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第945行： − Lévy过程是随机过程的一种类型，可以看作是连续时间中随机游动的推广=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=Cambridge University Press | isbn=978-0-521-64632-1 | page=viii}}</ref>这些过程在金融、流体力学等领域有着广泛的应用，自同年以来，自本次自微积分|网址=https://books.google.com/books？id=q7eDUjdJxIkC | year=2004 | publisher=Cambridge University Press | isbn=978-0-521-83263-2 | page=69}</ref>这些过程的主要特征是其平稳性和独立性，因此被称为“具有平稳和独立增量的过程”。换句话说，一个随机过程<math>X</math>是一个Lévy过程，如果<math>n</math>非负数，<math>0\leq t_1\leq\dots\leq t_n</math>相应的<math>n-1</math>递增+
→马尔可夫过程与链Markov processes and chains
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math>n</math>-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.<ref name="Rozanov2012page61">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=61}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=27}}</ref><ref name="Bremaud2013page253">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=253}}</ref>
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math>n</math>-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.<ref name="Rozanov2012page61">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=61}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=27}}</ref><ref name="Bremaud2013page253">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=253}}</ref>
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。<ref name=“Rozanov2012page61”>{引用图书|作者=Y.A.Rozanov | title=Markov随机场 |网址=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>
==='''<font color="#ff8000">鞅Martingale</font>'''===
==='''<font color="#ff8000">鞅Martingale</font>'''===
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>
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>
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<ref name=“Williams1991page93”/><ref name=“Steele2012page11”>{cite book | author=J.Michael Steele | title=随机微积分和金融应用|网址=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=11}</ref></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>在给定鞅矩的条件下，鞅会收敛，因此经常使用鞅得到收敛结果，这主要是由于[[鞅收敛定理]]s。<ref name="HallHeyde2014page2"/><ref name="Steele2012page22">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=22}}</ref><ref name="GrimmettStirzaker2001page336">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=336}}</ref>
Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref> They have found applications in areas in probability theory such as queueing theory and Palm calculus<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref> and other fields such as economics<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/>
Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref> They have found applications in areas in probability theory such as queueing theory and Palm calculus<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref> and other fields such as economics<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/>
Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref> These processes have many applications in fields such as finance, fluid mechanics, physics and biology.<ref name="Applebaum2004page1336">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336}}</ref><ref name="ApplebaumBook2004page69">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=69}}</ref> The main defining characteristics of these processes are their stationarity and independence properties, so they were known as ''processes with stationary and independent increments''. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments
Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref> These processes have many applications in fields such as finance, fluid mechanics, physics and biology.<ref name="Applebaum2004page1336">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336}}</ref><ref name="ApplebaumBook2004page69">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=69}}</ref> The main defining characteristics of these processes are their stationarity and independence properties, so they were known as ''processes with stationary and independent increments''. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments
莱维Lévy过程是随机过程的一种类型，可以看作是连续时间中随机游动的推广<ref name=“Applebaum2004page1337”/><ref name=“Bertoin1998pageVIII”>{引用图书|作者=Jean Bertoin | title=莱维过程 |网址=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=Cambridge University Press | isbn=978-0-521-64632-1 | page=viii}}</ref>这些过程在金融、流体力学等领域有着广泛的应用，<ref name="Applebaum2004page1336">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336}}</ref><ref name="ApplebaumBook2004page69">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=69}}</ref> 这些过程和过程的独立性被称为平稳过程的主要特征。换句话说，一个随机过程<math>X</math>是一个Lévy过程，如果对非负数<math>n</math>，<math>0\leq t_1\leq \dots \leq t_n</math>，当<math>n-1</math>递增
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