Editing Stochastic process (section)

===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>