Generalized Volatility

Let (X_t) be a stochastic process. Define \nu(X,t,I) = (X|_{[0,t]})^{-1}(I) for t > 0, I\subseteq\bm{R}. 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.