modification of a stochastic processModificationOfAStochasticProcess2006-08-12 04:19:532008-12-30 16:14:35Definitionmodification% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
Let $\lbrace X_t \rbrace_{t\geq 0}$, $\lbrace Y_t\rbrace_{t\geq 0}$ be stochastic processes on $(\Omega, \mathcal{F}, P)$. $\lbrace X_t \rbrace_{t\geq 0}$ is a \emph{modification} of {$\lbrace Y_t\rbrace_{t\geq 0}$ if
$$P[\{\omega:X_t(\omega)=Y_t(\omega)\}]=1$$ for all $t \in [0, \infty).$
\begin{thebibliography}{9}
\bibitem{Oksendal:several}
Bernt \O ksendal.
{\em \PMlinkescapetext{Stochastic Differential Equations,
(An Introduction with Applications)}, 5th ed Springer 1998.}
\end{thebibliography}