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This is a problem on sub-martingales.

Given : $X_n = X_0 \mathrm{e}^{\mu S_n}$, $n= 1,2,3,\ldots$, where $X_0 > 0$ and where $S_n$ is a symmetric random walk and $\mu$ is greater than zero. We have to prove that $\{X_n\}$ is a sub martingale.

How do we go about this?

I understand that, if $Z_n$ is a stochastic process, with $\mathbb{E}(Z_n) < \infty$ , then it is a sub martingale if $\mathbb{E}(Z_{n+1} | Z_1,Z_2,Z_3,\ldots,Z_n) \geqslant Z_n$. So how do I proceed with the problem?

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