 Solution

For a positive starting value, Itô's formula applied to $\log S_u$ gives $d\log S_u=(\mu -{\sigma }^2/2)du+\sigma d{W}_u$. Hence the geometric Brownian motion isThe future Brownian increment $W_T-W_t$ is independent of ${\mathcal{F}}_t$ and has normal distribution $N(0,\tau )$. If ${\tilde{W}}_v=W_{t+v}-W_t$, then $\tilde{W}$ is a fresh Brownian motion and the increment is exactly ${\tilde{W}}_{\tau }$. 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 $S>0$.