Optional sampling for bounded stopping times

Put $H_k=1_{\{\sigma <k\le \tau \}}$ for $1\le k\le N$. Both $\{\sigma <k\}=\{\sigma \le k-1\}$ and $\{\tau \ge k\}$ lie in ${\mathcal{F}}_{k-1}$, so $H$ is nonnegative, bounded, and predictable. On the probability-one event $\{\sigma \le \tau \le N\}$, the finite pathwise telescope is $X_{\tau }-X_{\sigma }={\sum }_{k=1}^N{H}_k(X_k-X_{k-1})\mathrm{.}$ The integral identities below use this almost-sure equality; no equality is asserted on a null outcome with $\tau >N$.

Fix $A\in {\mathcal{F}}_{\sigma }$. On $\{\sigma \ge k\}$, $H_k=0$; hence $1_A{H}_k=1_{A\cap \{\sigma \le k-1\}}{H}_k,$ and the first factor on the right is ${\mathcal{F}}_{k-1}$-measurable by F1. Thus $1_{A}H$ is another bounded nonnegative predictable process.

Apply F3's one-step conditional calculation and sum: for a submartingale, ${\int }_A(X_{\tau }-X_{\sigma })dP\ge 0;$ for a martingale equality holds, and for a supermartingale the inequality reverses. Almost-sure boundedness identifies every stopped value almost surely with a finite sum of integrable variables, while F2 gives ${\mathcal{F}}_{\sigma }$-measurability of $X_{\sigma }$. F4 therefore identifies the stated conditional relation. Taking $A=\mathrm{\Omega }$ gives the expectation relation. AC has exactly the role in F5.