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Stochastic Processes

Field: Probability & Statistics

Mathematical objects usually defined as families of random variables.

Sequence of Expressions

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or nn -dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is nn -dimensional Euclidean space, then the stochastic process is called a nn -dimensional vector process or nn -vector process. The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin, though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931. According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888. The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. The terms random process and stochastic process are considered synonyms and are used interchangeably, without the index set being precisely specified. Both "collection", or "family" are used while instead of "index set", sometimes the terms "parameter set" or "parameter space" are used. The term random function is also used to refer to a stochastic or random process, though sometimes it is only used when the stochastic process takes real values. This term is also used when the index sets are mathematical spaces other than the real line, while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is nn -dimensional Euclidean space Rn\mathbb {R} ^{n} or a manifold. A stochastic process can be denoted, among other ways, by {X(t)}t∈T\{X(t)\}_{t\in T} , {Xt}t∈T\{X_{t}\}_{t\in T} , {Xt}\{X_{t}\} {X(t)}\{X(t)\} or simply as XX . Some authors mistakenly write X(t)X(t) even though it is an abuse of function notation. For example, X(t)X(t) or XtX_{t} are used to refer to the random variable with the index tt , and not the entire stochastic process. If the index set is T=[0,∞)T=[0,\infty ) , then one can write, for example, (Xt,t≥0)(X_{t},t\geq 0) to denote the stochastic process. - ^ ^{a}^{b}Cite error: The named reference Parzen1999 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}^{d}^{e}^{f}Cite error: The named reference GikhmanSkorokhod1969page1 was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference doob1953stochasticP46to47 was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference KarlinTaylor2012page27 was invoked but never defined (see the help page). - ^Cite error: The named reference Applebaum2004page1337 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}^{d}^{e}Cite error: The named reference Lamperti1977page1 was invoked but never defined (see the help page). - ^Cite error: The named reference RogersWilliams2000page121b was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference Florescu2014page294 was invoked but never defined (see the help page). - ^ ^{a}^{b}Samuel Karlin; Howard E. Taylor (2012). A First Course in Stochastic Processes. Academic Press. p. 26. ISBN978-0-08-057041-9. - ^Donald L. Snyder; Michael I. Miller (2012). Random Point Processes in Time and Space. Springer Science & Business Media. pp. 24, 25. ISBN978-1-4612-3166-0. - ^Cite error: The named reference Billingsley2008page482 was invoked but never defined (see the help page). - ^ ^{a}^{b}Alexander A. Borovkov (2013). Probability Theory. Springer Science & Business Media. p. 527. ISBN978-1-4471-5201-9. - ^ ^{a}^{b}Cite error: The named reference Brémaud2014page120 was invoked but never defined (see the help page). - ^Jeffrey S Rosenthal (2006). A First Look at Rigorous Probability Theory. World Scientific Publishing Co Inc. pp. 177–178. ISBN978-981-310-165-4. - ^Peter E. Kloeden; Eckhard Platen (2013). Numerical Solution of Stochastic Differential Equations. Springer Science & Business Media. p. 63. ISBN978-3-662-12616-5. - ^Davar Khoshnevisan (2006). Multiparameter Processes: An Introduction to Random Fields. Springer Science & Business Media. pp. 153–155. ISBN978-0-387-21631-7. - ^ ^{a}^{b}"Stochastic". Oxford English Dictionary (Online ed.). Oxford University Press.(Subscription or participating institution membership required.) - ^O. B. Sheĭnin (2006). Theory of probability and statistics as exemplified in short dictums. NG Verlag. p. 5. ISBN978-3-938417-40-9. - ^Oscar Sheynin; Heinrich Strecker (2011). Alexandr A. Chuprov: Life, Work, Correspondence. V&R unipress GmbH. p. 136. ISBN978-3-89971-812-6. - ^Cite error: The named reference Doob1934 was invoked but never defined (see the help page). - ^Khintchine, A. (1934). "Korrelationstheorie der stationeren stochastischen Prozesse". Mathematische Annalen. 