On the rate of convergence of estimating the Hurst parameter of rough stochastic volatility models

In [Han & Schied, 2023, $\mathit{\text{arXiv}}\mathit{\text{2307.02582}}$], an easily computable scale-invariant estimator ${\widehat{\mathcal{R}}}_n^s$ 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 ${\widehat{\mathcal{R}}}_n^s$ 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.