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第167行：
第167行： − 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]]:+ − − 第177行：
第175行： − 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> − + − For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives 第191行：
第187行： − − so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/> 第202行：
第196行： − 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> − 随机过程或随机变量的规律也被称为“概率定律”，“概率分布”，或“分布”.<ref name=“Borovkov2013page528”/><ref name=“FrizVictoir2010page571”/><ref name=“Whitt2006page23”>{cite book | author=Ward Whitt | title=随机过程限制：随机过程限制及其在队列中的应用简介=图书https://books.com/？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 |作者=David Applebaum | title=Lévy过程和随机演算| 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=连续鞅和布朗运动| url=https://books.google.com/books？id=Oybncaaqbaj |年份=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-06400-9 | page=10}</ref>+ − − For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distributions''' are defined as: 第219行：
第210行： + − − − − For any measurable subset <math>C</math> of the <math>n</math>-fold [[Cartesian power]] <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as:<ref name="Lamperti1977page1"/> 第235行：
第223行： − The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.<ref name="Rosenthal2006page177"/>+ − + − − '''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 第251行：
第237行： − all have the same [[probability distribution]]. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.<ref name="Lamperti1977page6">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=6 and 7}}</ref><ref name="GikhmanSkorokhod1969page4">{{cite book|author1=Iosif I. Gikhman |author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=4}}</ref> But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.<ref name="Lamperti1977page6"/><ref name="Adler2010page14">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA263|year=2010|publisher=SIAM|isbn=978-0-89871-693-1|pages=14, 15}}</ref><ref name="ChiuStoyan2013page112">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=112}}</ref> − When the index set <math>T</math> can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations.<ref name="Lamperti1977page6"/> The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same.<ref name="Doob1990page94">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=94–96}}</ref> A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.<ref name="Lamperti1977page6"/>+ − − 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”/> − A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process <math>X</math> is said to be stationary in the wide sense, then the process <math>X</math> has a finite second moment for all <math>t\in T</math> and the covariance of the two random variables <math>X_t</math> and <math>X_{t+h}</math> depends only on the number <math>h</math> for all <math>t\in T</math>.<ref name="Doob1990page94"/><ref name="Florescu2014page298">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=298, 299}}</ref> [[Aleksandr Khinchin|Khinchin]] introduced the related concept of '''stationarity in the wide sense''', which has other names including '''covariance stationarity''' or '''stationarity in the broad sense'''.<ref name="Florescu2014page298"/><ref name="GikhmanSkorokhod1969page8">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=8}}</ref>+ − 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> − + − 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> − − 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 − 第285行：
第261行： − 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> − Instead of modification, the term '''version''' is also used,<ref name="Adler2010page14"/><ref name="Klebaner2005page48">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=48}}</ref><ref name="Øksendal2003page14">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=14}}</ref><ref name="Florescu2014page472">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=472}}</ref> however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.<ref name="RevuzYor2013page18">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|pages=18–19}}</ref><ref name="FrizVictoir2010page571"/>+ − − 除了修改，还使用了“版本”一词，<ref name=“Adler2010page14”/><ref name=“Klebaner2005page48”>{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 | page=48}</ref><ref name=“Øksendal2003page14”>{cite book | author=BerntØksendal | title=随机微分方程：应用简介| url=图书https://books.com/？id=VgQDWyihxKYC | year=2003 | publisher=Springer Science&Business Media | isbn=978-3-540-04758-2 | page=14}</ref><ref name=“Florescu2014page472”>{cite book |作者=Ionut Florescu | title=概率与随机过程| url=图书https://books.com/？id=z5sebqaaqabaj&pg=PR22 | year=2014 | publisher=John Wiley&Sons | isbn=978-1-118-59320-2 | pages=472}}</ref>然而，当两个随机过程具有相同的有限维分布，但它们可能定义在不同的概率空间上，因此两个过程是相互修改的，在后一种意义上，它们也是彼此的版本，但不是相反=图书https://books.com/？id=Oybncaaqbaj |年份=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-06400-9 | pages=18–19}</ref><ref name=“FrizVictoir2010page571” − − If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the [[Kolmogorov continuity theorem]] says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.