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Asymptotic expansion for the Hartman-Watson distribution | | Dan Pirjol
; | | Date: |
Mon, 27 Jan 2020 04:04:33 GMT (1824kb) | | Abstract: | 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 $ heta(r,t)$ which is difficult to evaluate numerically for small
$t o 0$. Using saddle point methods, we obtain the first two terms of the
$t o 0$ expansion of $ heta(
ho/t,t)$ at fixed $
ho >0$. As an application
we derive, under an uniformity assumption in $
ho$, the leading asymptotics of
the density of the time average of the geometric Brownian motion as $t o 0$.
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. | | Source: | arXiv, 2001.9579 | | Services: | Forum | Review | PDF | Favorites | |
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