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.


1 Answer 1


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .