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马尔可夫特性的概念最初是针对连续和离散时间的随机过程，但它也适用于其它指标集，如<math>n</math>维欧氏空间，这导致随机变量的集合被称为马尔可夫随机场=https://books.google.com/books？id=wguecaaqbaj | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4613-8190-7 | page=61}</ref><ref>{cite book | author1=Donald L.Snyder | author2=Michael I.Miller | title=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=27}</ref><ref name=“bremaud2013 page253”>{cite book |作者=Pierre Bremaud | title=Markov Chains:Gibbs Fields，montecarlo Simulation，and Queues |网址=https://books.google.com/books？id=jrpvwwaaqbaj |年份=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4757-3124-8 | page=253}</ref>

马尔可夫特性的概念最初是针对连续和离散时间的随机过程，但它也适用于其它指标集，如<math>n</math>维欧氏空间，这导致随机变量的集合被称为马尔可夫随机场=https://books.google.com/books？id=wguecaaqbaj | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4613-8190-7 | page=61}</ref><ref>{cite book | author1=Donald L.Snyder | author2=Michael I.Miller | title=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=27}</ref><ref name=“bremaud2013 page253”>{cite book |作者=Pierre Bremaud | title=Markov Chains:Gibbs Fields，montecarlo Simulation，and Queues |网址=https://books.google.com/books？id=jrpvwwaaqbaj |年份=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4757-3124-8 | page=253}</ref>

===Martingale鞅===

==='''<font color="#ff8000">鞅Martingale</font>'''===

{{Main|Martingale (probability theory)}}

{{Main|Martingale (probability theory)}}

A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,<ref name="Klebaner2005page65">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=65}}</ref><ref name="KaratzasShreve2014page11">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=11}}</ref><ref name="Williams1991page93">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5|pages=93, 94}}</ref> but they can also be complex-valued<ref name="Doob1990page292">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref> or even more general.<ref name="Pisier2016">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref>

A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,<ref name="Klebaner2005page65">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=65}}</ref><ref name="KaratzasShreve2014page11">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=11}}</ref><ref name="Williams1991page93">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5|pages=93, 94}}</ref> but they can also be complex-valued<ref name="Doob1990page292">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref> or even more general.<ref name="Pisier2016">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref>

鞅是一个离散时间或连续时间的随机过程，其性质是在给定过程的当前值和所有过去值的情况下，每个未来值的条件期望值等于当前值。在离散时间中，如果此属性适用于下一个值，则它适用于所有未来值。鞅的精确数学定义需要另外两个条件与过滤的数学概念相结合，这与随时间推移增加可用信息的直觉有关。鞅通常被定义为实值，<ref name=“Klebaner2005page65”>{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=65}</ref><ref name=“KaratzasShreve2014page11”>{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=11}</ref><ref name=“Williams1991page93”>{引用图书|作者=David Williams | title=Probability with鞅| url=https://books.google.com/books？id=e9saZ0YSi AC | year=1991 | publisher=Cambridge University Press | isbn=978-0-521-40605-5 | pages=93，94}</ref>但是它们也可以是复杂值<ref name=“Doob1990page292”>{cite book | author=Joseph L.Doob | title=randouses | url=https://books.google.com/books？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=2929293}</ref>或更一般的。<ref name=“Pisier2016”>{cite book | author=Gilles Pisier | title=Banach空格中的鞅| url=https://books.google.com/books？id=n3JNDAAAQBAJ&pg=PR4 | year=2016 | publisher=Cambridge University Press | isbn=978-1-316-67946-3}</ref>

'''<font color="#ff8000">鞅Martingale</font>'''是一个离散时间或连续时间的随机过程，其性质是在给定过程的当前值和所有过去值的情况下，每个未来值的条件期望值等于当前值。在离散时间中，如果此属性适用于下一个值，则它适用于所有未来值。'''<font color="#ff8000">鞅Martingale</font>'''的精确数学定义需要另外两个条件与过滤的数学概念相结合，这与随时间推移增加可用信息的直觉有关。'''<font color="#ff8000">鞅Martingale</font>'''通常被定义为实值，<ref name=“Klebaner2005page65”>{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=65}</ref><ref name=“KaratzasShreve2014page11”>{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=11}</ref><ref name=“Williams1991page93”>{引用图书|作者=David Williams | title=Probability with鞅| url=https://books.google.com/books？id=e9saZ0YSi AC | year=1991 | publisher=Cambridge University Press | isbn=978-0-521-40605-5 | pages=93，94}</ref>但是它们也可以是复杂值<ref name=“Doob1990page292”>{cite book | author=Joseph L.Doob | title=randouses | url=https://books.google.com/books？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=2929293}</ref>或更一般的。<ref name=“Pisier2016”>{cite book | author=Gilles Pisier | title=Banach空格中的鞅| url=https://books.google.com/books？id=n3JNDAAAQBAJ&pg=PR4 | year=2016 | publisher=Cambridge University Press | isbn=978-1-316-67946-3}</ref>

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.<ref name="Klebaner2005page65"/><ref name="KaratzasShreve2014page11"/> For a [[sequence]] of [[independent and identically distributed]] random variables <math>X_1, X_2, X_3, \dots</math> with zero mean, the stochastic process formed from the successive partial sums <math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> is a discrete-time martingale.<ref name="Steele2012page12">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|pages=12, 13}}</ref> In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.<ref name="HallHeyde2014page2">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</ref>

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.<ref name="Klebaner2005page65"/><ref name="KaratzasShreve2014page11"/> For a [[sequence]] of [[independent and identically distributed]] random variables <math>X_1, X_2, X_3, \dots</math> with zero mean, the stochastic process formed from the successive partial sums <math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> is a discrete-time martingale.<ref name="Steele2012page12">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|pages=12, 13}}</ref> In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.<ref name="HallHeyde2014page2">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</ref>

对称随机游动和Wiener过程（具有零漂移）分别是离散时间和连续时间的鞅的例子。<ref name=“Klebaner2005page65”/><ref name=“KaratzasShreve2014page11”/>对于一个[[独立且同分布]]随机变量的[[sequence]]<math>X_1，X_2，X_3，dots</math>且平均值为零，由连续部分和构成的随机过程是一个离散时间鞅=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | pages=12，13}</ref>，离散时间鞅推广了独立随机变量的部分和的概念=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=2}</ref>

对称随机游动和Wiener过程（具有零漂移）分别是离散时间和连续时间的'''<font color="#ff8000">鞅Martingale</font>'''的例子。<ref name=“Klebaner2005page65”/><ref name=“KaratzasShreve2014page11”/>对于一个[[独立且同分布]]随机变量的[[序列]]<math>X_1, X_2, X_3, \dots</math> 且平均值为零，由连续部分和<math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> 构成的随机过程是一个离散时间'''<font color="#ff8000">鞅Martingale</font>'''<ref name="Steele2012page12">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|pages=12, 13}}</ref>，离散时间鞅推广了独立随机变量的部分和的概念。<ref name="HallHeyde2014page2">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</ref>

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the ''compensated Poisson process''.<ref name="KaratzasShreve2014page11"/> Martingales can also be built from other martingales.<ref name="Steele2012page12"/> For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.<ref name="Klebaner2005page65"/><ref name="Steele2012page115">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=115}}</ref>

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the ''compensated Poisson process''.<ref name="KaratzasShreve2014page11"/> Martingales can also be built from other martingales.<ref name="Steele2012page12"/> For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.<ref name="Klebaner2005page65"/><ref name="Steele2012page115">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=115}}</ref>

通过应用适当的变换，也可以从随机过程中产生鞅，这就是齐次泊松过程（在实线上）的情形，其结果是一个称为“补偿泊松过程”的鞅。<ref name=“karatzashreserve2014page11”/>也可以从其他鞅中构建鞅。<ref name=“Steele2012page12”/>例如，有基于鞅的鞅Wiener过程，形成连续时间鞅。<ref name=“Klebaner2005page65”/><ref name=“Steele2012page115”>{cite book | author=J.Michael Steele | title=随机微积分与金融应用=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=115}</ref>

通过应用适当的变换，也可以从随机过程中产生'''<font color="#ff8000">鞅Martingale</font>'''，这就是齐次泊松过程（在实线上）的情形，其结果是一个称为“补偿泊松过程”的鞅。<ref name=“karatzashreserve2014page11”/>也可以从其他鞅中构建鞅。<ref name=“Steele2012page12”/>例如，有基于鞅的鞅Wiener过程，形成连续时间鞅。<ref name=“Klebaner2005page65”/><ref name=“Steele2012page115”>{cite book | author=J.Michael Steele | title=随机微积分与金融应用=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=115}</ref>

Martingales mathematically formalize the idea of a fair game,<ref name="Ross1996page295">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=295}}</ref> and they were originally developed to show that it is not possible to win a fair game.<ref name="Steele2012page11"/> But now they are used in many areas of probability, which is one of the main reasons for studying them.<ref name="Williams1991page93"/><ref name="Steele2012page11">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=11}}</ref><ref name="Kallenberg2002page96">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=96}}</ref> Many problems in probability have been solved by finding a martingale in the problem and studying it.<ref name="Steele2012page371">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=371}}</ref> Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to [[martingale convergence theorem]]s.<ref name="HallHeyde2014page2"/><ref name="Steele2012page22">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=22}}</ref><ref name="GrimmettStirzaker2001page336">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=336}}</ref>