109 (1): 604–615. doi:10.1007/BF01449156. ISSN0025-5831. S2CID122842868. - ^Kolmogoroff, A. (1931). "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung". Mathematische Annalen. 104 (1): 1. doi:10.1007/BF01457949. ISSN0025-5831. S2CID119439925. - ^"Random". Oxford English Dictionary (Online ed.). Oxford University Press.(Subscription or participating institution membership required.) - ^Bert E. Fristedt; Lawrence F. Gray (2013). A Modern Approach to Probability Theory. Springer Science & Business Media. p. 580. ISBN978-1-4899-2837-5. - ^Cite error: The named reference RogersWilliams2000page121 was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference Asmussen2003page408 was invoked but never defined (see the help page). - ^Cite error: The named reference Kallenberg2002page24 was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference ChaumontYor2012 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}Cite error: The named reference AdlerTaylor2009page7 was invoked but never defined (see the help page). - ^ ^{a}^{b}David Stirzaker (2005). Stochastic Processes and Models. Oxford University Press. p. 45. ISBN978-0-19-856814-8. - ^Murray Rosenblatt (1962). Random Processes. Oxford University Press. p. 91. - ^John A. Gubner (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. p. 383. ISBN978-1-139-45717-0. - ^ ^{a}^{b}Kiyosi Itō (2006). Essentials of Stochastic Processes. American Mathematical Soc. p. 13. ISBN978-0-8218-3898-3. - ^M. Loève (1978). Probability Theory II. Springer Science & Business Media. p. 163. ISBN978-0-387-90262-3. - ^Pierre Brémaud (2014). Fourier Analysis and Stochastic Processes. Springer. p. 133. ISBN978-3-319-09590-5. - ^ ^{a}^{b}Gusak et al. (2010), p. 1 - ^Richard F. Bass (2011). Stochastic Processes. Cambridge University Press. p. 1. ISBN978-1-139-50147-7. - ^ ^{a}^{b},John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. p. 3. ISBN978-3-540-90275-1. - ^Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 55. ISBN978-1-86094-555-7.
Definition

Definitions

A stochastic process is defined as a collection of random variables defined on a common probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) , where Ω\Omega is a sample space, F{\mathcal {F}} is a σ\sigma -algebra, and PP is a probability measure; and the random variables, indexed by some set TT , all take values in the same mathematical space SS , which must be measurable with respect to some σ\sigma -algebra Σ\Sigma . In other words, for a given probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) and a measurable space (S,Σ)(S,\Sigma ) , a stochastic process is a collection of SS -valued random variables, which can be written as: {X(t):t∈T}.\{X(t):t\in T\}. Historically, in many problems from the natural sciences a point t∈Tt\in T had the meaning of time, so X(t)X(t) is a random variable representing a value observed at time tt . A stochastic process can also be written as {X(t,ω):t∈T}\{X(t,\omega ):t\in T\} to reflect that it is actually a function of two variables, t∈Tt\in T and ω∈Ω\omega \in \Omega . There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a STS^{T} -valued random variable, where STS^{T} is the space of all the possible functions from the set TT into the space SS . However this alternative definition as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined. The set TT is called the index set or parameter set 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 TT the interpretation of time. In addition to these sets, the index set TT can be another set with a total order or a more general set, such as the Cartesian plane R2\mathbb {R} ^{2} or nn -dimensional Euclidean space, where an element t∈Tt\in T can represent a point in space. That said, many results and theorems are only possible for stochastic processes with a totally ordered index set. The mathematical space SS of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, nn -dimensional Euclidean spaces, 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. A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if {X(t,ω):t∈T}\{X(t,\omega ):t\in T\} is a stochastic process, then for any point ω∈Ω\omega \in \Omega , the mapping X(⋅,ω):T→S,X(\cdot ,\omega ):T\rightarrow S, is called a sample function, a realization, or, particularly when TT is interpreted as time, a sample path of the stochastic process {X(t,ω):t∈T}\{X(t,\omega ):t\in T\} . This means that for a fixed ω∈Ω\omega \in \Omega , there exists a sample function that maps the index set TT to the state space SS . Other names for a sample function of a stochastic process include trajectory, path function or path. 