= Solution
{c}
For a positive starting value, <Itô's formula> applied to $\log S_u$ gives $d\log S_u=(\mu-\sigma^2/2)du+\sigma dW_u$. Hence the <geometric Brownian motion> is
$$
\boxed{S_T=S\exp\!\left[(\mu-\sigma^2/2)\tau+\sigma(W_T-W_t)\right].}
$$
The future <Brownian increment> $W_T-W_t$ is independent of $\mathcal F_t$ and has <normal distribution> $N(0,\tau)$. If $\widetilde W_v=W_{t+v}-W_t$, then $\widetilde W$ is a fresh <Brownian motion> and the increment is exactly $\widetilde W_\tau$. This specifies the meaning of the paper's abbreviated $W_\tau$; the original process value $W_{T-t}$ is not generally that future increment. The <stochastic exponential> solution also verifies positivity for $S>0$.
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