Exponential Martingale Problem

Question

I'm trying to prove $M_n = 2^{n}e^{-X_n}$ is a Martingale where $X_{n+1} = X_n + Y_{n+1}$ (Random Walk)

My initial thoughts were along the line of: $$E[2^{n+1}e^{-(X_n+Y_{n+1})}|F_n]$$ $$2^{n+1}E[e^{-(X_n+Y_{n+1})}|F_n]$$ $$2^{n+1}e^{-X_n}E[e^{-Y_{n+1}}|F_n]$$

Then I get stuck because usually I would try to see if I can equate the $e^{-Y_{n+1}}$ term to $h(u)$ in $e^{uX_n+h(u)}$ Not too sure where I should go from here

Answer 1

Hint: $Y_{n+1}$ is independent of ${\cal F}_n$.