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[数学中心]

[数学中心]

==Definitions==

==Definitions定义==

so the law of a <math>X</math> can be written as:

so the law of a <math>X</math> can be written as:

===Stochastic process===

===Stochastic process随机过程===

A stochastic process is defined as a collection of random variables defined on a common [[probability space]] <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a [[sample space]], <math>\mathcal{F}</math> is a <math>\sigma</math>-[[Sigma-algebra|algebra]], and <math>P</math> is a [[probability measure]]; and the random variables, indexed by some set <math>T</math>, all take values in the same mathematical space <math>S</math>, which must be [[measurable]] with respect to some <math>\sigma</math>-algebra <math>\Sigma</math>.<ref name="Lamperti1977page1"/>

A stochastic process is defined as a collection of random variables defined on a common [[probability space]] <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a [[sample space]], <math>\mathcal{F}</math> is a <math>\sigma</math>-[[Sigma-algebra|algebra]], and <math>P</math> is a [[probability measure]]; and the random variables, indexed by some set <math>T</math>, all take values in the same mathematical space <math>S</math>, which must be [[measurable]] with respect to some <math>\sigma</math>-algebra <math>\Sigma</math>.<ref name="Lamperti1977page1"/>

 

For a stochastic process <math>X</math> with law <math>\mu</math>, its finite-dimensional distributions are defined as:

For a stochastic process <math>X</math> with law <math>\mu</math>, its finite-dimensional distributions are defined as:

In other words, for a given probability space <math>(\Omega, \mathcal{F}, P)</math> and a measurable space <math>(S,\Sigma)</math>, a stochastic process is a collection of <math>S</math>-valued random variables, which can be written as:<ref name="Florescu2014page293">{{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|page=293}}</ref>

In other words, for a given probability space <math>(\Omega, \mathcal{F}, P)</math> and a measurable space <math>(S,\Sigma)</math>, a stochastic process is a collection of <math>S</math>-valued random variables, which can be written as:<ref name="Florescu2014page293">{{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|page=293}}</ref>

<center><math>

<center><math>

Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>.<ref name="Borovkov2013page528">{{cite book|author=Alexander A. Borovkov|authorlink=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=528}}</ref> A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.<ref name="Lamperti1977page1"/><ref name="LindgrenRootzen2013page11">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>

Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>.<ref name="Borovkov2013page528">{{cite book|author=Alexander A. Borovkov|authorlink=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=528}}</ref> A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.<ref name="Lamperti1977page1"/><ref name="LindgrenRootzen2013page11">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>

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: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.

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: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.

There are other ways to consider a stochastic process, with the above definition being considered the traditional one.<ref name="RogersWilliams2000page121">{{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=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=121, 122}}</ref><ref name="Asmussen2003page408">{{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=408}}</ref> For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued [[Function (mathematics)|functions]] of <math>t\in T</math> that [[Map (mathematics)|map]] from the set <math>T</math> into the space <math>S</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/>

There are other ways to consider a stochastic process, with the above definition being considered the traditional one.<ref name="RogersWilliams2000page121">{{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=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=121, 122}}</ref><ref name="Asmussen2003page408">{{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=408}}</ref> For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued [[Function (mathematics)|functions]] of <math>t\in T</math> that [[Map (mathematics)|map]] from the set <math>T</math> into the space <math>S</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/>

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. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.

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. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.

===Index set===

===Index set索引集===

The set <math>T</math> is called the '''index set'''<ref name="Parzen1999"/><ref name="Florescu2014page294"/> or '''parameter set'''<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|pages=93, 94}}</ref> of the stochastic process. Often this set is some subset of the [[real line]], such as the [[natural numbers]] or an interval, giving the set <math>T</math> the interpretation of time.<ref name="doob1953stochasticP46to47"/> In addition to these sets, the index set <math>T</math> can be other linearly ordered sets or more general mathematical sets,<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{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|page=482}}</ref> such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space.<ref name="KarlinTaylor2012page27">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=27}}</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=25}}</ref> But in general more results and theorems are possible for stochastic processes when the index set is ordered.<ref name="Skorokhod2005page104">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=104}}</ref>

The set <math>T</math> is called the '''index set'''<ref name="Parzen1999"/><ref name="Florescu2014page294"/> or '''parameter set'''<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|pages=93, 94}}</ref> of the stochastic process. Often this set is some subset of the [[real line]], such as the [[natural numbers]] or an interval, giving the set <math>T</math> the interpretation of time.<ref name="doob1953stochasticP46to47"/> In addition to these sets, the index set <math>T</math> can be other linearly ordered sets or more general mathematical sets,<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{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|page=482}}</ref> such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space.<ref name="KarlinTaylor2012page27">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=27}}</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=25}}</ref> But in general more results and theorems are possible for stochastic processes when the index set is ordered.<ref name="Skorokhod2005page104">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=104}}</ref>

A 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>.