<ref name="Øksendal2003page14"/><ref name="Florescu2014page472"/><ref name="ApplebaumBook2004page20">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=20}}</ref> The theorem can also be generalized to random fields so the index set is <math>n</math>-dimensional Euclidean space<ref name="Kunita1997page31">{{cite book|author=Hiroshi Kunita|title=Stochastic Flows and Stochastic Differential Equations|url=https://books.google.com/books?id=_S1RiCosqbMC|year=1997|publisher=Cambridge University Press|isbn=978-0-521-59925-2|page=31}}</ref> as well as to stochastic processes with [[metric spaces]] as their state spaces.<ref name="Kallenberg2002page">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=35}}</ref> − − 如果一个连续时间的实值随机过程在其增量上满足一定的矩条件，则[[Kolmogorov连续性定理]]指出，该过程存在一个修正，其具有概率为1的连续样本路径，因此随机过程有一个连续的修改或版本=图书https://books.com/？id=q7eDUjdJxIkC | year=2004 | publisher=Cambridge University Press | isbn=978-0-521-83263-2 | page=20}</ref>该定理也可以推广到随机域，因此索引集是<math>n</math>-维欧几里德空间<ref name=“Kunita1997page31”>{cite book | author=Hiroshi Kunita|title=随机流和随机微分方程式| url=图书https://books.com/？id=_S1RiCosqbMC | year=1997 | publisher=Cambridge University Press | isbn=978-0-521-59925-2 | page=31}</ref>以及以[[度量空间]]为状态空间的随机过程=图书https://books.com/？id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | page=35}</ref> − − − =========================================================================================================10.26 − ====Indistinguishable无法识别====+ − 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>定义在同一概率空间<math>（\Omega，\mathcal{F}，P）</math>上，具有相同的索引集<math>T</math>和集空间<math>S</math>上的两个随机过程称为“不可识别的”，如果 − After Cardano, Jakob Bernoulli wrote Ars Conjectandi, which is considered a significant event in the history of probability theory. But despite some renowned mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev, most of the mathematical community did not consider probability theory to be part of mathematics until the 20th century.+ − 在 Cardano 之后，Jakob Bernoulli 写了 Ars Conjectandi，这被认为是概率论历史上的一个重大事件。但是，尽管一些著名的数学家为概率论做出了贡献，比如皮埃尔-西蒙·拉普拉斯，亚伯拉罕·棣莫弗，Carl Gauss，Siméon Poisson 和巴夫尼提·列波维奇·切比雪夫，大多数数学界直到20世纪才认为概率论是数学的一部分。+ 第318行：
第283行： − In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius, most of the work had little or no randomness. − − 在物理科学领域，科学家们在19世纪发展了统计力学学科，在这个学科中，物理系统，例如装满气体的容器，可以被看作或者从数学上被当作许多运动粒子的集合。尽管有一些科学家，比如鲁道夫 · 克劳修斯，试图将随机性纳入统计物理学，但大多数工作几乎没有或根本没有随机性。 − − holds.<ref name="FrizVictoir2010page571"/><ref name="RogersWilliams2000page130"/> If two <math>X</math> and <math>Y</math> are modifications of each other and are almost surely continuous, then <math>X</math> and <math>Y</math> are indistinguishable.<ref name="JeanblancYor2009page11">{{cite book|author1=Monique Jeanblanc|author1-link= Monique Jeanblanc |author2=Marc Yor|author2-link=Marc Yor|author3=Marc Chesney|title=Mathematical Methods for Financial Markets|url=https://books.google.com/books?id=ZhbROxoQ-ZMC|year=2009|publisher=Springer Science & Business Media|isbn=978-1-85233-376-8|page=11}}</ref> − − 保留。<ref name=“FrizVictoir2010page571”/><ref name=“Rogerswillams2000page130”/>如果两个<math>X</math>和<math>Y</math>是相互修改的，几乎肯定是连续的，那么<math>X</math>和<math>Y</math>是无法区分的。<ref name=“JeanblancYor2009page11”>{cite book | author1=Monique Jeanblanc | author2=Marc Yor | author2 link=Marc Yor | author3=Marc Chesney | title=金融市场数学方法| url=图书https://books.com/？id=ZhbROxoQ ZMC |年=2009 | publisher=Springer Science&Business Media | isbn=978-1-85233-376-8 | page=11}</ref> − − This changed in 1859 when James Clerk Maxwell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities. The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement. − − 这种情况在1859年发生了改变，当时詹姆斯·克拉克·麦克斯韦对这个领域做出了重大贡献，更具体地说，他提出了假设气体粒子以随机速度向随机方向运动的工作，这对分子运动论研究有重大贡献。分子运动论和统计物理学在19世纪下半叶继续发展，主要由克劳修斯、路德维希·玻尔兹曼和约西亚吉布斯完成的工作，后来对阿尔伯特爱因斯坦的布朗运动的数学模型产生了影响。 − − ====Separability可分性==== − − '''Separability''' is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a [[separable space]],{{efn|The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic sense), in addition to other conditions.<ref name="Skorokhod2005page93"/>}} which means that the index set has a dense countable subset.<ref name="Adler2010page14"/><ref name="Ito2006page32">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|pages=32–33}}</ref> − − “可分离性”是随机过程的一种性质，它基于与概率测度有关的指标集。假设随机过程或具有不可数指标集的随机场的泛函可以形成随机变量。对于一个随机过程是可分离的，除了其他条件外，它的指标集必须是一个[[可分离空间]]，{efn |术语“可分离”在这里出现了两次，有两种不同的含义，第一种含义来自概率，第二种含义来自拓扑和分析。对于一个随机过程是可分的（概率意义上），它的指标集必须是一个可分空间（在拓扑或分析意义上），除了其他条件。<ref name=“Skorokhod2005page93”/>}}，这意味着索引集有一个稠密的可数子集。<ref name=“Adler2010page14”/><ref name=“Ito2006page32”>{cite book | author=Kiyosi Itōtitle=Essentials of randomic Processes|url=图书https://books.com/？id=pY5|DkvI-CcC&pg=PR4 | year=2006 | publisher=美国数学学会| isbn=978-0-8218-3898-3 |页=32–33}</ref> − − At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms.}} and Andrei Kolmogorov. In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line. − − 1900年在巴黎的国际数学家大会，David Hilbert 展示了一系列数学问题，其中他的第六个问题要求对物理学和涉及公理的概率进行数学处理和安德烈 · 科尔莫戈罗夫。