= Solution
{c}
Apply <Itô's formula> to the <logarithm> of the positive <geometric Brownian motion>. The quadratic-variation correction gives
$$
d\log S_t=(r-\sigma^2/2)dt+\sigma dW_t.
$$
Integrating yields
$$
\boxed{S_T=S_t\exp\!\left[(r-\sigma^2/2)(T-t)+\sigma(W_T-W_t)\right].}
$$
The increment is a <Gaussian random variable> with <variance> $T-t$, independent of current information. Writing it as $W_\tau$ means a fresh future <Brownian motion> $\widetilde W_v=W_{t+v}-W_t$, evaluated at $\tau$; it need not equal the original process at time $T-t$.
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