= Solution
{c}
Use a <filtration (probability theory)> with respect to which $B$ is Brownian. The process is adapted and continuous because it is a continuous function of $B_t$ and $t$. The <moment-generating function> gives
$$
\mathbb E M_t=1,\qquad
\mathbb E M_t^2
=e^{-\alpha^2t}\mathbb E e^{2\alpha B_t}
=e^{\alpha^2t}<\infty.
$$
For $0\leq s<t$, write
$$
M_t=M_s\exp\{\alpha(B_t-B_s)-\alpha^2(t-s)/2\}.
$$
The increment is independent of $\mathcal F_s$ and normal with variance $t-s$, so the <conditional expectation> of the second factor is one. Thus
$$
\boxed{\mathbb E(M_t\mid\mathcal F_s)=M_s.}
$$
This verifies the <martingale> property from the definition and proves square integrability at each finite time, as well as <continuity>. It does not assert uniform square integrability over an infinite time horizon.
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