Let $X=(X\_t:t\ge 0)$ be a stochastic process (martingale in general) starting at $X\_0=0$. For $T>0$ and $a<b$, let $U^T\_{a,b}(X)$ be the number of upcrossings of $X$ across the interval $[a,b]$ over $[0,T]$, i.e. $U^T\_{a,b}(X)$ is the supremum of the nonnegative integers $n\in\mathbb N$ such that there exist times $s\_k,t\_k\in [0,T]$ satisfying
$$s\_1<t\_1<s\_2<t\_2<\cdots<s\_n<t\_n$$
and for which $X\_{s\_k}\le a<b\le X\_{t\_k}$. My question is the following: For any continuous and non-decreasing function $h:\mathbb R\_+\to [0,\infty]$ with $h(0)=0$ (may not be strictly increasing), set $Y=(Y\_t:=X\_{h(t)}: t\ge 0)$. Does $U^T\_{a,b}(X)=U^{h(T)}\_{a,b}(Y)$ hold?



Is the number of uncrossing invariant under time-change?
We have $U\_{a,b}^T(Y) = U\_{a,b}^{h(T)}(X)$.


By construction, the function $h$ maps $[0,T]$ onto $[0,h(T)]$. More precisely, each $t' \in [0,h(T)]$ can be written $h(h^\leftarrow(t'))$,
where $h^\leftarrow(t') := \inf\{t \in [0,T] : h(t) \ge t'\} \in [0,T]$.
The function $h^\leftarrow$ thus defined is strictly increasing (but not necessarily continuous).


Hence, if $s'\_1<t'\_1<\ldots<s'\_n<t'\_n$ provide $n$ upcroassings of $X$ in $[0,h(T)]$ then $h^\leftarrow(s'\_1)<h^\leftarrow(t'\_1)<\ldots<h^\leftarrow(s'\_n)<h^\leftarrow(t'\_n)$ provide $n$ upcroassings of $Y$ in $[0,T]$.


Conversely, if $s\_1<t\_1<\ldots<s\_n<t\_n$ provide $n$ upcroassings of $Y$ in $[0,T]$,
then $h(s\_1)<h(t\_1)<\ldots<h(s\_n)<h(t\_n)$ provide $n$ upcroassings of $X$ in $[0,h(T)]$ (the inequalities cannot be equalities because of the conditions $X(h(s\_i)) \le a < b \le X(h(t\_i))$.


