= Solution
{c}
For $v\leq t$, the two stock paths agree, so their geometric averages agree. For $v>t$, the future log-price is
$$
\log\widetilde S_u=\log S_t+(r-\sigma^2/2)(u-t)+\sigma(W_u-W_t).
$$
Split the average's log-integral into $[0,t]$ and $[t,v]$. The first part is $t\log A_t$, while integration of the displayed formula gives
$$
v\log\widetilde A_v=t\log A_t+(v-t)\log S_t+\tfrac12(r-\sigma^2/2)(v-t)^2
+\sigma\int_t^v(W_u-W_t)du.
$$
Dividing by $v$ and exponentiating proves
$$
\boxed{\widetilde A_v=A_t^{t/v}S_t^{(v-t)/v}
\exp\left[\left(r-\frac{\sigma^2}{2}\right)\frac{(v-t)^2}{2v}
+\frac\sigma v\int_t^v(W_u-W_t)du\right]\quad(v>t).}
$$
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