MathcalF_tMeasurableFunction 2006-09-22 18:25:17 2007-02-26 12:15:53 Definition % 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 $\lbrace X_t \mid t\in T\rbrace$ be a stochastic process defined on a probability space $(\Omega,\mathcal{F},P)$ and $\lbrace \mathcal{F}_t \mid t\in T\rbrace$ a filtration (an increasing sequence of sigma subalgebras of $\mathcal{F}$), where $T$ is a linearly ordered subset of $\mathbb{R}$ with a minimum $t_0$. Then the process $\lbrace X_t\rbrace$ is said to be \emph{adapted to} the filtration $\lbrace \mathcal{F}_t\rbrace$ if for each $t\ge t_0$, $X_t$ is \PMlinkname{$\mathcal{F}_t$-measurable}{MathcalFMeasurableFunction}: $$X_t^{-1}(B)\in \mathcal{F}_t\mbox{ for each Borel set }B\in\mathbb{R}.$$ A stochastic process is an \emph{adapted process} if it is adapted to some filtration.