= Solution
{c}
Put $Q_t=\langle M,M\rangle_t$ and $F(x,q)=e^{\sigma x-\sigma^2q/2}$. In the two-variable <Itô formula>, $Q$ has finite variation, while the only nonzero quadratic term is $d\langle M\rangle=dQ$. Therefore
$$
dF(M_t,Q_t)=\sigma F\,dM_t-\tfrac12\sigma^2F\,dQ_t
+\tfrac12\sigma^2F\,dQ_t=\sigma F\,dM_t.
$$
This proves the <stochastic exponential> solves the equation. Its initial value is $e^{\sigma M_0}$; no normalization $M_0=0$ was stipulated.
The resulting process is initially a <continuous local martingale>. Since $Q_t\geq0$ and $|M_t|<C$,
$$
0<Z_t=e^{\sigma M_t-\sigma^2Q_t/2}\leq e^{|\sigma|C}.
$$
A bounded local martingale is a true <martingale>: apply the martingale identity to a localizing sequence and use <dominated convergence> in conditional expectations. The deterministic bound works on every finite horizon, proving \b[$Z$ is a martingale on every $[0,T]$]. The bound on $Q$ also gives a positive deterministic lower bound $e^{-|\sigma|C-\sigma^2C/2}$, although the upper bound already proves the martingale assertion.
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