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 $tto 0$. Using saddle point methods, we obtain the first two terms of the $tto 0$ 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 $tto 0$. This has the form $mathbb{P}(frac{1}{t} int_0^t e^{2(B_s+mu s)} ds in da) = (2pi t)^{-1/2} g(a,mu) e^{-frac{1}{t} J(a)} (1 + O(t))$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.