 Solution

A martingale is an adapted integrable process $M_n$ with $\mathbb{E}(M_{n+1}\mid {\mathcal{F}}_n)=M_n$. A stopping time $T$ satisfies $\{T\le n\}\in {\mathcal{F}}_n$. The bounded optional sampling theorem states that for bounded stopping times $S\le T$, $\mathbb{E}(M_T\mid {\mathcal{F}}_S)=M_S$.

To prove it, let $A\in {\mathcal{F}}_S$ and take a common deterministic bound $N$. Thensince each indicator is ${\mathcal{F}}_j$-measurable. This proves the conditional identity. The theorem extends to almost surely finite stopping times for a uniformly integrable martingale by truncating the times and passing in $L^1$; without such a hypothesis unrestricted optional sampling is false.

For the proposed exponential process, conditioning on one increment gives the martingale condition $z(p{e}^{\theta }+q{e}^{-\theta })=1$. Thus, when $q>0$, the complete set isThese are the two real logarithms: ${\theta }_{-}<0<{\theta }_{+}$. If $q=0$, the sole finite value is $\theta =-\log z$. Stop the ${\theta }_{+}$ martingale at $n\wedge {\tau }_k$. Before the first hit, $S_n<k$, so the stopped process is bounded above by $e^{k{\theta }_{+}}$. On failure to hit, its value tends to zero because of $z^n$. Bounded convergence after bounded optional sampling gives $1=e^{k{\theta }_{+}}\mathbb{E}[z^{{\tau }_k};{\tau }_k<\infty ]$, henceIts limit at $z=1$ is one, proving almost-sure hitting as well. Differentiating at one givesFor instance implicit differentiation of the martingale equation gives ${\theta }_{+}^{\prime }(1)=-1/(p-q)$.

For the first subsequent return ${\tau }_k^{\prime }$, the walk may first rise above $k$, so the uniform bound used in the stopping-limit argument fails. In fact after reaching $k$ it returns with probability $q+p(q/p)=2q<1$: a downward first step is followed by an almost-sure upward hit, while an upward first step hits the level below only with probability $q/p$. Thus ${\tau }_k^{\prime }=\infty$ with positive probability and its unconditional mean is infinite. This pinpoints why repeating the earlier differentiation argument is invalid; bounded-time optional sampling itself remains valid.