Editing Stochastic process (section)

===Stochastic process===
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"/>

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>
<div class="center"><math>
\{X(t):t\in T \}.
</math></div>

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|author-link=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=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>

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 [[Function (mathematics)|functions]] from the set <math>T</math> into the space <math>S</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/> However this alternative definition as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined.<ref name="aumann">{{cite journal |last1=Aumann |first1=Robert |title=Borel structures for function spaces |journal=Illinois Journal of Mathematics |date=December 1961 |volume=5 |issue=4 |doi=10.1215/ijm/1255631584 |s2cid=117171116 |doi-access=free }}</ref>