| title |
author |
Generalized Volatility |
Keith A. Lewis |
\newcommand\RR{\bm{R}}
Let $(X_t)$ be a stochastic process.
Define $\nu(X,t,I) = (X|_{[0,t]})^{-1}(I)$ for $t > 0$, $I\subseteq\RR$.
Let $\sim$ be the relation $x R y$ if and only if the interval
$[x,y]\subseteq \nu_t(I)$. It is an equivalence relation on its domain
and the cardinality of the quotient space is then number of crossings
of $I$ up to time $t$.
Define $\nu(X,t,a) = {s\le t\mid X_s\le a}$.
Let $R_a$ be the relation $x R_a y$ if and only if the interval
$[x,y]\subseteq \nu_t(X,t,a)$. It is an equivalence relation on its domain
and the cardinality of the quotient space is then number of crossings
of $I$ up to time $t$.