From Jacod-Protter:
DEFINITION Let $(X_n)_{n\geq 0}$ be a submartingale and $a<b$. The number of upcrossings of an interval $[a,b]$ is the number of times a process crosses from below $a$ and to above $b$ at a later time. Using stopping times, this translates into defining: \begin{equation} T_0=0 \end{equation} and inductively for $j\geq0$: \begin{equation} S_{j+1}=\min\{k>T_j:X_k\leq a\} \end{equation} \begin{equation} T_{j+1}=\min\{k>S_{j+1}:X_k\geq b\} \end{equation} with convention that $\min\{\emptyset\}=+\infty$ and $\max\{\emptyset\}=0$. Finally: \begin{equation} U_{n}=\max\{j:T_j\leq n\} \end{equation} with $U_n$ representing the number of upcrossings of $[a,b]$ before time $n$.
I have tried to verify this definition by means of a simple example and I get unexpected results. The example unfolds as follows:
MY EXAMPLE
- Fix $a= 0,5$, $b=1,2$;
- $X_0=0,2$, $X_1=0,7$, $X_2=1,4$, $X_3=0,65$; $X_4=0,1$, $X_5=1,5$;
which means that there are just two upcrossings of the interval $[a,b]$ before time $n=5$, just one upcrossing before time $n=4$ and time $n=3$ and time $n=2$, and no upcrossing before time $n=1$ and $n=0$ (See the figure attached below so as to have a clear visualization of the example).
Hence, by definition of $T_{j+1}$ and $S_{j+1}$, one has that:
- $T_0=0$;
- $S_1=4$, $T_1=5$;
- $S_j=T_j=+\infty$ for $j\geq 2$.
This means that, by definition of $U_n$:
- $U_0=0$ (AS EXPECTED);
- $U_1=0$ (AS EXPECTED);
- $U_2=0$ (UNEXPECTED, I WOULD EXPECT $U_2=1$);
- $U_3=0$ (UNEXPECTED, I WOULD EXPECT $U_3=1$);
- $U_4=0$ (UNEXPECTED, I WOULD EXPECT $U_4=1$);
- $U_5=1$ (UNEXPECTED, I WOULD EXPECT $U_5=2$);
At this point, if instead one considers the definition of $S_{j+1}$ as follows: \begin{equation} S_{j+1}=\min\{k\geq T_j:X_k\leq a\} \end{equation} (that is with $k\geq T_j$ in place of $k > T_j$), by definition of $T_{j+1}$ and by the new definition of $S_{j+1}$, one would have:
- $T_0=0$;
- $S_1=0$, $T_1=2$;
- $S_2=4$, $T_2=5$;
- $S_j=T_j=+\infty$ for $j\geq 3$;
and so
- $U_0=0$ (AS EXPECTED);
- $U_1=0$ (AS EXPECTED);
- $U_2=1$ (AS EXPECTED);
- $U_3=1$ (AS EXPECTED);
- $U_4=1$ (AS EXPECTED);
- $U_5=2$ (AS EXPECTED).
In conclusion, it seems to me that $U_n=\max\{j:T_j\leq n\}$ is the correct definition of the number of upcrossings of $[a,b]$ before time $n$ ONLY IF $S_{j+1}=\min\{k\geq T_j:X_k\leq a\}$ and NOT IF $S_{j+1}=\min\{k > T_j:X_k\leq a\}$.
Is my observation correct?
VISUALIZATION OF THE ABOVE EXAMPLE