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 {X(t):t∈T}\{X(t):t\in T\} is a stochastic process with state space SS and index set T=[0,∞)T=[0,\infty ) , then for any two non-negative numbers t1∈[0,∞)t_{1}\in [0,\infty ) and t2∈[0,∞)t_{2}\in [0,\infty ) such that t1≤t2t_{1}\leq t_{2} , the difference Xt2−Xt1X_{t_{2}}-X_{t_{1}} is a SS -valued random variable known as an increment. When interested in the increments, often the state space SS is the real line or the natural numbers, but it can be nn -dimensional Euclidean space or more abstract spaces such as Banach spaces. For a stochastic process X ⁣:Ω→STX\colon \Omega \rightarrow S^{T} defined on the probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) , the law of stochastic process XX is defined as the pushforward measure: μ=P∘X−1,\mu =P\circ X^{-1}, where PP is a probability measure, the symbol ∘\circ denotes function composition and X−1X^{-1} is the pre-image of the measurable function or, equivalently, the STS^{T} -valued random variable XX , where STS^{T} is the space of all the possible SS -valued functions of t∈Tt\in T , so the law of a stochastic process is a probability measure. For a measurable subset BB of STS^{T} , the pre-image of XX gives X−1(B)={ω∈Ω:X(ω)∈B},X^{-1}(B)=\{\omega \in \Omega :X(\omega )\in B\}, so the law of a XX can be written as: μ(B)=P({ω∈Ω:X(ω)∈B}).\mu (B)=P(\{\omega \in \Omega :X(\omega )\in B\}). The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution. For a stochastic process XX with law μ\mu , its finite-dimensional distribution for t1,…,tn∈Tt_{1},\dots ,t_{n}\in T is defined as: μt1,…,tn=P∘(X(t1),…,X(tn))−1,\mu _{t_{1},\dots ,t_{n}}=P\circ (X({t_{1}}),\dots ,X({t_{n}}))^{-1}, This measure μt1,..,tn\mu _{t_{1},..,t_{n}} is the joint distribution of the random vector (X(t1),…,X(tn))(X({t_{1}}),\dots ,X({t_{n}})) ; it can be viewed as a "projection" of the law μ\mu onto a finite subset of TT . For any measurable subset CC of the nn -fold Cartesian power Sn=S×⋯×SS^{n}=S\times \dots \times S , the finite-dimensional distributions of a stochastic process XX can be written as: μt1,…,tn(C)=P({ω∈Ω:(Xt1(ω),…,Xtn(ω))∈C}).\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 )}. The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions. 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 XX is a stationary stochastic process, then for any t∈Tt\in T the random variable XtX_{t} has the same distribution, which means that for any set of nn index set values t1,…,tnt_{1},\dots ,t_{n} , the corresponding nn random variables Xt1,…Xtn,X_{t_{1}},\dots X_{t_{n}}, 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. But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time. When the index set TT 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. The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. 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 XX is said to be stationary in the wide sense, then the process XX has a finite second moment for all t∈Tt\in T and the covariance of the two random variables XtX_{t} and Xt+hX_{t+h} depends only on the number hh for all t∈Tt\in T .Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense. 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 {Ft}t∈T\{{\mathcal {F}}_{t}\}_{t\in T} , on a probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) is a family of sigma-algebras such that Fs⊆Ft⊆F{\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}\subseteq {\mathcal {F}} for all s≤ts\leq t , where t,s∈Tt,s\in T and ≤\leq denotes the total order of the index set TT . With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process XtX_{t} at t∈Tt\in T , which can be interpreted as time tt . The intuition behind a filtration Ft{\mathcal {F}}_{t} is that as time tt passes, more and more information on XtX_{t} is known or available, which is captured in Ft{\mathcal {F}}_{t} , resulting in finer and finer partitions of Ω\Omega . A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process XX that has the same index set TT , state space SS , and probability space (Ω,F,P)(\Omega ,{\cal {F}},P) as another stochastic process YY is said to be a modification of XX if for all t∈Tt\in T the following P(Xt=Yt)=1,P(X_{t}=Y_{t})=1, holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be stochastically equivalent or equivalent. 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. 