====
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>
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>
[[过滤（概率论）|过滤]]是定义在某个概率空间中的sigma代数的递增序列和具有某种[[总阶]]关系的索引集，例如在索引集是实数的某个子集的情况下。更为正式的是，如果随机过程有一个指数集总排序的随机过程，则如果随机过程有一个指数集的总序为总序，那么在概率空间上概率空间<math>（\Omega，\mathcal{F{F}u t}{t}{math>\{\mathcal{F{F}，P，P）</math>上是一个西格玛代数家族，这样一个西格玛代数家族使得<math>\mathcal{mathcal{F{F}mathcal{F{F{F{F{F{F{F{F{F}数学>为所有<数学>s\s\s\subteq\mathcal{F}{F}{leq t</math>，其中，t中的<math>t，s\in t</math>和<math>\leq</math>表示索引集<math>t</math>的总顺序<ref name=“Florescu2014page294”/>通过过滤的概念，可以研究t</math>中随机过程<math>X\t</math>所包含的信息量，这可以解释为时间<math>t</math><ref name=“Florescu2014page294”/><ref name=“Williams1991page93”/>过滤背后的直觉是，随着时间的流逝，关于<math>t</math>的更多信息是已知的或可用的，这些信息可以在<math>\mathcal{F}t</math>中获得，使<math>\Omega</math>的分区越来越细{cite book | author=Fima C.Klebaner | title=Introduction to Ratical Calculation with Applications |网址=图书https://books.com/？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | pages=22–23}</ref><ref name=“Mörters2010page37”>{cite book | author1=Peter Mörters | author2=Yuval Peres | title=布朗运动| url=图书https://books.com/？id=e-TbA-dSrzYC | year=2010 | publisher=剑桥大学出版社| isbn=978-1-139-48657-6 | page=37}</ref>
[[过滤（概率论）|过滤]]是定义在某个概率空间中的sigma代数的递增序列和具有某种[[总阶]]关系的索引集，例如在索引集是实数的某个子集的情况下。更为正式的是，如果随机过程有一个指数集总排序的随机过程，则如果随机过程有一个指数集的总序为总序，那么在概率空间上概率空间<math>（\Omega，\mathcal{F{F}u t}{t}{math>\{\mathcal{F{F}，P，P）</math>上是一个西格玛代数家族，这样一个西格玛代数家族使得<math>\mathcal{mathcal{F{F}mathcal{F{F{F{F{F{F{F{F{F}数学>为所有<数学>s\s\s\subteq\mathcal{F}{F}{leq t</math>，其中，t中的<math>t，s\in t</math>和<math>\leq</math>表示索引集<math>t</math>的总顺序<ref name=“Florescu2014page294”/>通过过滤的概念，可以研究t</math>中随机过程<math>X\t</math>所包含的信息量，这可以解释为时间<math>t</math><ref name=“Florescu2014page294”/><ref name=“Williams1991page93”/>过滤背后的直觉是，随着时间的流逝，关于<math>t</math>的更多信息是已知的或可用的，这些信息可以在<math>\mathcal{F}t</math>中获得，使<math>\Omega</math>的分区越来越细{cite book | author=Fima C.Klebaner | title=Introduction to Ratical Calculation with Applications |网址=图书https://books.com/？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | pages=22–23}</ref><ref name=“Mörters2010page37”>{cite book | author1=Peter Mörters | author2=Yuval Peres | title=布朗运动| url=图书https://books.com/？id=e-TbA-dSrzYC | year=2010 | publisher=剑桥大学出版社| isbn=978-1-139-48657-6 | page=37}</ref>
====修正 Modification====
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
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
随机过程的“修正”是另一个随机过程，它与原始随机过程密切相关。更确切地说，一个随机过程<math>X</math>，与另一个随机过程<math>Y</math> 具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>（\Omega，{\cal F}，P）</math>具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>(\Omega,{\cal F},P)</math>，被称为<math>Y</math>的修改，如果对所有<math>t\in T</math>有
随机过程的“修正”是另一个随机过程，它与原始随机过程密切相关。更确切地说，一个随机过程<math>X</math>，与另一个随机过程<math>Y</math> 具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>（\Omega，{\cal F}，P）</math>具有相同的索引集<math>T</math>、集空间<math>S</math>和概率空间<math>(\Omega,{\cal F},P)</math>，被称为<math>Y</math>的修改，如果对所有<math>t\in T</math>有
<center><math>
<center><math>
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>
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>
持有。两个相互修正的随机过程具有相同的有限维法则<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>它们被称为“随机等价”或“等价物”<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>
=========================================================================================================10.26
====Indistinguishable无法识别====
====Indistinguishable无法识别====
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.
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Joseph Doob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。
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.
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Joseph Doob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不相关====
====Uncorrelatedness不相关====
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"/>
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"/>
1953年，Joseph Doob出版了《随机过程》一书，这本书对随机过程理论产生了重大影响，并强调了测度理论在概率论中的重要性。<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"/>
<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"/>