Paper: arXiv 2504.09276

Abstract

In [“Han & Schied, 2023, \textit{“arXiv 2307.02582”}”], an easily computable scale-invariant estimator $\widehat{"\mathscr{R"}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its antiderivative. This paper extends this convergence result by proving that $\widehat{"\mathscr{R"}}^s_n$ also consistently estimates the Hurst parameter when applied to the antiderivative of $g \circ X$ for a general nonlinear function $g$. We also establish an almost sure rate of convergence in this general setting. Our result applies, in particular, to the estimation of the Hurst parameter of a wide class of rough stochastic volatility models from discrete observations of the integrated variance, including the rough fractional stochastic volatility model.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly mathematical, featuring advanced stochastic calculus, fractional Brownian motion, and a rigorous asymptotic analysis of an estimator, placing it in the high math category. However, it focuses purely on theoretical convergence rates and pathwise properties without any backtesting, implementation details, or empirical data analysis, making it low in empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Estimate Hurst Parameter H"] --> B["Methodology<br>Extend Scale-Invariant Estimator"]
  B --> C["Input Data<br>Discrete Observations of Antiderivative"]
  C --> D{"Compute Estimator<br>\\widehat{\\mathscr{R"}}^s_n<br>on Log-Returns}
  D --> E["Theoretical Analysis<br>Proof of Consistency & Convergence Rate"]
  E --> F["Key Outcome 1<br>Estimation of H in fBM"]
  E --> G["Key Outcome 2<br>Estimation of H in Rough SV Models"]