Asymptotic expansion for the Hartman-Watson distribution
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arXiv
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| Natura: | Preprint |
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2020
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| _version_ | 1866913617198186496 |
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| author | Pirjol, Dan |
| author_facet | Pirjol, Dan |
| contents | 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 $θ(r,t)$ 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 $θ(ρ/t,t)$ at fixed $ρ>0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $ρ$. 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 $\mathbb{P}(\frac{1}{t} \int_0^t e^{2(B_s+μs)} ds \in da) \sim (2πt)^{-1/2} g(a,μ) e^{-\frac{1}{t} J(a)} da/a$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_09579 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansion for the Hartman-Watson distribution Pirjol, Dan Probability Numerical Analysis Mathematical Finance 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 $θ(r,t)$ 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 $θ(ρ/t,t)$ at fixed $ρ>0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $ρ$. 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 $\mathbb{P}(\frac{1}{t} \int_0^t e^{2(B_s+μs)} ds \in da) \sim (2πt)^{-1/2} g(a,μ) e^{-\frac{1}{t} J(a)} da/a$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory. |
| title | Asymptotic expansion for the Hartman-Watson distribution |
| topic | Probability Numerical Analysis Mathematical Finance |
| url | https://arxiv.org/abs/2001.09579 |