= Solution
The <natural filtration> of $S$ is the <natural Brownian filtration>: since $\sigma>0$, the equation $W_t=(\log(S_t/S_0)-\mu t)/\sigma$ recovers the entire <Brownian motion> history from the <stock> history. For $s<t$, <independent increments> and the <moment-generating function of a normal distribution> give
$$
\mathbb E[S_t\mid\mathcal F_s^S]
=S_s\mathbb E e^{\mu(t-s)+\sigma(W_t-W_s)}
=S_s e^{(\mu+\sigma^2/2)(t-s)}.
$$
All these <expectations> are finite. Since $S_s>0$, the <martingale> identity holds for every $s<t$ exactly when the last exponential equals one. \b[The required logarithmic drift is]
$$
\boxed{\mu=-\frac{\sigma^2}{2}.}
$$
This is the distinction between the <drift> of $\log S$ and the <drift> of $S$: here the latter is zero.
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