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. The theorem can also be generalized to random fields so the index set is nn -dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces. Two stochastic processes XX and YY defined on the same probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) with the same index set TT and set space SS are said be indistinguishable if the following P(Xt=Yt for all t∈T)=1,P(X_{t}=Y_{t}{\text{ for all }}t\in T)=1, holds. If two XX and YY are modifications of each other and are almost surely continuous, then XX and YY are indistinguishable. 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, which means that the index set has a dense countable subset. More precisely, a real-valued continuous-time stochastic process XX on a probability space (Ω,F,P)(\Omega ,{\cal {F}},P) is separable iff its index set TT has a dense countable subset U⊂TU\subset T and there is a set Ω0⊂Ω\Omega _{0}\subset \Omega of probability zero, so P(Ω0)=0P(\Omega _{0})=0 , such that for every open set G⊂TG\subset T and every closed set F⊂R=(−∞,∞)F\subset \mathbb {R} =(-\infty ,\infty ) , the two events {Xt∈F for all t∈G∩U}\{X_{t}\in F{\text{ for all }}t\in G\cap U\} and {Xt∈F for all t∈G}\{X_{t}\in F{\text{ for all }}t\in G\} differ from each other at most on a subset of Ω0\Omega _{0} . 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 nn -dimensional Euclidean space. The concept of separability of a stochastic process was introduced by Joseph Doob. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process. Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable. 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. Two stochastic processes XX and YY defined on the same probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) with the same index set TT are said be independent if for all n∈Nn\in \mathbb {N} and for every choice of epochs t1,…,tn∈Tt_{1},\ldots ,t_{n}\in T , the random vectors (X(t1),…,X(tn))\left(X(t_{1}),\ldots ,X(t_{n})\right) and (Y(t1),…,Y(tn))\left(Y(t_{1}),\ldots ,Y(t_{n})\right) are independent. Two stochastic processes {Xt}\left\{X_{t}\right\} and {Yt}\left\{Y_{t}\right\} are called uncorrelated if their cross-covariance K⁡XY(t1,t2)=E⁡[(X(t1)−μX(t1))(Y(t2)−μY(t2))]\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] is zero for all times. Formally: {Xt},{Yt} uncorrelated  ⟺  K⁡XY(t1,t2)=0∀t1,t2\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} . If two stochastic processes XX and YY are independent, then they are also uncorrelated. Two stochastic processes {Xt}\left\{X_{t}\right\} and {Yt}\left\{Y_{t}\right\} are called orthogonal if their cross-correlation R⁡XY(t1,t2)=E⁡[X(t1)Y(t2)‾]\operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} [X(t_{1}){\overline {Y(t_{2})}}] is zero for all times. Formally: {Xt},{Yt} orthogonal  ⟺  R⁡XY(t1,t2)=0∀t1,t2\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} . 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 [0,1][0,1] or [0,∞)[0,\infty ) , and take values on the real line or on some metric space. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French phrase continue à droite, limite à gauche. A Skorokhod function space, introduced by Anatoliy Skorokhod, is often denoted with the letter DD , so the function space is also referred to as space DD . The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, D[0,1]D[0,1] denotes the space of càdlàg functions defined on the unit interval [0,1][0,1] . 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. 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. 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. 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ISBN978-0-521-83263-2. - ^Daniel Revuz; Marc Yor (2013). Continuous Martingales and Brownian Motion. Springer Science & Business Media. p. 10. ISBN978-3-662-06400-9. - ^L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. p. 123. ISBN978-1-107-71749-7. - ^Cite error: The named reference Rosenthal2006page177 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}^{d}John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. pp. 6 and 7. ISBN978-3-540-90275-1. - ^Iosif I. Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. p. 4. ISBN978-0-486-69387-3. - ^ ^{a}^{b}^{c}^{d}Robert J. Adler (2010). The Geometry of Random Fields. SIAM. pp. 14, 15. ISBN978-0-89871-693-1. - ^Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (2013). Stochastic Geometry and Its Applications. John Wiley & Sons. p. 112. ISBN978-1-118-65825-3. - ^ ^{a}^{b}Joseph L. Doob (1990). Stochastic processes. Wiley. pp. 94–96. - ^ ^{a}^{b}Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 298, 299. ISBN978-1-118-59320-2. - ^Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. p. 8. ISBN978-0-486-69387-3. - ^Cite error: The named reference Williams1991page93 was invoked but never defined (see the help page). - ^Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. pp. 22–23. ISBN978-1-86094-555-7. - ^Peter Mörters; Yuval Peres (2010). Brownian Motion. Cambridge University Press. p. 37. ISBN978-1-139-48657-6. - ^ ^{a}^{b}L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. p. 130. ISBN978-1-107-71749-7. - ^Alexander A. Borovkov (2013). Probability Theory. Springer Science & Business Media. p. 530. ISBN978-1-4471-5201-9. - ^Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 48. ISBN978-1-86094-555-7. - ^ ^{a}^{b}Bernt Øksendal (2003). Stochastic Differential Equations: An Introduction with Applications. Springer Science & Business Media. p. 14. ISBN978-3-540-04758-2. - ^ ^{a}^{b}Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. p. 472. ISBN978-1-118-59320-2. - ^Daniel Revuz; Marc Yor (2013). Continuous Martingales and Brownian Motion. Springer Science & Business Media. pp. 18–19. ISBN978-3-662-06400-9. - ^David Applebaum (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press. p. 20. ISBN978-0-521-83263-2. - ^Hiroshi Kunita (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge University Press. p. 31. ISBN978-0-521-59925-2. - ^Olav Kallenberg (2002). Foundations of Modern Probability. Springer Science & Business Media. p. 35. ISBN978-0-387-95313-7. - ^Monique Jeanblanc; Marc Yor; Marc Chesney (2009). Mathematical Methods for Financial Markets. Springer Science & Business Media. p. 11. ISBN978-1-85233-376-8. - ^ ^{a}^{b}^{c}Kiyosi Itō (2006). Essentials of Stochastic Processes. American Mathematical Soc. pp. 32–33. ISBN978-0-8218-3898-3. - ^Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. p. 150. ISBN978-0-486-69387-3. - ^ ^{a}^{b}Petar Todorovic (2012). An Introduction to Stochastic Processes and Their Applications. Springer Science & Business Media. pp. 19–20. ISBN978-1-4613-9742-7. - ^Ilya Molchanov (2005). Theory of Random Sets. Springer Science & Business Media. p. 340. ISBN978-1-85233-892-3. - ^ ^{a}^{b}Patrick Billingsley (2008). Probability and Measure. Wiley India Pvt. Limited. pp. 526–527. ISBN978-81-265-1771-8. - ^Alexander A. Borovkov (2013). Probability Theory. Springer Science & Business Media. p. 535. ISBN978-1-4471-5201-9. - ^Gusak et al. (2010), p. 22 - ^Cite error: The named reference AdlerTaylor2009page7 was invoked but never defined (see the help page). - ^Joseph L. Doob (1990). Stochastic processes. Wiley. p. 56. - ^Davar Khoshnevisan (2006). Multiparameter Processes: An Introduction to Random Fields. Springer Science & Business Media. p. 155. ISBN978-0-387-21631-7. - ^Lapidoth, Amos, A Foundation in Digital Communication, Cambridge University Press, 2009. - ^ ^{a}^{b}^{c}Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3 - ^ ^{a}^{b}^{c}^{d}Ward Whitt (2006). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Science & Business Media. pp. 78–79. ISBN978-0-387-21748-2. - ^ ^{a}^{b}Gusak et al. (2010), p. 24 - ^ ^{a}^{b}^{c}^{d}Vladimir I. Bogachev (2007). Measure Theory (Volume 2). Springer Science & Business Media. p. 53. ISBN978-3-540-34514-5. - ^ ^{a}^{b}^{c}Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 4. ISBN978-1-86094-555-7. - ^ ^{a}^{b}Søren Asmussen (2003). Applied Probability and Queues. Springer Science & Business Media. p. 420. ISBN978-0-387-00211-8. - ^ ^{a}^{b}^{c}Patrick Billingsley (2013). Convergence of Probability Measures. John Wiley & Sons. p. 121. ISBN978-1-118-62596-5. - ^Richard F. Bass (2011). Stochastic Processes. Cambridge University Press. p. 34. ISBN978-1-139-50147-7. - ^Nicholas H. Bingham; Rüdiger Kiesel (2013). Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. Springer Science & Business Media. p. 154. ISBN978-1-4471-3856-3. - ^Alexander A. Borovkov (2013). Probability Theory. Springer Science & Business Media. p. 532. ISBN978-1-4471-5201-9. - ^Davar Khoshnevisan (2006). Multiparameter Processes: An Introduction to Random Fields. Springer Science & Business Media. pp. 148–165. ISBN978-0-387-21631-7. - ^Petar Todorovic (2012). An Introduction to Stochastic Processes and Their Applications. Springer Science & Business Media. p. 22. ISBN978-1-4613-9742-7. - ^Ward Whitt (2006). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Science & Business Media. p. 79. ISBN978-0-387-21748-2. Cite error: There are tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).