在20世纪30年代早期，钦钦和科尔莫戈罗夫设立了概率研讨会，参加研讨会的研究人员有 Eugene Slutsky 和 Nikolai Smirnov，钦钦给出了第一个数学定义，随机过程是一组由实数线索引的随机变量。 − − − − More precisely, a real-valued continuous-time stochastic process <math>X</math> with a probability space <math>(\Omega,{\cal F},P)</math> is separable if its index set <math>T</math> has a dense countable subset <math>U\subset T</math> and there is a set <math>\Omega_0 \subset \Omega</math> of probability zero, so <math>P(\Omega_0)=0</math>, such that for every open set <math>G\subset T</math> and every closed set <math>F\subset \textstyle R =(-\infty,\infty) </math>, the two events <math>\{ X_t \in F \text{ for all } t \in G\cap U\}</math> and <math>\{ X_t \in F \text{ for all } t \in G\}</math> differ from each other at most on a subset of <math>\Omega_0</math>.<ref name="GikhmanSkorokhod1969page150">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=150}}</ref><ref name="Todorovic2012page19">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|pages=19–20}}</ref><ref name="Molchanov2005page340">{{cite book|author=Ilya Molchanov|title=Theory of Random Sets|url=https://books.google.com/books?id=kWEwk1UL42AC|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-892-3|page=340}}</ref> − − 更精确地说，具有概率空间<math>（\Omega，{\calf}，P）</math>的实值连续时间随机过程<math>X</math>是可分离的，如果它的指数集<math>T</math>有一个稠密的可数子集<math>U\subset\Omega</math>，因此<math>P（\Omega_0）=0</math>，这样对于每个开集<math>G\subset T</math>和每个闭集<math>F\subset\textstyle R=（-\infty，\infty）</math>，在F\text{FORALALL}t\in G\cap U\}</math>和F\text{FORALALL}\t\G\cap U\}</math>和F\text{FORALALL}t\in G\}</math>这两个事件在<math>\Omega</math><math><ref name=“Gikhmankorokod1969Page150”>{（引自《引证图书| author1=IOSIIF IlichiGikhman | author2=Anatoly Vladimirovich Vladimirovich author2=Anatol2=AnatolyVladimal斯科罗霍德| title=介绍随机过程理论=图书https://books.com/？id=yJyLzG7N7r8C&pg=PR2 | year=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=150}</ref><ref name=“Todorovic2012page19”>{cite book | author=Petar Todorovic | title=随机过程及其应用简介| url=图书https://books.com/？第1249页{jqbn=1240页{jqbn=1240第1页{jqbn=1240第1页，第1页，第1页=图书https://books.com/？id=kWEwk1UL42AC |年份=2005 | publisher=Springer Science&Business Media | isbn=978-1-85233-892-3 | page=340}</ref> − − The definition of separability{{efn|The definition of separability for a continuous-time real-valued stochastic process can be stated in other ways.<ref name="Billingsley2008page526">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=526–527}}</ref><ref name="Borovkov2013page535">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=535}}</ref>}} can also be stated for other index sets and state spaces,<ref name="GusakKukush2010page22">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 22</ref> such as in the case of random fields, where the index set as well as the state space can be <math>n</math>-dimensional Euclidean space.<ref name="AdlerTaylor2009page7"/><ref name="Adler2010page14"/> − − 可分离性的定义{efn |连续时间实值随机过程的可分性定义可以用其他方式表述。<ref name=“Billingsley2008page526”>{cite book | author=Patrick Billingsley | title=Probability and Measure | url=图书https://books.com/？id=QyXqOXyxEeIC |年份=2008 | publisher=Wiley India私人有限公司| isbn=978-81-265-1771-8 | pages=526-527}</ref><ref name=“Borovkov2013page535”>{引用图书|作者=Alexander A.Borovkov | title=Probability Theory |网址=图书https://books.com/？id=hRk_AAAAQBAJ&pg | year=2013；publisher=Springer Science&Business Media | isbn=978-1-4471-5201-9 | page=535}}</ref>}}}也可以为其他索引集和状态空间而声明，<ref name=“gusakukukukukukukukukukukukukukukukukukukukukukukush201Page22”>{{{harvvxt Gusak kukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukukuku在随机场的情况下，其中，索引集和状态空间可以是<math>n</math>维欧几里德空间。<ref name=“adlertaylor2009 page7”/><ref name=“Adler2010page14”/> − − In 1933 Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,{{efn|Later translated into English and published in 1950 as Foundations of the Theory of Probability − − 1933年 Andrei Kolmogorov 出版了一本关于概率论基础的书，名为 grundbigriffe der Wahrscheinlichkeitsrechnung，1950年被翻译成英文并出版为概率论基础 + − The concept of separability of a stochastic process was introduced by [[Joseph Doob]],<ref name="Ito2006page32"/>. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.<ref name="Billingsley2008page526"/> Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.<ref name="Doob1990page56">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=56}}</ref> A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.<ref name="Ito2006page32"/><ref name="Todorovic2012page19"/><ref name="Khoshnevisan2006page155">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|page=155}}</ref> Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.<ref name="Skorokhod2005page93"/>+ − 随机过程可分性的概念是由[[Joseph Doob]]，<ref name=“Ito2006page32”/>提出的。可分性的基本思想是使指标集的可数点集决定随机过程的性质，因此离散时间随机过程总是可分离的=图书https://books.com/？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=56}</ref>Doob的一个定理，有时被称为Doob的可分性定理，表示任何实值连续时间随机过程都有一个可分离的修改。<ref name=“Ito2006page32”/><ref name=“Todorovic2012page19”/><ref name=“Khoshnivesan2006page155”>{cite book | author=Davar khoshnivesan | title=Multiparameter Processes:随机字段简介| url=图书https://books.com/？