 Solution

Apply Itô's formula to the logarithm of the positive geometric Brownian motion. The quadratic-variation correction givesIntegrating yieldsThe 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 ${\tilde{W}}_v=W_{t+v}-W_t$, evaluated at $\tau$; it need not equal the original process at time $T-t$.