This process is called Wald's martingale. ($W_{t}$ is a wiener process).
And I'm having trouble proving that it is in fact a martingale.
What I have:
$E(X_{t} | F_{s}) = E(\exp(\lambda (W_{t} - W_{s} + W_{s}) - \frac{1}{2}\lambda^{2}t)|F_{s})= E\left[\exp(\lambda(W_{t} - W_{s})\right] \exp(\lambda W_{s})\exp(-\frac{1}{2}\lambda^{2}t) = \exp(\lambda W_{s} - \frac{1}{2}\lambda^{2}t) $
Which is not equal to $\exp(\lambda W_{s} - \frac{1}{2}\lambda^{2}s)$, which it should have been equal to if it was a martingale.
Where am I wrong here?
As suggested in a comment above, note that for $s<t,$ $$\begin{align*} E(X_t|F_s)&=E\left[\exp\left(\lambda(W_t-W_s)+\lambda W_s-\frac{1}{2}\lambda^2t\right)|F_s\right]\\ &=\exp(\lambda W_s-\frac{1}{2}\lambda^2t)E\left[\exp(\lambda(W_t-W_s))|F_s\right]. \end{align*}$$
Now note that $W_t-W_s\sim N(0, t-s)\sim \sqrt{t-s}N(0, 1)$ and is independent of $F_s.$
Therefore, $E\left[\exp(\lambda(W_t-W_s))|F_s\right]=E\left[\exp(\lambda\sqrt{t-s}N(0, 1)\right].$
The problem reduces to showing that the exponential moment $$E[\exp(\mu N(0, 1)]=\exp(\frac{1}{2}\mu^2).$$