id=XADpBwAAQBAJ | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21631-7 | page=155}</ref>该定理的版本也适用于具有索引集和状态空间而非实线的更一般的随机过程。<ref name=“skorokood205page93”/> − 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。 + − + − Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theory, with early ideas by Shizuo Kakutani and then later work by Joseph Doob. − 同样从20世纪40年代开始，随机过程，特别是鞅，和势场理论的数学领域之间建立了联系，早期的思想由 Shizuo Kakutani 提出，后来由 Joseph Doob 提出。+ + − Two stochastic processes <math>X</math> and <math>Y</math> defined on the same probability space <math>(\Omega,\mathcal{F},P)</math> with the same index set <math>T</math> are said be '''independent''' if for all <math>n \in \mathbb{N}</math> and for every choice of epochs <math>t_1,\ldots,t_n \in T</math>, the random vectors <math>\left( X(t_1),\ldots,X(t_n) \right)</math> and <math>\left( Y(t_1),\ldots,Y(t_n) \right)</math> are independent.<ref name=Lapidoth>Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.</ref>{{rp|p. 515}} − 两个在相同的概率空间<math>(\Omega,\mathcal{F},P)</math>上定义，具有相同索引集<math>T</math>的随机过程<math>X</math>和<math>Y</math>被称为“相互独立”，如果对于所有<math>n \in \mathbb{N}</math>，以及每个特定的<math>t_1,\ldots,t_n \in T</math>，随机向量<math>\left( X(t_1),\ldots,X(t_n) \right)</math> 和<math>\left( Y(t_1),\ldots,Y(t_n) \right)</math>是独立的。<ref name=Lapidoth>Amos，“数字通信基础”，剑桥大学出版社，2009年。</ref>{rp | p.515}}+ − In 1953 Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-André Meyer. Earlier work had been carried out by Sergei Bernstein, Paul Lévy and Jean Ville, the latter adopting the term martingale for the stochastic process. Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes. which would later result in Varadhan winning the 2007 Abel Prize. In the 1990s and 2000s the theories of Schramm–Loewner evolution and rough paths were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin Werner in 2008 and to Martin Hairer in 2014.+ − − 1953年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==== − − Two stochastic processes <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are called '''uncorrelated''' if their cross-covariance <math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right]</math> is zero for all times.<ref name=KunIlPark>Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3</ref>{{rp|p. 142}} Formally: − − 两个随机过程<math>\left\{X_t\right\}</math>和<math>\left\{Y_t\right\}</math> 称为“不相关的”的，如果它们的互协方差<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>始终为零。<ref name=KunIlPark>Kun Il Park，《概率与随机过程基础与通信应用》，Springer，2018，978-3-319-68074-3</ref>{{rp|p. 142}} 最后： − − The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes. − − 随机过程理论仍然是研究的焦点，每年都有关于随机过程的国际会议。 + + − + − The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. Bernoulli's work, including the Bernoulli process, were published in his book Ars Conjectandi in 1713. − − 伯努利过程可以作为一个数学模型来抛出一个有偏见的硬币，它可能是第一个被研究的随机过程。伯努利的著作，包括《伯努利过程，于1713年在他的书《猜测》中出版。 − − − − If two stochastic processes <math>X</math> and <math>Y</math> are independent, then they are also uncorrelated.<ref name=KunIlPark/>{{rp|p. 151}} − In 1905 Karl Pearson coined the term random walk while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. and is an example of a random walk with absorbing barriers. Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods, and then more detailed solutions were presented by Jakob Bernoulli and Abraham de Moivre.+ − + − 1905年，卡尔 · 皮尔森在提出一个描述平面上随机漫步的问题时，创造了''' 随机漫步Random walk'''这个术语，这个问题的动机是生物学中的一个应用，但是这种涉及随机漫步的问题已经在其他领域得到了研究。几个世纪前研究过的某些赌博问题可以被认为是涉及随机漫步的问题。这是一个带有吸收屏障的随机漫步的例子。和 Huyens 都给出了这个问题的数值解，但没有详细介绍他们的方法，然后 Jakob Bernoulli 和亚伯拉罕·棣莫弗提供了更详细的解。 − − − − Two stochastic processes <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are called '''orthogonal''' if their cross-correlation <math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E}[X(t_1) \overline{Y(t_2)}]</math> is zero for all times.<ref name=KunIlPark/>{{rp|p. 142}} Formally: − − 由</math>和<math>\left\{Y Y{R}{mathbf{math>\ left\{YY{right\}</math>和<math>\left\{Y Y{right\}</math>两个随机过程<数学>\operatorname{R{mathbf{mathbf{X}\mathbf{Y}}（t 1、t U2）的=\operatorname{E{E}[X（t（t U1）1）\顶顶天{Y（t〈2）右}}</math>如果它们的相互关联<正交{正交ref name=KunIlPark/>{{rp | p.142}}形式上： − − For random walks in <math>n</math>-dimensional integer lattices, George Pólya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions. − − 对于 < math > n </math > 维整数格中的随机游动，George Pólya 在1919年和1921年发表的工作中，他研究了对称随机游动回到格中以前位置的概率。Pólya 证明了对称随机游动，它在格子中向任何方向前进的概率相等，将无限次地回到格子中的一个先前的位置，概率为1在一维和2维，但概率为0在三维或更高维。 − − + − The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics. The work is now considered as an early discovery of the statistical method known as Kalman filtering, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time. in order to model price changes on the Paris Bourse, a stock exchange, without knowing the work of Thiele. and Bachelier's thesis is now considered pioneering in the field of financial mathematics.+ − − 维纳过程或布朗运动过程起源于不同的领域，包括统计学、金融学和物理学。这项工作现在被认为是卡尔曼滤波统计方法的早期发现，但是这项工作在很大程度上被忽视了。