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Let $X_1 , X_2,...$ be a sequence of independent random variables such that $\mathbb{E} (X_n)=0$ for all $n=1,2,...$. Whether the sequence $Y_n = \exp (X_1 + ... + X_n)$ with respect to $\mathcal{F}_n = \sigma(X_1,...,X_n)$ is a martingale?

I tried to solve it and I don't know how to count $\mathbb{E} \exp(X_1)$.

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Note that $Y_{n+1}=Y_n e^{X_{n+1}}$ so $$ E(Y_{n+1}\mid \mathcal{F_n})=Y_{n}E(e^{X_{n+1}}\mid \mathcal{F}_n)=Y_nEe^{X_{n+1}}\geq Y_n\exp(EX_{n+1})=Y_n $$ where we used the fact that $Y_n$ is measurable w.r.t to $\mathcal{F_n}$, the fact that $X_{n+1}$ is independent of $\mathcal{F_n}$ and Jensen's inequality in the final step. So $Y_n$ is a sub-martingale and not a martingale in general (see the equality conditions of Jensen's inequality for when it is a martingale)

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