Hartman-Watson 分布的渐近展开
Asymptotic expansion for the Hartman-Watson distribution
AI总结：
本文利用鞍点方法推导了 Hartman-Watson 分布密度积分在 $t\to 0$ 时的前两项渐近展开及一致误差界，并据此获得了几何布朗运动时间平均密度的主导渐近性，验证了大偏差理论的结果。
摘要：
密度为 $f_r(t)$ 的 Hartman-Watson 分布是定义在 $t \geq 0$ 上的概率分布，出现在应用概率的若干问题中。该分布的密度由积分 $θ(r,t)$ 表示，当 $t\to 0$ 时在数值上难以计算。使用鞍点方法，我们获得了在固定 $ρ>0$ 下 $θ(ρ/t,t)$ 的 $t\to 0$ 展开的前两项。通过对被积函数的数值估计得到了误差界，且该误差界在 $ρ$ 上是一致的。作为应用，我们获得了几何布朗运动时间平均密度在 $t\to 0$ 时的主导渐近性。其形式为 $\mathbb{P}(\frac{1}{t} \int_0^t e^{2(B_s+μs)} ds \in da) \sim (2πt)^{-1/2} g(a,μ) e^{-\frac{1}{t} J(a)} da/a$，其指数 $J(a)$ 重现了先前使用大偏差理论获得的已知结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
