Asymptotic expansion for the Hartman-Watson distribution

The Hartman-Watson distribution with density $f_r(t)$ is a probability distribution defined on $t \geq 0$ which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral $\theta(r,t)$ which is difficult to evaluate numerically for small $t\to 0$. Using saddle point methods, we obtain the first two terms of the $t\to 0$ expansion of $\theta(\rho/t,t)$ at fixed $\rho >0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $\rho$. As an application we obtain the leading asymptotics of the density of the time average of the geometric Brownian motion as $t\to 0$. This has the form $\mathbb{P}(\frac{1}{t} \int_0^t e^{2(B_s+\mu s)} ds \in da) \sim (2\pi t)^{-1/2} g(a,\mu) e^{-\frac{1}{t} J(a)} da/a$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.