Martingales mathematically formalize the idea of a fair game,<ref name="Ross1996page295">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=295}}</ref> and they were originally developed to show that it is not possible to win a fair game.<ref name="Steele2012page11"/> But now they are used in many areas of probability, which is one of the main reasons for studying them.<ref name="Williams1991page93"/><ref name="Steele2012page11">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=11}}</ref><ref name="Kallenberg2002page96">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=96}}</ref> Many problems in probability have been solved by finding a martingale in the problem and studying it.<ref name="Steele2012page371">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=371}}</ref> Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to [[martingale convergence theorem]]s.<ref name="HallHeyde2014page2"/><ref name="Steele2012page22">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=22}}</ref><ref name="GrimmettStirzaker2001page336">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=336}}</ref>

鞅数学形式化了公平博弈的概念，<ref name=“Ross1996page295”>{cite book | author=Sheldon M.Ross | title=random processes | url=https://books.google.com/books？id=ImUPAQAAMAAJ | year=1996 | publisher=Wiley | isbn=978-0-471-12062-9 | page=295}</ref>它们最初的开发目的是表明不可能赢得一场公平的比赛。<ref name=“Steele2012page11”/>但现在它们被用于许多概率领域，这是研究它们的主要原因之一=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=11}</ref><ref name=“Kallenberg 2002page96”>{cite book |作者=Olav Kallenberg | title=Foundations of Modern Probability |网址=https://books.google.com/books？id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | pages=96}</ref>许多概率问题已经通过在问题中找到鞅并加以研究而得到解决。<ref name=“Steele2012page371”>{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=371}}</ref>鞅会收敛，因此通常使用它们来推导收敛结果，{124steelography=“2012steelography”{124steelography=“2012steelography”/>=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=22}</ref><ref name=“grimmetttstirzaker2001page336”>{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=2001 | publisher=OUP Oxford | isbn=978-0-19-857222-0 | page=336}</ref>

数学上的鞅形式化了公平博弈的概念，<ref name=“Ross1996page295”>{cite book | author=Sheldon M.Ross | title=random processes | url=https://books.google.com/books？id=ImUPAQAAMAAJ | year=1996 | publisher=Wiley | isbn=978-0-471-12062-9 | page=295}</ref>它们最初的开发目的是表明不可能赢得一场公平的比赛。<ref name=“Steele2012page11”/>但现在它们被用于许多概率领域，这是研究它们的主要原因之一=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=11}</ref><ref name=“Kallenberg 2002page96”>{cite book |作者=Olav Kallenberg | title=Foundations of Modern Probability |网址=https://books.google.com/books？id=L6fhXh13OyMC | year=2002 | publisher=Springer Science&Business Media | isbn=978-0-387-95313-7 | pages=96}</ref>许多概率问题已经通过在问题中找到鞅并加以研究而得到解决。<ref name=“Steele2012page371”>{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=371}}</ref>鞅会收敛，因此通常使用它们来推导收敛结果，{124steelography=“2012steelography”{124steelography=“2012steelography”/>=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4684-9305-4 | page=22}</ref><ref name=“grimmetttstirzaker2001page336”>{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=2001 | publisher=OUP Oxford | isbn=978-0-19-857222-0 | page=336}</ref>

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref> They have found applications in areas in probability theory such as queueing theory and Palm calculus<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref> and other fields such as economics<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/>

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref> They have found applications in areas in probability theory such as queueing theory and Palm calculus<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref> and other fields such as economics<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/>

鞅在统计学中有许多应用，但有人指出，它的使用和应用并不像它在统计学领域那样广泛，尤其是统计推断，293 | issn=0883-4237 | doi=10.1214/088342306000000088 | arxiv=math/0609294 | bibcode=2006math……9294G}</ref>他们在排队论和棕榈微积分等概率论领域找到了应用微积分与随机循环=https://books.google.com/books？id=dh3pcaaqbaj&pg=PR2 | year=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-11657-9}</ref>和其他领域，如经济学<ref name=“HallHeyde2014pageX”{cite book | author1=P.Hall | author2=C.C.Heyde | title=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 | year=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=x}</ref>和财务。<ref name=“Musielarukowski2006”/>

'''<font color="#ff8000">鞅Martingale</font>'''在统计学中有许多应用，但有人指出，它的使用和应用并不像它在统计学领域那样广泛，尤其是统计推断，293 | issn=0883-4237 | doi=10.1214/088342306000000088 | arxiv=math/0609294 | bibcode=2006math……9294G}</ref>他们在排队论和棕榈微积分等概率论领域找到了应用微积分与随机循环=https://books.google.com/books？id=dh3pcaaqbaj&pg=PR2 | year=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-11657-9}</ref>和其他领域，如经济学<ref name=“HallHeyde2014pageX”{cite book | author1=P.Hall | author2=C.C.Heyde | title=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 | year=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=x}</ref>和财务。<ref name=“Musielarukowski2006”/>

===Lévy process莱维过程===

===Lévy process莱维过程===