Editing Stochastic process (section)

==Mathematical construction==
In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically.<ref name="Rosenthal2006page177"/> There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.<ref name="Adler2010page13">{{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|page=13}}</ref>

Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using [[Kolmogorov extension theorem|Kolmogorov's existence theorem]]{{efn|The theorem has other names including Kolmogorov's consistency theorem,<ref name="AthreyaLahiri2006">{{cite book|author1=Krishna B. Athreya|author2=Soumendra N. Lahiri|title=Measure Theory and Probability Theory|url=https://books.google.com/books?id=9tv0taI8l6YC|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-32903-1}}</ref>  Kolmogorov's extension theorem<ref name="Øksendal2003page11">{{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=11}}</ref> or the Daniell–Kolmogorov theorem.<ref name="Williams1991page124">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5|page=124}}</ref>}} to prove a corresponding stochastic process exists.<ref name="Rosenthal2006page177"/><ref name="Adler2010page13"/> This theorem, which is an existence theorem for measures on infinite product spaces,<ref name="Durrett2010page410">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=410}}</ref> says that if any finite-dimensional distributions satisfy two conditions, known as ''consistency conditions'', then there exists a stochastic process with those finite-dimensional distributions.<ref name="Rosenthal2006page177"/>

===Construction issues===
When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes.<ref name="KloedenPlaten2013page63"/><ref name="Khoshnevisan2006page153"/> One problem is that it is possible to have more than one stochastic process with the same finite-dimensional distributions. For example, both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions.<ref name="Billingsley2008page493to494">{{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=493–494}}</ref> This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.<ref name="Adler2010page13"/><ref name="Borovkov2013page529">{{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|pages=529–530}}</ref>

Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined.<ref name="Ito2006page32"/> For example, the supremum of a stochastic process or random field is not necessarily a well-defined random variable.<ref name="AdlerTaylor2009page7"/><ref name="Khoshnevisan2006page153"/> For a continuous-time stochastic process <math>X</math>, other characteristics that depend on an uncountable number of points of the index set <math>T</math> include:<ref name="Ito2006page32"/>
* a sample function of a stochastic process <math>X</math> is a [[continuous function]] of <math>t\in T</math>;
* a sample function of a stochastic process <math>X</math> is a [[bounded function]] of <math>t\in T</math>; and
* a sample function of a stochastic process <math>X</math> is an [[increasing function]] of <math>t\in T</math>.
where the symbol '''∈''' can be read "a member of the set", as in <math>t</math> a member of the set <math>T</math>.

To overcome the two difficulties described above, i.e., "more than one..." and "functionals of...", different assumptions and approaches are possible.<ref name="Asmussen2003page408"/>

===Resolving construction issues===
One approach for avoiding mathematical construction issues of stochastic processes, proposed by [[Joseph Doob]], is to assume that the stochastic process is separable.<ref name="AthreyaLahiri2006page221">{{cite book|author1=Krishna B. Athreya|author2=Soumendra N. Lahiri|title=Measure Theory and Probability Theory|url=https://books.google.com/books?id=9tv0taI8l6YC|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-32903-1|page=221}}</ref> Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set.<ref name="AdlerTaylor2009page14">{{cite book|author1=Robert J. Adler|author2=Jonathan E. Taylor|title=Random Fields and Geometry|url=https://books.google.com/books?id=R5BGvQ3ejloC|year=2009|publisher=Springer Science & Business Media|isbn=978-0-387-48116-6|page=14}}</ref> Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied.<ref name="Ito2006page32"/><ref name="AdlerTaylor2009page14"/>

Another approach is possible, originally developed by [[Anatoliy Skorokhod]] and [[Andrei Kolmogorov]],<ref name="AthreyaLahiri2006page211">{{cite book|author1=Krishna B. Athreya|author2=Soumendra N. Lahiri|title=Measure Theory and Probability Theory|url=https://books.google.com/books?id=9tv0taI8l6YC|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-32903-1|page=211}}</ref> for a continuous-time stochastic process with any metric space as its state space. For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption,<ref name="Asmussen2003page408"/><ref name="Getoor2009">{{cite journal|last1=Getoor|first1=Ronald|title=J. L. Doob: Foundations of stochastic processes and probabilistic potential theory|journal=The Annals of Probability|volume=37|issue=5|year=2009|page=1655|issn=0091-1798|doi=10.1214/09-AOP465|arxiv=0909.4213|bibcode=2009arXiv0909.4213G|s2cid=17288507}}</ref> but such a stochastic process based on this approach will be automatically separable.<ref name="Borovkov2013page536">{{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=536}}</ref>

Although less used, the separability assumption is considered more general because every stochastic process has a separable version.<ref name="Getoor2009"/> It is also used when it is not possible to construct a stochastic process in a Skorokhod space.<ref name="Borovkov2013page535"/> For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as <math>n</math>-dimensional Euclidean space.<ref name="AdlerTaylor2009page7"/><ref name="Yakir2013page5">{{cite book|author=Benjamin Yakir|title=Extremes in Random Fields: A Theory and Its Applications|url=https://books.google.com/books?id=HShwAAAAQBAJ&pg=PT97|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-72062-2|page=5}}</ref>