更改

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"/>

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"/>

 

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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>

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====

====难以区分 Indistinguishable====