人们认为，蒂勒论文中的观点太过先进，当时更广泛的数学和统计学界无法理解。为了模拟巴黎证券交易所的价格变化，不知道蒂勒的工作。巴切利耶的论文现在被认为是金融数学领域的先驱。 − − ====Skorokhod space斯科罗霍德空间==== − − {{Main|Skorokhod space}} − {{Main |斯科罗霍德空间}} − − Einstein's work, as well as experimental results obtained by Jean Perrin, later inspired Norbert Wiener in the 1920s to use a type of measure theory, developed by Percy Daniell, and Fourier analysis to prove the existence of the Wiener process as a mathematical object. There are a number of claims for early uses or discoveries of the Poisson − − 爱因斯坦的工作，以及 Jean Perrin 获得的实验结果，后来激发了 Norbert Wiener 在20世纪20年代使用一种由 Percy Daniell 和傅立叶变换家族中的关系提出的测量理论来证明 Wiener 过程作为一个数学对象的存在。关于泊松鱼的早期用途和发现，有许多说法 − − − − ''skorokod space''也写为''Skorohod space''，是所有右连续左极限的函数的数学空间，定义在实线的某个区间上，例如<math>[0,1]</math>或<math>[0，\infty）</math>，取实线或度量空间上的值=图书https://books.com/？id=LkQOBwAAQBAJ&pg=PR5 | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21748-2 |页=78–79}</ref><ref name=“gusakkush2010page24”>{harvxt | Gusak | kush | Kulik | Mishura | 2010}，p、 24</ref><ref name=“Bogachev2007Vol2page53”>{引用图书|作者=Vladimir I.Bogachev | title=测量理论（第2卷）|网址=图书https://books.com/？id=CoSIe7h5mTsC | year=2007 | publisher=Springer Science&Business Media | isbn=978-3-540-34514-5 | page=53}</ref>这些函数被称为cádLag或cadlag函数，这是基于法语表达式“continue a droite，limiteégauche”的首字母缩略词，因为这些函数是右连续的，具有左极限。<ref name=“Whitt2006page78”/><refname=“Klebaner2005page4”>{cite book | author=Fima C.Klebaner | title=随机微积分及其应用简介|网址=图书https://books.com/？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=4}</ref>由[[Anatoliy Skorokod]]引入的Skorokod函数空间，<ref name=“Bogachev2007Vol2page53”/>通常用字母<math>D</math>表示，<ref name=“Whitt2006page78”/><ref name=“GusakKukush2010page24”/><ref name=“Bogachev2007Vol2page53”/><ref name=“Klebaner2005page4”/>因此函数空间也被称为空间<math>D</math><ref name=“Whitt2006page78”/><ref name=“Asmussen2003page420”>{cite book | author=S|ren Asmussen | title=应用概率和队列| url=图书https://books.com/？id=BeYaTxesKy0C | year=2003 | publisher=Springer Science&Business Media | isbn=978-0-387-00211-8 | page=420}</ref><ref name=“Billingsley2013page121”>{cite book |作者=Patrick Billingsley | title=Convergence of Probability Measures|网址=图书https://books.com/？id=6ItqtwaWZZQC | year=2013 | publisher=John Wiley&Sons | isbn=978-1-118-62596-5 | page=121}</ref>此函数空间的表示法还可以包括定义所有cádlág函数的间隔，因此，例如，<math>D[0,1]</math>表示在[[单位间隔]]<math>[0上定义的c|dla g函数的空间，1] </math><ref name=“Klebaner2005page4”/><ref name=“Billingsley2013page121”/><ref name=“Bass2011page34”>{cite book | author=Richard F.Bass | title=random Processes |网址=图书https://books.com/？id=Ll0T7PIkcKMC |年=2011 | publisher=Cambridge University Press | isbn=978-1-139-50147-7 | page=34}</ref> − − process. − − 过程。 − − − − Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.<ref name="Bogachev2007Vol2page53"/><ref name="Asmussen2003page420"/> Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.<ref name="Billingsley2013page121"/><ref name="BinghamKiesel2013page154">{{cite book|author1=Nicholas H. Bingham|author2=Rüdiger Kiesel|title=Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives|url=https://books.google.com/books?id=AOIlBQAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-3856-3|page=154}}</ref> − − 在随机过程理论中，由于通常假定连续时间随机过程的样本函数属于一个Skorokod空间，因此经常使用Skorokod函数空间，对应于Wiener过程的样本函数。但是空间也有间断函数，这意味着随机过程的样本函数具有跳跃性，例如泊松过程（在实线上），同时也是这一领域的成员。<ref name=“Billingsley2013page121”/><ref name=“BinghamKiesel2013page154”>{cite book | author1=Nicholas H.Bingham | author2=Rüdiger Kiesel | title=风险中性估值：金融衍生品的定价和对冲| url=图书https://books.com/？id=AOIlBQAAQBAJ |年份=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4471-3856-3 | page=154}</ref> − − − Another discovery occurred in Denmark in 1909 when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution. Markov was interested in studying an extension of independent random sequences. which had been commonly regarded as a requirement for such mathematical laws to hold. Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains. − − 另一个发现发生在1909年的丹麦。在开发一个有限时间间隔内接听电话数量的数学模型时，Erlang 得出了这个泊松分佈。当时 Erlang 并不知道 Poisson 的早期工作，并且假设每个时间间隔内到达的号码电话是相互独立的。然后他发现了极限情况，这是有效地重铸泊松分佈作为一个二项分布的限制。马尔科夫对研究独立随机序列的推广很感兴趣。这被普遍认为是这样的数学定律的一个必要条件。从1928年开始，莫里斯 · 弗雷切特对马尔可夫链产生了兴趣，最终导致他在1938年发表了一篇关于马尔可夫链的详细研究。 − − ====Regularity规律性==== − Andrei Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s. In addition to Lévy, Khinchin and Kolomogrov, early fundamental contributions to the theory of Lévy processes were made by Bruno de Finetti and Kiyosi Itô.+ − 安德烈 · 科尔莫戈罗夫在1931年的一篇论文中发展了早期连续时间马尔可夫过程理论的很大一部分。他介绍并研究了一组特殊的马尔可夫过程，称为扩散过程，在这组过程中他推导出了一组描述这些过程的微分方程。在研究布朗运动时，Sydney Chapman 在1928年的一篇论文中，独立于 Kolmogorov 的工作，用一种比 Kolmogorov 更不严密的数学方法，推导出了一个方程，现在称为 Chapman-Kolmogorov 方程。这些微分方程现在被称为 Kolmogorov 方程或 Kolmogorov-Chapman 方程。其他对马尔可夫过程的基础做出了重大贡献的数学家包括威廉 · 费勒，从20世纪30年代开始，然后是尤金 · 戴金，从20世纪50年代开始。除了 Lévy，Khinchin 和 Kolomogrov，早期对 Lévy 过程理论的根本性贡献是由德福内梯和 Kiyosi itô。 − In the context of mathematical construction of stochastic processes, the term '''regularity''' is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.