Editing Stochastic process (section)

===Further definitions===

====Law====
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]]:
<div class="center"><math>
\mu=P\circ X^{-1},
</math></div>
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>

For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives
<div class="center"><math>
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
</math></div>
so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>
<div class="center"><math>
\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).
</math></div>

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====
{{Main|Finite-dimensional distribution}}
For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distribution''' for <math>t_1,\dots,t_n\in T</math> is defined as:
<div class="center"><math>
\mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1},
</math></div>
This measure <math>\mu_{t_1,..,t_n}</math> is the joint distribution of the random vector <math>
(X({t_1}),\dots, X({t_n}))
</math>; it can be viewed as a "projection" of the law <math>\mu</math> onto a finite subset of <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>

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"/>
<div class="center"><math>
\mu_{t_1,\dots,t_n}(C) =P \Big(\big\{\omega\in \Omega: \big( X_{t_1}(\omega), \dots, X_{t_n}(\omega) \big) \in C \big\} \Big).
</math></div>
The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.<ref name="Rosenthal2006page177"/>

====Stationarity====
{{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
<div class="center"><math>
X_{t_1}, \dots X_{t_n},
</math></div>
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>

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"/>

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>

====Filtration====
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>

====Modification====
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>, state 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>X</math> if for all <math>t\in T</math> the following
<div class="center"><math>
P(X_t=Y_t)=1 ,
</math></div>
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|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>

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>

====Indistinguishable====
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
<div class="center"><math>
P(X_t=Y_t   \text{ for all }  t\in T )=1 ,
</math></div>
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>

====Separability====
'''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>

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|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 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"/>

====Independence====
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}}

====Uncorrelatedness====
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:

:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.

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

====Orthogonality====
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:

:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ orthogonal} \quad \iff \quad \operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.

====Skorokhod space====
{{Main|Skorokhod space}}
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 phrase ''continue à droite, limite à gauche''.<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>

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>

====Regularity====
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|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>