 Solution

The moment-generating function of a normal distribution givesIndependence of $X_{n+1}$ from ${\mathcal{F}}_n$ impliesEvery exponential here is integrable. Thus the process is a martingale exactly when $-\lambda +{\lambda }^2/2=0$. Its two roots are $0$ and $2$, and the unique positive root isThe resulting exponential martingale of a random walk is nonnegative and has expectation $1$ at every finite time.