Consider the sequence $\{X_n\}_{n\geq 1}$ of independent random variables with law $N(0, \sigma^2)$. Define the sequence $Y_n= \exp \bigg(a\sum_{i=1}^n X_i-n\sigma^2\bigg), n\geq 1,$ for $a$ a real parameter, and $Y_0=1$.
Now How to find the value of $a$ such that sequence $\{Y_n\}_{n\geq 1}$ is a Martingale, Sub-martingale, Super-martingale?
My attempt
I have generated 10 normal random variables with 0 mean and variance 4. These numbers are 1.9,4.8,-0.43,1.3,-1.5,1.5,1.2,1.1,-1.6,1.9. So i got $\sum_1^{10} X_i-4*n=-209.73$. So $Y_{10}=e^{-209.73a}$
Now, how to proceed further?