= Solution
The independent Gaussian increments of <Brownian motion> give $E[Y_t\mid\mathcal F_s]=Y_s e^{\mu(t-s)}$; the exponential is integrable at every finite time. Therefore
$$
\boxed{Y\text{ is a supermartingale}\iff\mu\le0,\qquad Y\text{ is a martingale}\iff\mu=0.}
$$
Similarly $E[Y_t^{-1}\mid\mathcal F_s]=Y_s^{-1}e^{(\sigma^2-\mu)(t-s)}$, so $\boxed{Y^{-1}\text{ is a martingale}\iff\mu=\sigma^2}$. This uses actual conditional expectations, not only the local stochastic differential equation.
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