On the Rate of Convergence of Estimating the Hurst Parameter of Rough Stochastic Volatility Models

In [Han & Schied, 2023, arXiv 2307.02582], an easily computable scale-invariant estimator Rˢₙ 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 Rˢₙ also consistently estimates the Hurst parameter when applied to the antiderivative of g ∘ 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.