<ref name="Borovkov2013page532">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=532}}</ref><ref name="Khoshnevisan2006page148to165">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=148–165}}</ref> For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.<ref name="Todorovic2012page22">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|page=22}}</ref><ref name="Whitt2006page79">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|page=79}}</ref>+ − 在随机过程的数学构造中，当讨论和假设随机过程的某些条件以解决可能的构造问题时，使用术语“正则性”。<ref name=“Borovkov2013page532”>{cite book | author=Alexander a.Borovkov | title=Probability Theory | url=图书https://books.com/？id=hRk|AAAAQBAJ&pg | year=2013 | publisher=Springer Science&Business Media | isbn=978-1-4471-5201-9 | page=532}</ref><ref name=“khoshnivesan2006page148to165”>{cite book | author=Davar khoshnivesan | title=multiple Processes:An Introduction to Random Fields |网址=图书https://books.com/？id=XADpBwAAQBAJ | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21631-7 | pages=148–165}}</ref>例如，研究具有不可数索引集的随机过程，假设随机过程服从某种正则条件，例如样本函数是连续的=图书https://books.com/？id=XpjqBwAAQBAJ&pg=PP5 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4613-9742-7 | page=22}</ref><ref name=“Whitt2006page79”>{cite book | author=Ward Whitt | title=随机过程限制：随机过程限制及其在队列中的应用简介| url=图书https://books.com/？id=LkQOBwAAQBAJ&pg=PR5 | year=2006 | publisher=Springer科学与商业媒体| isbn=978-0-387-21748-2 | page=79}</ref> + − Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem to prove a corresponding stochastic process exists. says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions. This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.+ − 另一种方法是定义一组具有特定有限维分布的随机变量，然后用 Kolmogorov 的存在性定理证明相应的随机过程存在。他说，如果任何有限维分布满足两个条件，也就是所谓的一致性条件，那么就存在这些有限维分布的随机过程。这意味着随机过程的分布并不一定唯一地指定随机过程的样本函数的属性。+
→进一步定义
====定律====
====定律====
对于定义在概率空间<math>（\Omega，\mathcal{F}，P）</math>上的随机过程<math>X\colon\Omega\rightarrow S^T</math>，随机过程X</math>的定律被定义为'''前推度量 Pushforward measure'''：
对于定义在概率空间<math>（\Omega，\mathcal{F}，P）</math>上的随机过程<math>X\colon\Omega\rightarrow S^T</math>，随机过程X</math>的“定律”被定义为[[前推度量|图像度量]：
<center><math>
<center><math>
</math></center>
</math></center>
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<math>t\in 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。<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>
对于<math>S^T</math>的可测子集<math>B</math>，预图像<math>X</math>给出
对于<math>S^T</math>的可测子集<math>B</math>，预图像<math>X</math>给出
</math></center>
</math></center>
所以a<math>X</math>定律可以写成：<ref name=“Lamperti1977page1”/>
所以a<math>X</math>定律可以写成：<ref name=“Lamperti1977page1”/>
</math></center>
</math></center>
随机过程或随机变量的规律也被称为“概率定律 probability law”，“概率分布 probability distribution”，或“分布”。<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>
====有限维概率分布 Finite-dimensional probability distributions====
====有限维概率分布 Finite-dimensional probability distributions====
对于随机过程<math>X</math>，其“有限维分布”定义为：
对于随机过程<math>X</math>，其“有限维分布”定义为：
</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>
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”/>
</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====
====稳定性 Stationarity====
“稳定性”是当随机过程的所有随机变量都是相同分布时随机过程所具有的数学性质。换言之，如果<math>X</math>是一个平稳随机过程，那么对于任何<math>t\in T</math>，随机变量<math>X_t</math>具有相同的分布，这意味着对于任何一组<math>n</math>索引集值<math>t_1,\dots, t_n</math>而言，对应的<math>n</math>随机变量
“稳定性”是当随机过程的所有随机变量都是相同分布时随机过程所具有的数学性质。换言之，如果<math>X</math>是一个平稳随机过程，那么对于任何<math>t\in T</math>，随机变量<math>X_t</math>具有相同的分布，这意味着对于任何一组<math>n</math>索引集值<math>t_1,\dots, t_n</math>而言，对应的<math>n</math>随机变量
</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=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|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><ref name="Doob1990page94"/><ref name="Florescu2014page298">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=298, 299}}</ref> [[Aleksandr Khinchin | Khinchin]]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 Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=8}}</ref>
====过滤 Filtration====
====过滤 Filtration====
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。<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>
====修正 Modification====
====修正 Modification====
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
</math></center>
</math></center>
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>
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>
除了修改，还使用了“版本”一词，<ref name="Adler2010page14"/><ref name="Klebaner2005page48">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=48}}</ref><ref name="Øksendal2003page14">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=14}}</ref><ref name="Florescu2014page472">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=472}}</ref>然而，当两个随机过程具有相同的有限维分布，但它们可能定义在不同的概率空间上，因此两个过程是相互修改的，在后一种意义上，它们也是彼此的版本，但不是相反。<ref name="RevuzYor2013page18">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|pages=18–19}}</ref><ref name="FrizVictoir2010page571"/>
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the [[Kolmogorov continuity theorem]] says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version. 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.
