 Solution

Put $Q_t=⟨M,M{⟩}_t$ and $F(x,q)=e^{\sigma x-{\sigma }^{2}q/2}$. In the two-variable Itô formula, $Q$ has finite variation, while the only nonzero quadratic term is $d⟨M⟩=dQ$. ThereforeThis 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 $Q_t\ge 0$ and $|M_t|<C$,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 $Z$ is a martingale on every $[0,T]$. The bound on $Q$ also gives a positive deterministic lower bound $e^{-|\sigma |C-{\sigma }^{2}C/2}$, although the upper bound already proves the martingale assertion.