For a stochastic process X ⁣:Ω→STX\colon \Omega \rightarrow S^{T} defined on the probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) , the law of stochastic process XX is defined as the pushforward measure: μ=P∘X−1,\mu =P\circ X^{-1}, where PP is a probability measure, the symbol ∘\circ denotes function composition and X−1X^{-1} is the pre-image of the measurable function or, equivalently, the STS^{T} -valued random variable XX , where STS^{T} is the space of all the possible SS -valued functions of t∈Tt\in T , so the law of a stochastic process is a probability measure. For a measurable subset BB of STS^{T} , the pre-image of XX gives X−1(B)={ω∈Ω:X(ω)∈B},X^{-1}(B)=\{\omega \in \Omega :X(\omega )\in B\}, so the law of a XX can be written as: μ(B)=P({ω∈Ω:X(ω)∈B}).\mu (B)=P(\{\omega \in \Omega :X(\omega )\in B\}). The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution. For a stochastic process XX with law μ\mu , its finite-dimensional distribution for t1,…,tn∈Tt_{1},\dots ,t_{n}\in T is defined as: μt1,…,tn=P∘(X(t1),…,X(tn))−1,\mu _{t_{1},\dots ,t_{n}}=P\circ (X({t_{1}}),\dots ,X({t_{n}}))^{-1}, This measure μt1,..,tn\mu _{t_{1},..,t_{n}} is the joint distribution of the random vector (X(t1),…,X(tn))(X({t_{1}}),\dots ,X({t_{n}})) ; it can be viewed as a "projection" of the law μ\mu onto a finite subset of TT . For any measurable subset CC of the nn -fold Cartesian power Sn=S×⋯×SS^{n}=S\times \dots \times S , the finite-dimensional distributions of a stochastic process XX can be written as: μt1,…,tn(C)=P({ω∈Ω:(Xt1(ω),…,Xtn(ω))∈C}).\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 )}. The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions. 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 XX is a stationary stochastic process, then for any t∈Tt\in T the random variable XtX_{t} has the same distribution, which means that for any set of nn index set values t1,…,tnt_{1},\dots ,t_{n} , the corresponding nn random variables Xt1,…Xtn,X_{t_{1}},\dots X_{t_{n}}, 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. But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time. When the index set TT 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. The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. 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 XX is said to be stationary in the wide sense, then the process XX has a finite second moment for all t∈Tt\in T and the covariance of the two random variables XtX_{t} and Xt+hX_{t+h} depends only on the number hh for all t∈Tt\in T .Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense. 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 {Ft}t∈T\{{\mathcal {F}}_{t}\}_{t\in T} , on a probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) is a family of sigma-algebras such that Fs⊆Ft⊆F{\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}\subseteq {\mathcal {F}} for all s≤ts\leq t , where t,s∈Tt,s\in T and ≤\leq denotes the total order of the index set TT . With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process XtX_{t} at t∈Tt\in T , which can be interpreted as time tt . The intuition behind a filtration Ft{\mathcal {F}}_{t} is that as time tt passes, more and more information on XtX_{t} is known or available, which is captured in Ft{\mathcal {F}}_{t} , resulting in finer and finer partitions of Ω\Omega . A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process XX that has the same index set TT , state space SS , and probability space (Ω,F,P)(\Omega ,{\cal {F}},P) as another stochastic process YY is said to be a modification of XX if for all t∈Tt\in T the following P(Xt=Yt)=1,P(X_{t}=Y_{t})=1, holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be stochastically equivalent or equivalent. 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. 