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Let $X_n$ be a martingale such that $|X_n-X_{n-1}|\leq 1$. How would one bound something like $E[e^{\lambda(X_n-X_{n-1})}|\mathcal{F_{n-1}}]$ where $\mathcal{F}_{n}$ is the standard filtration and $\lambda$ a real parameter? I can tell that an $e^{-\lambda X_{n-1}}$ is measurable and can be taken out, but wouldn't know how to proceed. Any hint would do.

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Yes, you are right that $e^{-\lambda X_{n-1}}$ is $\mathcal{F}_{n-1}$ measurable, but you do not need to use that fact.

The bound can be found as a direct consequence of Hoeffding's lemma,

$$\mathbb{E}[e^{\lambda(X_n-X_{n-1})}|\mathcal{F}_{n-1}] \leq e^{\lambda^2/2}$$

Note that the standard form of the Hoeffding's lemma applies to total expectation (not to conditional expectation), but the result holds when the bounds for $|X_n-X_{n-1}|$ are themselves measurable by the conditioning variable (as is the case here). See this for reference.

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