# Martingale - $Y_n = \exp (X_1 + … + X_n)$

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 $$\mathbb{E} \exp(X_1)$$.

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)