Asymptotic expansion for the Hartman-Watson distribution

TL;DR

This work derives a saddle-point expansion for the Hartman-Watson integral in the small-$t$ regime with fixed $\rho=rt$, yielding a leading exponential factor $e^{-\frac{1}{t}(F(\rho)-\frac{\pi^2}{2})}$ and a calculable pre-factor $G(\rho)$, plus a uniform error bound. The leading term provides a practical, uniform approximation for the Hartman-Watson density and enables the derivation of the small-$t$ asymptotics for the time-average of the geometric Brownian motion via Yor's formula. The authors apply this expansion to obtain the asymptotic density $f(a,t)\frac{da}{a}$ of $\frac{1}{t}A_t^{(\mu)}$, with an explicit rate function $J(a)$ and prefactor $g(a,\mu)$, and connect $J(a)$ to the large-deviation rate function $\mathcal{J}_{BS}$ by $J(a)=\frac{1}{4}\mathcal{J}_{BS}(a)$. The analysis links analytical, probabilistic, and numerical perspectives, providing uniform error control and insights into the LD regime for Asian-option-type quantities.

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 $θ(r,t)$ 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.

Figures (5)

• Figure 1: Left: Plot of $F(\rho)$ given in Eq. (\ref{['Fsol']}). The two branches in Eq. (\ref{['Fsol']}) are shown as the blue and red curves, respectively. The function $F(\rho)$ has a minimum at $\rho=\frac{\pi}{2}$, with $F(\frac{\pi}{2})=\frac{3\pi^2}{8}$. The dashed line shows the asymptotic line $F(\rho) \sim \rho$ for $\rho\to \infty$. Right: Plot of $G(\rho)$ defined in Eq. (\ref{['Gsol']}).
• Figure 2: Integration contours for $I_+(\rho,t)$ in the $\xi$ complex plane for the application of the asymptotic expansion. The contours for $I_-(\rho,t)$ are obtained by changing the sign of $y$ (reflection in the real axis). The red dots show the saddle points. Left: contour for $0<\rho<1$. The contour passes through the saddle points $B(\xi=-x_1+i\pi)$ and $A(\xi=x_1+i\pi)$. Middle: contour for $\rho>1$. The contour passes through the saddle point $S(\xi=i y_1)$. Right: the contour for $\rho=1$ passes through the fourth order saddle point at $S(\xi = i\pi)$.
• Figure 3: Plot of $\hat{\theta}(r,t)$ vs $t$ from the asymptotic expansion of Proposition \ref{['prop:1']} (black) and from the Theorem 1 of Gerhold Gerhold2011 (blue). The red curves show the results of direct numerical integration of $\theta(r,t)$. The three panels correspond to the three values of $r=0.5, 1.0,1.5$.
• Figure 4: Plot of $\hat{\theta}(r,t)$ vs $t$ defined in (\ref{['tapprox']}) giving the leading asymptotic result from Proposition \ref{['prop:1']} for three values of $r=0.5, 1.0,1.5$ (solid, dashed, dotted). The widths of the curves reflect the error bound (\ref{['errbound']}).
• Figure 5: Plot of the function $\tilde{g}_2(\rho)$ appearing in the subleading correction to the asymptotic expansion of the Hartman-Watson integral (\ref{['thetasub']}).

Theorems & Definitions (18)

• Proposition 1
• proof : Proof of Proposition \ref{['prop:1']}
• Remark 2
• Remark 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6