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. The theorem can also be generalized to random fields so the index set is nn -dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces. Two stochastic processes XX and YY defined on the same probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) with the same index set TT and set space SS are said be indistinguishable if the following P(Xt=Yt for all t∈T)=1,P(X_{t}=Y_{t}{\text{ for all }}t\in T)=1, holds. If two XX and YY are modifications of each other and are almost surely continuous, then XX and YY are indistinguishable. 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, which means that the index set has a dense countable subset. More precisely, a real-valued continuous-time stochastic process XX on a probability space (Ω,F,P)(\Omega ,{\cal {F}},P) is separable iff its index set TT has a dense countable subset U⊂TU\subset T and there is a set Ω0⊂Ω\Omega _{0}\subset \Omega of probability zero, so P(Ω0)=0P(\Omega _{0})=0 , such that for every open set G⊂TG\subset T and every closed set F⊂R=(−∞,∞)F\subset \mathbb {R} =(-\infty ,\infty ) , the two events {Xt∈F for all t∈G∩U}\{X_{t}\in F{\text{ for all }}t\in G\cap U\} and {Xt∈F for all t∈G}\{X_{t}\in F{\text{ for all }}t\in G\} differ from each other at most on a subset of Ω0\Omega _{0} . 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 nn -dimensional Euclidean space. The concept of separability of a stochastic process was introduced by Joseph Doob. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process. Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable. 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. Two stochastic processes XX and YY defined on the same probability space (Ω,F,P)(\Omega ,{\mathcal {F}},P) with the same index set TT are said be independent if for all n∈Nn\in \mathbb {N} and for every choice of epochs t1,…,tn∈Tt_{1},\ldots ,t_{n}\in T , the random vectors (X(t1),…,X(tn))\left(X(t_{1}),\ldots ,X(t_{n})\right) and (Y(t1),…,Y(tn))\left(Y(t_{1}),\ldots ,Y(t_{n})\right) are independent. Two stochastic processes {Xt}\left\{X_{t}\right\} and {Yt}\left\{Y_{t}\right\} are called uncorrelated if their cross-covariance K⁡XY(t1,t2)=E⁡[(X(t1)−μX(t1))(Y(t2)−μY(t2))]\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] is zero for all times. Formally: {Xt},{Yt} uncorrelated  ⟺  K⁡XY(t1,t2)=0∀t1,t2\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} . If two stochastic processes XX and YY are independent, then they are also uncorrelated. Two stochastic processes {Xt}\left\{X_{t}\right\} and {Yt}\left\{Y_{t}\right\} are called orthogonal if their cross-correlation R⁡XY(t1,t2)=E⁡[X(t1)Y(t2)‾]\operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} [X(t_{1}){\overline {Y(t_{2})}}] is zero for all times. Formally: {Xt},{Yt} orthogonal  ⟺  R⁡XY(t1,t2)=0∀t1,t2\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} . 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 [0,1][0,1] or [0,∞)[0,\infty ) , and take values on the real line or on some metric space. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French phrase continue à droite, limite à gauche. A Skorokhod function space, introduced by Anatoliy Skorokhod, is often denoted with the letter DD , so the function space is also referred to as space DD . The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, D[0,1]D[0,1] denotes the space of càdlàg functions defined on the unit interval [0,1][0,1] . 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. 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. 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. - ^ ^{a}^{b}Cite error: The named reference Kallenberg2002page24 was invoked but never defined (see the help page). - ^Cite error: The named reference RogersWilliams2000page121 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}^{d}Cite error: The named reference FrizVictoir2010page571 was invoked but never defined (see the help page). - ^Sidney I. Resnick (2013). Adventures in Stochastic Processes. Springer Science & Business Media. pp. 40–41. ISBN978-1-4612-0387-2. - ^ ^{a}^{b}Cite error: The named reference Lamperti1977page1 was invoked but never defined (see the help page). - ^Cite error: The named reference Borovkov2013page528 was invoked but never defined (see the help page). - ^Ward Whitt (2006). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Science & Business Media. p. 23. ISBN978-0-387-21748-2. - ^David Applebaum (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press. p. 4. ISBN978-0-521-83263-2. - ^Daniel Revuz; Marc Yor (2013). Continuous Martingales and Brownian Motion. Springer Science & Business Media. p. 10. ISBN978-3-662-06400-9. - ^L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. p. 123. ISBN978-1-107-71749-7. - ^Cite error: The named reference Rosenthal2006page177 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}^{d}John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. pp. 6 and 7. ISBN978-3-540-90275-1. - ^Iosif I. Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. p. 4. ISBN978-0-486-69387-3. - ^ ^{a}^{b}^{c}^{d}Robert J. Adler (2010). The Geometry of Random Fields. SIAM. pp. 14, 15. ISBN978-0-89871-693-1. - ^Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (2013). Stochastic Geometry and Its Applications. John Wiley & Sons. p. 112. ISBN978-1-118-65825-3. - ^ ^{a}^{b}Joseph L. Doob (1990). Stochastic processes. Wiley. pp. 94–96. - ^ ^{a}^{b}Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 298, 299. ISBN978-1-118-59320-2. - ^Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. p. 8. ISBN978-0-486-69387-3. - ^ ^{a}^{b}Cite error: The named reference Florescu2014page294 was invoked but never defined (see the help page). - ^Cite error: The named reference Williams1991page93 was invoked but never defined (see the help page). - ^Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. pp. 22–23. ISBN978-1-86094-555-7. - ^Peter Mörters; Yuval Peres (2010). Brownian Motion. Cambridge University Press. p. 37. ISBN978-1-139-48657-6. - ^ ^{a}^{b}L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. p. 130. ISBN978-1-107-71749-7. - ^Alexander A. Borovkov (2013). Probability Theory. Springer Science & Business Media. p. 530. ISBN978-1-4471-5201-9. - ^Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 48. ISBN978-1-86094-555-7. - ^ ^{a}^{b}Bernt Øksendal (2003). Stochastic Differential Equations: An Introduction with Applications. Springer Science & Business Media. p. 14. ISBN978-3-540-04758-2. - ^ ^{a}^{b}Ionut Florescu (2014). Probability and Stochastic Processes. John Wiley & Sons. p. 472. ISBN978-1-118-59320-2. - ^Daniel Revuz; Marc Yor (2013). 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Let (Ω,F,P)(\Omega, \mathcal{F}, P) be a complete probability space, and let SS be a measurable state space (e.g., S=RdS = \mathbb{R}^d). A stochastic process XX indexed by a set TT (the index set, typically T=[0,∞)T = [0, \infty) or T=ZT = \mathbb{Z}) is formally defined as a collection of random variables X={Xt}t∈TX = \{X_t \}_{t \in T} such that:\n\n1. **Measurability:** For every t∈Tt \in T, the random variable Xt:Ω→SX_t: \Omega \to S is F\mathcal{F}-measurable.\n2. **Joint Measurability:** The mapping X:Ω×T→SX: \Omega \times T \to S defined by X(ω,t)=Xt(ω)X(\omega, t) = X_t(\omega) must be measurable with respect to the product σ\sigma-algebra F⊗B(T)\mathcal{F} \otimes \mathcal{B}(T), where B(T)\mathcal{B}(T) is the Borel σ\sigma-algebra on TT. \n\nAlternatively, the process XX induces a filtration FT=σ{Xt}t∈T\mathcal{F}_T = \sigma\{X_t \}_{t \in T} on F\mathcal{F}, which represents the information available up to time TT. If TT is continuous, the process is often defined as a measurable function X:Ω×T→SX: \Omega \times T \to S such that for any measurable set A⊂SA \subset S, the mapping (ω,t)↦1A(Xt(ω))(\omega, t) \mapsto \mathbf{1}_A(X_t(\omega)) is measurable with respect to F⊗B(T)\mathcal{F} \otimes \mathcal{B}(T).