如果一个连续时间的实值随机过程在其增量上满足一定的矩条件，则[[Kolmogorov连续性定理]]指出，该过程存在一个修正，其具有概率为1的连续样本路径，因此随机过程有一个连续的修改或版本<ref name="Øksendal2003page14"/><ref name="Florescu2014page472"/><ref name="ApplebaumBook2004page20">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=20}}</ref>该定理也可以推广到随机域，因此索引集是<math>n</math>-维欧几里德空间<ref name="Kunita1997page31">{{cite book|author=Hiroshi Kunita|title=Stochastic Flows and Stochastic Differential Equations|url=https://books.google.com/books?id=_S1RiCosqbMC|year=1997|publisher=Cambridge University Press|isbn=978-0-521-59925-2|page=31}}</ref>以及以[[度量空间]]为状态空间的随机过程。<ref name="Kallenberg2002page">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=35}}</ref>
====难以区分 Indistinguishable====
两个随机过程<math>X</math>和<math>Y</math>定义在同一概率空间<math>（\Omega，\mathcal{F}，P）</math>上，具有相同的索引集<math>T</math>和集空间<math>S</math>上的两个随机过程如果
<center><math>
<center><math>
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</math></center>
成立，则称为“难以区分的”。<ref name="FrizVictoir2010page571"/><ref name="RogersWilliams2000page130"/>如果两个<math>X</math>和<math>Y</math>是相互修改的，几乎肯定是连续的，那么<math>X</math>和<math>Y</math>是无法区分的。<ref name="JeanblancYor2009page11">{{cite book|author1=Monique Jeanblanc|author1-link= Monique Jeanblanc |author2=Marc Yor|author2-link=Marc Yor|author3=Marc Chesney|title=Mathematical Methods for Financial Markets|url=https://books.google.com/books?id=ZhbROxoQ-ZMC|year=2009|publisher=Springer Science & Business Media|isbn=978-1-85233-376-8|page=11}}</ref>
====可分性 Separability====
“可分性”是随机过程的一种性质，它基于与概率测度有关的指标集。假设随机过程或具有不可数指标集的随机场的泛函可以形成随机变量。对于一个随机过程是可分离的，除了其他条件外，它的指标集必须是一个[[可分离空间]]，{efn |术语“可分离”在这里出现了两次，有两种不同的含义，第一种含义来自概率，第二种含义来自拓扑和分析。对于一个随机过程是可分的（概率意义上），它的指标集必须是一个可分空间（在拓扑或分析意义上），除了其他条件。<ref name="Skorokhod2005page93"/>}}，这意味着索引集有一个稠密的可数子集。<ref name="Skorokhod2005page93"/>}} which means that the index set has a dense countable subset.<ref name="Adler2010page14"/><ref name="Ito2006page32">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|pages=32–33}}</ref>
更精确地说，具有概率空间<math>（\Omega，{\calf}，P）</math>的实值连续时间随机过程<math>X</math>是可分离的，如果它的指数集<math>T</math>有一个稠密的可数子集<math>U\subset\Omega</math>，因此<math>P（\Omega_0）=0</math>，这样对于每个开集<math>G\subset T</math>和每个闭集<math>F\subset\textstyle R=（-\infty，\infty）</math>，F\text{FORALALL}t\in G\cap U\}</math>和F\text{FORALALL}\t\G\cap U\}</math>和F\text{FORALALL}t\in G\}</math>这两个事件最多在<math>\Omega</math>的一个子集上不同。<ref name="GikhmanSkorokhod1969page150">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=150}}</ref><ref name="Todorovic2012page19">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|pages=19–20}}</ref><ref name="Molchanov2005page340">{{cite book|author=Ilya Molchanov|title=Theory of Random Sets|url=https://books.google.com/books?id=kWEwk1UL42AC|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-892-3|page=340}}</ref>
====Independence独立性====
可分离性的定义（连续时间实值随机过程的可分性定义可以用其他方式表述。<ref name="Billingsley2008page526">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=526–527}}</ref><ref name="Borovkov2013page535">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=535}}</ref>）也可以为其他索引集和状态空间而声明，<ref name="GusakKukush2010page22">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 22</ref>例如在随机场的情况下，索引集和状态空间可以是<math>n</math>维欧几里德空间。<ref name="AdlerTaylor2009page7"/><ref name="Adler2010page14"/>
随机过程可分性的概念是由[[Joseph Doob]]，<ref name="Ito2006page32"/>提出的。可分性的基本思想是使指标集的可数点集决定随机过程的性质，<ref name="Billingsley2008page526"/>因此离散时间随机过程总是可分离的。<ref name="Doob1990page56">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=56}}</ref>Doob的一个定理，有时被称为Doob的可分性定理，表示任何实值连续时间随机过程都有一个可分离的修改。<ref name="Ito2006page32"/><ref name="Todorovic2012page19"/><ref name="Khoshnevisan2006page155">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|page=155}}</ref> Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.<ref name="Skorokhod2005page93"/>
对于具有索引集和状态空间而不是实线的更一般的随机过程，也存在该定理的版本。<ref name="Skorokhod2005page93"/>
====独立性 Independence====
两个在相同的概率空间<math>(\Omega,\mathcal{F},P)</math>上定义，具有相同索引集<math>T</math>的随机过程<math>X</math>和<math>Y</math>被称为“相互独立”，如果对于所有<math>n \in \mathbb{N}</math>，以及每个特定的<math>t_1,\ldots,t_n \in T</math>，随机向量<math>\left( X(t_1),\ldots,X(t_n) \right)</math> 和<math>\left( Y(t_1),\ldots,Y(t_n) \right)</math>是独立的。<ref name=Lapidoth>Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.</ref>
====不相关 Uncorrelatedness====
两个随机过程<math>\left\{X_t\right\}</math>和<math>\left\{Y_t\right\}</math> 称为“不相关的”的，如果它们的互协方差<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>始终为零。<ref name=KunIlPark>Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3</ref>最后：
:<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>.
