Editing Stochastic process (section)

====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>