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Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments

鞅在统计学中有许多应用，但有人指出，鞅的使用和应用并不象在统计学领域，特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性，因此它们被称为具有平稳增量和独立增量的过程。换句话说，如果对于 < math > n </math > 非负数，< math > 0 leq t 1 leq dots leq t n </math > ，相应的 < math > n-1 </math > 递增值是一个 Lévy 过程

'''<font color="#ff8000"> 鞅Martingales</font>'''在统计学中有许多应用，但有人指出，鞅的使用和应用并不象在统计学领域，特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性，因此它们被称为具有平稳增量和独立增量的过程。换句话说，如果对于 <math>n</math> 非负数，<math>0\leq t_1\leq \dots \leq t_n</math> ，相应的 <math>n-1</math> 递增值是一个列维 Lévy 过程

A [[Filtration (probability theory)|filtration]] is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some [[total order]] relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math>\{\mathcal{F}_t\}_{t\in T} </math>, on a probability space <math>(\Omega, \mathcal{F}, P)</math> is a family of sigma-algebras such that <math>  \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq  \mathcal{F} </math> for all <math>s \leq t</math>, where <math>t, s\in T</math> and <math>\leq</math> denotes the total order of the index set <math>T</math>.<ref name="Florescu2014page294"/> With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process <math>X_t</math> at <math>t\in T</math>, which can be interpreted as time <math>t</math>.<ref name="Florescu2014page294"/><ref name="Williams1991page93"/> The intuition behind a filtration <math>\mathcal{F}_t</math> is that as time <math>t</math> passes, more and more information on <math>X_t</math> is known or available, which is captured in <math>\mathcal{F}_t</math>, resulting in finer and finer partitions of <math>\Omega</math>.<ref name="Klebaner2005page22">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|pages=22–23}}</ref><ref name="MörtersPeres2010page37">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=37}}</ref>

A [[Filtration (probability theory)|filtration]] is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some [[total order]] relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math>\{\mathcal{F}_t\}_{t\in T} </math>, on a probability space <math>(\Omega, \mathcal{F}, P)</math> is a family of sigma-algebras such that <math>  \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq  \mathcal{F} </math> for all <math>s \leq t</math>, where <math>t, s\in T</math> and <math>\leq</math> denotes the total order of the index set <math>T</math>.<ref name="Florescu2014page294"/> With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process <math>X_t</math> at <math>t\in T</math>, which can be interpreted as time <math>t</math>.<ref name="Florescu2014page294"/><ref name="Williams1991page93"/> The intuition behind a filtration <math>\mathcal{F}_t</math> is that as time <math>t</math> passes, more and more information on <math>X_t</math> is known or available, which is captured in <math>\mathcal{F}_t</math>, resulting in finer and finer partitions of <math>\Omega</math>.<ref name="Klebaner2005page22">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|pages=22–23}}</ref><ref name="MörtersPeres2010page37">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=37}}</ref>

<center><math>

<center><math>

2}-x _ { t _ 1} ，点，x _ { t _ { n-1}-x _ { t _ n } ,

2}-x _ { t _ 1} ，点，x _ { t _ { n-1}-x _ { t _ n } ,

====Modification调整====

====Modification修正====

</math></center>

</math></center>

A '''modification''' of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process <math>X</math> that has the same index set <math>T</math>, set space <math>S</math>, and probability space <math>(\Omega,{\cal F},P)</math> as another stochastic process <math>Y</math> is said to be a modification of <math>Y</math> if for all <math>t\in T</math> the following

A '''modification''' of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process <math>X</math> that has the same index set <math>T</math>, set space <math>S</math>, and probability space <math>(\Omega,{\cal F},P)</math> as another stochastic process <math>Y</math> is said to be a modification of <math>Y</math> if for all <math>t\in T</math> the following

“随机过程”是另一个与随机过程密切相关的随机过程。更确切地说，一个随机过程<math>X</math>具有相同的索引集<math>T</math>、集空间<math>和概率空间<math>（\Omega，{\cal F}，P）</math>作为另一个随机过程<math>Y</math>的随机过程被称为<math>Y</math>的修改，如果T</math>中的所有<math>T\

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

 

are all independent of each other, and the distribution of each increment only depends on the difference in time. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.

are all independent of each other, and the distribution of each increment only depends on the difference in time. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.

holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law<ref name="RogersWilliams2000page130">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=130}}</ref> and they are said to be '''stochastically equivalent''' or '''equivalent'''.<ref name="Borovkov2013page530">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>

holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law<ref name="RogersWilliams2000page130">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=130}}</ref> and they are said to be '''stochastically equivalent''' or '''equivalent'''.<ref name="Borovkov2013page530">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>

持有。两个相互修正的随机过程具有相同的有限维定律=图书https://books.com/？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”=图书https://books.com/？id=hRk_AAAAQBAJ&pg | year=2013 | publisher=Springer科学与商业媒体| isbn=978-1-4471-5201-9 | page=530}</ref>

注意。两个相互修正的随机过程具有相同的有限维法则<ref name="RogersWilliams2000page130">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=130}}</ref>它们被称为“随机等价”或“等价物”<ref name="Borovkov2013page530">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>

 

A point process is a collection of points randomly located on some mathematical space such as the real line, <math>n</math>-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or <math>n</math>-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.

A point process is a collection of points randomly located on some mathematical space such as the real line, <math>n</math>-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or <math>n</math>-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.

点过程是一个点的集合，这些点随机地分布在一些数学空间上，比如实数直线、 n 维欧氏空间或者更多的抽象空间。有时，词汇点过程并不是首选，因为历史上词汇过程表示某个系统在时间上的演变，所以点过程也称为随机点场。一个点过程有不同的解释，比如随机计数测度或随机集合。有些作者把点过程和随机过程过程看作是两个不同的对象，例如，点过程是一个随机的对象，它起源于或与随机过程过程相关联，尽管有人指出点过程和随机过程之间的区别并不清楚。它对应于随机过程术语中的索引集。}它被定义在其上，例如实线或者 < math > n </math > 维欧氏空间。在点过程理论中研究了更新和计数过程等其他随机过程。

点过程是一个点的集合，这些点随机地分布在一些数学空间上，比如实数直线、 n 维欧氏空间或者更多的抽象空间。有时，词汇点过程并不是首选，因为历史上词汇过程表示某个系统在时间上的演变，所以点过程也称为随机点场。一个点过程有不同的解释，比如随机计数测度或随机集合。有些作者把点过程和随机过程过程看作是两个不同的对象，例如，点过程是一个随机的对象，它起源于或与随机过程过程相关联，尽管有人指出点过程和随机过程之间的区别并不清楚。它对应于随机过程术语中的索引集。}它被定义在其上，例如实线或者<math>n</math> 维欧氏空间。在点过程理论中研究了更新和计数过程等其他随机过程。