:<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>.
====独立意味着不相关 Independence implies uncorrelatedness====
====Independence implies uncorrelatedness独立意味着不相关====
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
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
====正交性 Orthogonality====
如果两个随机过程<math>\left\{X_t\right\}</math>和<math>\left\{Y_t\right\}</math>的互相关<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E}[X(t_1) \overline{Y(t_2)}]</math>一直为0，则称为“正交”，形式为<ref name=KunIlPark/>
====Orthogonality正交性====
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ orthogonal} \quad \iff \quad \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ orthogonal} \quad \iff \quad \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.
====斯科罗霍德空间 Skorokhod space====
A '''Skorokhod space''', also written as '''Skorohod space''', is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as <math>[0,1]</math> or <math>[0,\infty)</math>, and take values on the real line or on some metric space. 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]], is often denoted with the letter <math>D</math>,<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/> so the function space is also referred to as space <math>D</math>. The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, <math>D[0,1]</math> denotes the space of càdlàg functions defined on the [[unit interval]] <math>[0,1]</math>.<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>
A '''Skorokhod space''', also written as '''Skorohod space''', is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as <math>[0,1]</math> or <math>[0,\infty)</math>, and take values on the real line or on some metric space.<ref name="Whitt2006page78">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|pages=78–79}}</ref><ref name="GusakKukush2010page24">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 24</ref><ref name="Bogachev2007Vol2page53">{{cite book|author=Vladimir I. Bogachev|title=Measure Theory (Volume 2)|url=https://books.google.com/books?id=CoSIe7h5mTsC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-34514-5|page=53}}</ref> Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression ''continue à droite, limite à gauche'', due to the functions being right-continuous with left limits.<ref name="Whitt2006page78"/><ref name="Klebaner2005page4">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=4}}</ref> A Skorokhod function space, introduced by [[Anatoliy Skorokhod]],<ref name="Bogachev2007Vol2page53"/> is often denoted with the letter <math>D</math>,<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/> so the function space is also referred to as space <math>D</math>.<ref name="Whitt2006page78"/><ref name="Asmussen2003page420">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=420}}</ref><ref name="Billingsley2013page121">{{cite book|author=Patrick Billingsley|title=Convergence of Probability Measures|url=https://books.google.com/books?id=6ItqtwaWZZQC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-62596-5|page=121}}</ref> The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, <math>D[0,1]</math> denotes the space of càdlàg functions defined on the [[unit interval]] <math>[0,1]</math>.<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>
''skorokod space''也写为''Skorohod space''，是所有右连续左极限的函数的数学空间，定义在实线的某个区间上，例如<math>[0,1]</math>或<math>[0，\infty）</math>，取实线或度量空间上的值。<ref name="Whitt2006page78">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|pages=78–79}}</ref><ref name="GusakKukush2010page24">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 24</ref><ref name="Bogachev2007Vol2page53">{{cite book|author=Vladimir I. Bogachev|title=Measure Theory (Volume 2)|url=https://books.google.com/books?id=CoSIe7h5mTsC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-34514-5|page=53}}</ref>这些函数被称为cádLag或cadlag函数，这是基于法语表达式“continue a droite，limiteégauche”的首字母缩略词，因为这些函数是右连续的，具有左极限。<ref name="Whitt2006page78"/><ref name="Klebaner2005page4">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=4}}</ref>由[[Anatoliy Skorokod]]引入的Skorokod函数空间，<ref name="Bogachev2007Vol2page53"/>通常用字母<math>D</math>表示，<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/>因此函数空间也被称为空间<math>D</math><ref name="Whitt2006page78"/><ref name="Asmussen2003page420">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=420}}</ref><ref name="Billingsley2013page121">{{cite book|author=Patrick Billingsley|title=Convergence of Probability Measures|url=https://books.google.com/books?id=6ItqtwaWZZQC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-62596-5|page=121}}</ref>此函数空间的表示法还可以包括定义所有cádlág函数的间隔，因此，例如，<math>D[0,1]</math>表示在单位间隔<math>[0，1] </math>。<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>
在随机过程理论中，由于通常假定连续时间随机过程的样本函数属于一个Skorokod空间，<ref name="Bogachev2007Vol2page53"/><ref name="Asmussen2003page420"/>因此经常使用Skorokod函数空间，对应于Wiener过程的样本函数。但是空间也有间断函数，这意味着随机过程的样本函数具有跳跃性，例如泊松过程（在实线上），同时也是这一领域的成员。<ref name="Billingsley2013page121"/><ref name="BinghamKiesel2013page154">{{cite book|author1=Nicholas H. Bingham|author2=Rüdiger Kiesel|title=Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives|url=https://books.google.com/books?id=AOIlBQAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-3856-3|page=154}}</ref>
====规律性 Regularity====
在随机过程的数学构造中，当讨论和假设随机过程的某些条件以解决可能的构造问题时，使用术语“正则性”。<ref name="Borovkov2013page532">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=532}}</ref><ref name="Khoshnevisan2006page148to165">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=148–165}}</ref>例如，研究具有不可数索引集的随机过程，假设随机过程服从某种正则条件，例如样本函数是连续的。<ref name="Todorovic2012page22">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|page=22}}</ref><ref name="Whitt2006page79">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|page=79}}</ref>
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==Further examples更多示例==
==Further examples更多示例==