Editing Stochastic process (section)

====Law====
For a stochastic process <math>X\colon\Omega \rightarrow S^T</math> defined on the probability space <math>(\Omega, \mathcal{F}, P)</math>, the '''law''' of stochastic process <math>X</math> is defined as the [[pushforward measure]]:
<div class="center"><math>
\mu=P\circ X^{-1},
</math></div>
where <math>P</math> is a probability measure, the symbol <math>\circ </math> denotes function composition and <math>X^{-1}</math> is the pre-image of the measurable function or, equivalently, the <math>S^T</math>-valued random variable <math>X</math>, where <math>S^T</math> is the space of all the possible <math>S</math>-valued functions of <math>t\in T</math>, so the law of a stochastic process is a probability measure.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/><ref name="FrizVictoir2010page571"/><ref name="Resnick2013page40">{{cite book|author=Sidney I. Resnick|title=Adventures in Stochastic Processes|url=https://books.google.com/books?id=VQrpBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4612-0387-2|pages=40–41}}</ref>

For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives
<div class="center"><math>
X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},
</math></div>
so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>
<div class="center"><math>
\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).
</math></div>

The law of a stochastic process or a random variable is also called the '''probability law''', '''probability distribution''', or the '''distribution'''.<ref name="Borovkov2013page528"/><ref name="FrizVictoir2010page571"/><ref name="Whitt2006page23">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|page=23}}</ref><ref name="ApplebaumBook2004page4">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=4}}</ref><ref name="RevuzYor2013page10">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=10}}</ref>