Asymptotic expansion for the Hartman-Watson distribution
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The Hartman-Watson distribution with density f_r(t) is a probability distribution defined on t ≥ 0 which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral θ(r,t) which is difficult to evaluate numerically for small t→ 0. Using saddle point methods, we obtain the first two terms of the t→ 0 expansion of θ(ρ/t,t) at fixed ρ >0. As an application we derive, under an uniformity assumption in ρ, the leading asymptotics of the density of the time average of the geometric Brownian motion as t→ 0. This has the form P(1/t∫_0^t e^2(B_s+μ s) ds ∈ da) = (2π t)^-1/2 g(a,μ) 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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