I'm reading a paper where we consider a nonnegative, bounded supermartingale $((X_n)_{n \ge 1}, (\mathcal{F}_n)_{n\ge 1})$, where $X_k = 0$ for $k \ge N$, and stopping times given by a recursive formula: $$t_1 = 1; \qquad t_{m} = \inf \lbrace k > t_{m-1} : X_k = 0 \ \hbox{or} \ X_k \ge X_{t_{m-1}}\rbrace, \qquad m = 2,3,\ldots$$
I guess the formula is not important, as long as those are stopping times and they are finite for each $m$. Then it is said that for $Y_{m} = X_{t_{m}}$, $m = 1,2, \ldots, N$, the sequence $Y_1, Y_2, \ldots Y_N$ is also a supermartingale.
It is not specified and I want to be sure: it is a supermartingale to the filtration $(\mathcal{F}_{t_m})_{1 \le n \le N}$, not $(\mathcal{F}_n)_{1 \le n \le N}$, am I right?