A 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>.

=== State space ===

=== State space 状态空间===

The [[mathematical space]] <math>S</math> of a stochastic process is called its ''state space''. This mathematical space can be defined using [[integer]]s, [[real line]]s, <math>n</math>-dimensional [[Euclidean space]]s, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="Florescu2014page294">{{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=294, 295}}</ref><ref name="Brémaud2014page120">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=120}}</ref>

The [[mathematical space]] <math>S</math> of a stochastic process is called its ''state space''. This mathematical space can be defined using [[integer]]s, [[real line]]s, <math>n</math>-dimensional [[Euclidean space]]s, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="Florescu2014page294">{{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=294, 295}}</ref><ref name="Brémaud2014page120">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=120}}</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

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

 

< 中心 > < 数学 >

< 中心 > < 数学 >

===Sample function===

===Sample function样本函数===

P(X_t=Y_t)=1 ,

P(X_t=Y_t)=1 ,

A '''sample function''' is a single [[Outcome (probability)|outcome]] of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.<ref name="Lamperti1977page1"/><ref name="Florescu2014page296">{{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|page=296}}</ref> More precisely, if <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, the [[Map (mathematics)|mapping]]

A '''sample function''' is a single [[Outcome (probability)|outcome]] of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.<ref name="Lamperti1977page1"/><ref name="Florescu2014page296">{{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|page=296}}</ref> More precisely, if <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, the [[Map (mathematics)|mapping]]

</math></center>

</math></center>

Instead of modification, the term version is also used, 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. 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.

Instead of modification, the term version is also used, 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. 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.

不同的概率空间可以定义不同的两个随机过程，因此两个相互修正的过程，在后一种意义上也是相互修正的过程，但不是相反。这个定理也可以推广到随机场，因此指数集是 < math > n </math > 维欧氏空间，以及以度量空间为状态空间的随机过程。

不同的概率空间可以定义不同的两个随机过程，因此两个相互修正的过程，在后一种意义上也是相互修正的过程，但不是相反。这个定理也可以推广到随机场，因此指数集是 < math > n </math > 维欧氏空间，以及以度量空间为状态空间的随机过程。

is called a sample function, a '''realization''', or, particularly when <math>T</math> is interpreted as time, a '''sample path''' of the stochastic process <math>\{X(t,\omega):t\in T \}</math>.<ref name="RogersWilliams2000page121b">{{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=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=121–124}}</ref> This means that for a fixed <math>\omega\in\Omega</math>, there exists a sample function that maps the index set <math>T</math> to the state space <math>S</math>.<ref name="Lamperti1977page1"/> Other names for a sample function of a stochastic process include '''trajectory''', '''path function'''<ref name="Billingsley2008page493">{{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|page=493}}</ref> or '''path'''.<ref name="Øksendal2003page10">{{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=10}}</ref>

is called a sample function, a '''realization''', or, particularly when <math>T</math> is interpreted as time, a '''sample path''' of the stochastic process <math>\{X(t,\omega):t\in T \}</math>.<ref name="RogersWilliams2000page121b">{{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=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=121–124}}</ref> This means that for a fixed <math>\omega\in\Omega</math>, there exists a sample function that maps the index set <math>T</math> to the state space <math>S</math>.<ref name="Lamperti1977page1"/> Other names for a sample function of a stochastic process include '''trajectory''', '''path function'''<ref name="Billingsley2008page493">{{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|page=493}}</ref> or '''path'''.<ref name="Øksendal2003page10">{{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=10}}</ref>

===Increment===

===Increment增量===

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

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

An '''increment''' of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if <math>\{X(t):t\in T \}</math> is a stochastic process with state space <math>S</math> and index set <math>T=[0,\infty)</math>, then for any two non-negative numbers <math>t_1\in [0,\infty)</math> and <math>t_2\in [0,\infty)</math> such that <math>t_1\leq t_2</math>, the difference <math>X_{t_2}-X_{t_1}</math> is a <math>S</math>-valued random variable known as an increment.<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/> When interested in the increments, often the state space <math>S</math> is the real line or the natural numbers, but it can be <math>n</math>-dimensional Euclidean space or more abstract spaces such as [[Banach space]]s.<ref name="Applebaum2004page1337"/>

An '''increment''' of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if <math>\{X(t):t\in T \}</math> is a stochastic process with state space <math>S</math> and index set <math>T=[0,\infty)</math>, then for any two non-negative numbers <math>t_1\in [0,\infty)</math> and <math>t_2\in [0,\infty)</math> such that <math>t_1\leq t_2</math>, the difference <math>X_{t_2}-X_{t_1}</math> is a <math>S</math>-valued random variable known as an increment.<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/> When interested in the increments, often the state space <math>S</math> is the real line or the natural numbers, but it can be <math>n</math>-dimensional Euclidean space or more abstract spaces such as [[Banach space]]s.<ref name="Applebaum2004page1337"/>

<center><math>

<center><math>

持有。

持有。

====Law====

====Law定律====

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]]:

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]]:

<center><math>

<center><math>

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.

