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Let $S_n$ have integrable, mean-zero independent and identically distributed random variables as increments, taking integer values at most one, with $\mathbb{P}(X_1=1)>0$. For a positive integer $b$, put $T_b=\inf \{n:S_n=b\}$ and $M(\tau )=\mathbb{E}{e}^{\tau X_1}$, $\tau >0$. Then $T_b$ is finite almost surely. Indeed, $b-S_{n\wedge T_b}$ is a nonnegative martingale; finite convergence on $\{T_b=\infty \}$ would force its integer increments eventually to vanish, contrary to the Borel-Cantelli lemmas applied to $\{X_n=1\}$.

The exponential martingale of a random walk $e^{\tau S_n}/M(\tau {)}^n$ stopped at $T_b$ is bounded by $e^{\tau b}$ because $S_{n\wedge T_b}\le b$ and $M(\tau )\ge 1$. The dominated convergence theorem therefore preserves its mean one at the limit and proves the displayed first-passage time transform. This argument accommodates arbitrarily large negative jumps; the restriction concerns upward jumps. For a simple symmetric random walk, $M(\tau )=\cosh \tau$, giving $\mathbb{E}{e}^{-\alpha T_b}=(e^{\alpha }-\sqrt{e^{2\alpha }-1}{)}^b$.

The moment-generating function satisfies $0<\varphi (\lambda )<\infty$. The proposed process is nonnegative and adapted to the natural filtration of the random walk, and independence of the increments givesMoreover $X_{n+1}$ is independent of ${\mathcal{F}}_n$, soTherefore $M^{\lambda }$ is a mean-one nonnegative martingale, the exponential martingale of a random walk. For $\lambda =0$ it is the constant process one.

Here $X_1\le 1$, so for every $\tau >0$,The Jensen inequality for the strictly convex exponential function givesIn fact the inequality is strict: $\mathbb{P}(X_1=1)>0$ and $\mathbb{E}{X}_1=0$ exclude a constant random variable. In particular, $\boxed{1\le M(\tau )<\infty }$ as required.

For $Z_n=e^{\tau S_n}/M(\tau {)}^n$, independence and identical distribution giveThus $Z_n$ is positive, integrable with mean one, and adapted to the natural filtration. Conditioning on the next increment givesConsequently $\boxed{(Z_n)\text{ is a martingale}.}$ This is the exponential martingale of a random walk.

The moment-generating function of a normal distribution givesIndependence of $X_{n+1}$ from ${\mathcal{F}}_n$ impliesEvery exponential here is integrable. Thus the process is a martingale exactly when $-\lambda +{\lambda }^2/2=0$. Its two roots are $0$ and $2$, and the unique positive root isThe resulting exponential martingale of a random walk is nonnegative and has expectation $1$ at every finite time.

For $m\ge 1$, the increments have exponential momentThus $Z_n=2^{S_n}$ is an exponential martingale of a random walk relative to the natural filtration. We justify its stopping limit. From any surviving state in $(-m,m)$, a block of $2m$ successive increments all equal to $-1$ forces an exit. The block has probability $q=(6/7{)}^{2m}>0$, independently of the preceding increments. The geometric tail bound from a uniform escape probability givesso $T$ is finite almost surely and has finite expectation.

At the lower exit $S_T=-m$ exactly; at the upper exit $S_T$ is either $m$ or $m+1$. Before exit the same overall bound $-m\le S_{n\wedge T}\le m+1$ holds. Hence $Z_{n\wedge T}$ is bounded by $2^{m+1}$, uniformly in $n$. The bounded optional stopping theorem gives $\mathbb{E}{Z}_{n\wedge T}=Z_0=1$, and the dominated convergence theorem now yieldsAllowing the upper overshoot by one is essential; the terminal state need not equal $m$ on upper exit.