 Solution

The bounded-time optional stopping theorem states that if $X$ is an integrable discrete-time martingale and $\tau$ is a stopping time bounded by a deterministic integer $N$, then $X_{\tau }$ is integrable and $\mathbb{E}{X}_{\tau }=\mathbb{E}{X}_0$. Indeed $|X_{\tau }|\le {\sum }_{j=0}^N|X_j|$ proves integrability, andThe event $\{\tau \ge j\}=\{\tau >j-1\}$ belongs to ${\mathcal{F}}_{j-1}$. Conditional expectation of each summand given ${\mathcal{F}}_{j-1}$ is zero, soMore generally, if $\sigma \le \tau \le N$ are stopping times, $\mathbb{E}[X_{\tau }\mid {\mathcal{F}}_{\sigma }]=X_{\sigma }$. To prove this, test against $A\in {\mathcal{F}}_{\sigma }$ and apply the same argument to $1_A{1}_{\{\sigma <j\le \tau \}}(X_j-X_{j-1})$; its coefficient is ${\mathcal{F}}_{j-1}$-measurable. Boundedness of the stopping times makes the sums finite, so no limiting optional-stopping hypothesis is being omitted.