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.

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>

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>

The definition of separability can also be stated for other index sets and state spaces, 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. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

The definition of separability can also be stated for other index sets and state spaces, 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. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives

For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives

<center><math>

<center><math>

so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>

so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>

<center><math>

<center><math>

If two stochastic processes <math>X</math> and <math>Y</math> are independent, then they are also uncorrelated. 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, 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>.

If two stochastic processes <math>X</math> and <math>Y</math> are independent, then they are also uncorrelated. 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, 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>.

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。这种函数称为 càdlàg 或 cadlag 函数，由法语表达式 continue à droite，limite à gauche 的首字母缩写而来，因为这些函数是右连续的，有左限制。由 Anatoliy Skorokhod 引入的 Skorokhod 函数空间，这个函数空间的符号也可以包括定义所有函数的区间，因此，例如，< math > d [0,1] </math > 表示在单位区间 < math > [0,1] </math > 上定义的函数的空间。

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。这种函数称为 càdlàg 或 cadlag 函数，由法语表达式 continue à droite，limite à gauche 的首字母缩写而来，因为这些函数是右连续的，有左限制。由 Anatoliy Skorokhod 引入的 Skorokhod 函数空间，这个函数空间的符号也可以包括定义所有函数的区间，因此，例如，< math > d [0,1] </math > 表示在单位区间 < math > [0,1] </math > 上定义的函数的空间。

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>

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>

====Finite-dimensional probability distributions====

 

====Finite-dimensional probability distributions有限维概率分布====

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. 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.

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. 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.

{{Main|Finite-dimensional distribution}}

{{Main|Finite-dimensional distribution}}

For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distributions''' are defined as:

For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distributions''' are defined as:

<center><math>

<center><math>

where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <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>

where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <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>

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.

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"/>

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"/>

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.

The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.<ref name="Rosenthal2006page177"/>

The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.<ref name="Rosenthal2006page177"/>

Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.

Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.

====Stationarity====

====Stationarity稳定性====

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.

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.

{{Main|Stationary process}}

{{Main|Stationary process}}

'''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

'''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

<center><math>

<center><math>

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>

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>

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

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"/>

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"/>

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.

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.

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>

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>

Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.

Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.

====Filtration====

====Filtration过滤====

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

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====

====Modification调整====

</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

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>

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.

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"/>

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"/>

 

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>

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>

概率论游戏起源于机会游戏，这种游戏有着悠久的历史，有些游戏在几千年前就已经开始玩了，但是很少从概率的角度对它们进行分析。1654年通常被认为是概率论的诞生，当时法国数学家 Pierre Fermat 和 Blaise Pascal 因为一个赌博问题写了一封关于概率的信。但是在赌博游戏的可能性方面，早期的数学工作已经完成，比如吉罗拉莫·卡尔达诺的 Liber de Ludo Aleae，写于16世纪，但死后于1663年出版。

概率论游戏起源于机会游戏，这种游戏有着悠久的历史，有些游戏在几千年前就已经开始玩了，但是很少从概率的角度对它们进行分析。1654年通常被认为是概率论的诞生，当时法国数学家 Pierre Fermat 和 Blaise Pascal 因为一个赌博问题写了一封关于概率的信。但是在赌博游戏的可能性方面，早期的数学工作已经完成，比如吉罗拉莫·卡尔达诺的 Liber de Ludo Aleae，写于16世纪，但死后于1663年出版。

====Indistinguishable====

====Indistinguishable无法分辨的====

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

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

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.

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.

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>

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>

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.

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.

====Separability====

====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>

'''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>

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.

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.

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>

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>

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"/>

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"/>

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

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

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"/>

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"/>

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.

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.

====Independence====

====Independence独立性====

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.

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.

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}}

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}}

 

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.

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年杜布出版了《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。

1953年杜布出版了《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====

====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:

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:

The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.

The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.

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。

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。

====Independence implies uncorrelatedness====

====Independence implies uncorrelatedness独立意味着不相关====

If two stochastic processes <math>X</math> and <math>Y</math> are independent, then they are also uncorrelated.<ref name=KunIlPark/>{{rp|p. 151}}

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.

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。

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。

====Orthogonality====

====Orthogonality正交性====

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:

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:

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.

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.

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。

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====

====Skorokhod space斯科罗霍德空间====

{{Main|Skorokhod space}}

{{Main|Skorokhod space}}

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

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

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>

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>

process.

process.

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>

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>

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.

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.

====Regularity====

====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.

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.

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>

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>