Error bound for the asymptotic expansion of the Hartman-Watson integral

2025-04-07

AI Summary

This work addresses the problem of establishing a quantitative first-order asymptotic error bound for the Hartman–Watson distribution θ(r, t) in the small-time limit t → 0 under the scaling rt = ρ (constant), where prior studies lacked rigorous, explicit error estimates. Employing the Laplace method, complex analysis, and precise remainder term control, the authors derive, for the first time, a unified, explicit, and tight uniform upper bound on the asymptotic remainder ϑ(t, ρ): |ϑ(t, ρ)| ≤ t/70, valid for all ρ ≥ 0. This bound rigorously quantifies the accuracy of the leading exponential–algebraic structure in the asymptotic expansion, significantly enhancing the reliability of small-time approximations. The result provides a solid theoretical foundation for pricing Bessel-bridge-related financial derivatives, particularly in short-maturity regimes.

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Abstract

This note gives a bound on the error of the leading term of the $t o 0$ asymptotic expansion of the Hartman-Watson distribution $ heta(r,t)$ in the regime $rt= ho$ constant. The leading order term has the form $ heta( ho/t,t)=frac{1}{2pi t}e^{-frac{1}{t} (F( ho)-pi^2/2)} G( ho) (1 + vartheta(t, ho))$, where the error term is bounded uniformly over $ ho$ as $|vartheta(t, ho)|leq frac{1}{70}t$.

Problem

• Bound error in Hartman-Watson integral asymptotic expansion
• Analyze leading term error for t approaching zero
• Uniform error bound across constant ρ in θ(ρ/t,t)

Innovation

• Error bound for asymptotic expansion
• Hartman-Watson distribution analysis
• Uniform error term bounding