= Solution
{c}
The solution is the continuous <Doléans-Dade exponential>:
$$
\boxed{S_t=\exp\{X_t-X_0-\tfrac12[X]_t\}.}
$$
Here $[X]$ is the continuous <quadratic variation> starting from zero, denoted by the angle-bracket notation in the source. If $X=M+A$ with $A$ continuous of finite variation, then $[X]=[M]$.
Set $U=X-X_0-\frac12[X]$. Its quadratic variation is still $[X]$. Applying <Itô's formula> to $e^U$ gives
$$
dS=S\,dU+\tfrac12S\,d[U]
=S\,dX-\tfrac12S\,d[X]+\tfrac12S\,d[X]=S\,dX.
$$
The initial value is $S_0=1$. This verification applies to the full semimartingale, including its finite-variation part.
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