Definition (Upcrossings)

Let $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$, let $F = \{ S_{1},\dots,S_{n} \}\subset \mathbb{R}^{+}$ and let $a,b\in\mathbb{R}$ where $a<b$, with $S_{1}<\dots<S_{n}$. Now let $$\begin{align*}t_{1}&=\inf\{ s\in F:f(s)\le a \}\\t_{2}&=\inf\{ s\in F:s>t_{1},f(s)\ge b \}\\t_{3}&=\inf\{ s\in F:s>t_{2},f(s)\ge a \}\\t_{4}&=\inf\{ s\in F:s>t_{3},f(s)\ge b \}\end{align*}$$ Let $k\in\mathbb{N}$, be the largest integer for which $f(t_{2k-1})\le a$ and $f(t_{2k})\ge b$. If $\not\exists k$, then $k=0$. We call $k$ the number of upcrossings from $a$ to $b$ over $F$ and is denoted by $$M(f,F,[a,b])$$Let now $S\subset \mathbb{R}^{+}$ arbitrary. We define, $$M(f,S,[a,b])=\sup_{\mathclap{\substack{F\subset S \\ F\text{ finite}}}} M(f,F,[a,b])$$i.e. the highest number of upcrossings over all finite $F\subset S$.

Definition (Upcrossings over sample path)

Let $X=(X_{n})_{n\in\mathbb{N}}$ be a process on $(\Omega,\mathcal{F},P)$ and let $S\subset \mathbb{N}$ arbitrary. Let $a,b\in\mathbb{R}$, $a<b$, then $$M(X,S,[a,b])(\omega)=M(X(\omega),S,[a,b]), \ \forall\omega\in\Omega$$where $X(\omega)=(X_{n})_{n\in\mathbb{N}}$, i.e. $$M(X,S,[a,b])(\omega)=\sup_{\mathclap{\substack{F\subset S \\ F\text{ finite} }}}M(X(\omega),F,[a,b]), \ \forall\omega\in\Omega$$