Before revision:   As an application we derive, under an uniformity assumption in \rho, the leading asymptotics of the density of the time average of the geometric Brownian motion as t\to 0.
After revision:  An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in \rho. As an application we derive, under an uniformity assumption in \rho, the leading asymptotics of the density of the time average of the geometric Brownian motion as t\to 0.