Martingale and Poisson Process: Show $X_{n}$ defines a nonnegative martingale

Question

The question:

Let {$W_{n}$} be the sequence of waiting times in a Poisson process of intensity $\lambda = 1$. Show that $ X_{n} = 2^{n}e^{-W_{n}}$ defines a nonnegative martingale.

So I must demonstrate $E[X_{n+1}|X_{1},...,X_{n}] = X_{n}$.
The sequence of waiting times are composed of sojourn times $ S_{n}$ so we can write as
$ W_n = S_{0}+ ...+S_{n-1}$.

Now we start from the conditional expectation:
$ E[X_{n+1}|X_{1},...,X_{n}] = E[2^{n+1}e^{-W_{n+1}}|X_{1},...,X_{n}] \\ = E[(2e^{-S_{n}})(2^{n}e^{-W_{n}})|X_{1},...,X_{n}] = X_{n}E[2e^{-S_{n}}] $

At this point the expectation left over looks like an MGF and sojourn times are exponentially distributed, but I don't see how that will disappear to leave $X_{n}$.

Thanks for all your clarifications and help.

Answer 1

For every $S$ exponential with parameter $1$ such as the times $S_n$, $$ \mathbb E(\mathrm e^{-S})=\int_0^{+\infty}\mathrm e^{-x}f_S(x)\mathrm dx=\int_0^{+\infty}\mathrm e^{-2x}\mathrm dx=\frac12. $$