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.  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. This has the form P(1{t} \int_0^t e^{2(B_s+\mu s)} ds \in da) = (2\pi t)^{-1/2} g(a,\mu) e^{-1{t} J(a)} (1 + O(t)), with an exponent J(a) which reproduces the known result obtained previously using Large Deviations theory.
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 P(1{t} \int_0^t e^{2(B_s+\mu s)} ds \in da) = (2\pi t)^{-1/2} g(a,\mu) e^{-1{t} J(a)} (1 + O(t)), with an exponent J(a) which reproduces the known result obtained previously using Large Deviations theory.
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An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in \rho.
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derive, under an uniformity assumption in \rho,
obtain