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

ArXiv ID: 2504.04992 “View on arXiv”

Authors: Unknown

Abstract

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

Keywords: Asymptotic Expansion, Hartman-Watson Distribution, Stochastic Calculus, Error Bounds, Stochastic Volatility, Quantitative Methods

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is dominated by advanced asymptotic analysis and complex analytic methods (saddle point expansions, steepest descent, contour integration) to derive error bounds for a special function, but contains no data, backtests, or implementation details for practical quant strategies.

  flowchart TD
    A["Research Goal: <br>Bound error of asymptotic expansion of<br>Hartman-Watson integral <br><i>θ(r,t)</i> as <i>t → 0</i> with <i>ρ = rt</i> constant"] --> B

    subgraph B ["Key Methodology"]
        B1["Stochastic Calculus <br>(Brownian motion paths)"]
        B2["Rigorous Bounds <br>(Non-linear ODEs)"]
        B3["Asymptotic Analysis <br>(Method of Dominant Terms)"]
    end

    B --> C["Data / Inputs:<br>Asymptotic Form:<br><i>θ(ρ/t,t) ~ (1/2πt)e^{"-1/t F(ρ)"} G(ρ)</i>"]

    C --> D["Computational Process:<br>Derive uniform error bound<br><i>|θ(t,ρ)| ≤ 1/70 t</i><br>Valid for all <i>ρ > 0</i>"]

    D --> E["Key Outcomes:<br>1. Explicit error bound<br>2. Uniformity over <i>ρ</i><br>3. Validity for small <i>t</i>"]