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

2025-04-12

AI Summary

This paper addresses the nonparametric estimation of the Hurst parameter $H$ in rough stochastic volatility (rSV) models. It focuses on constructing scale-invariant estimators from discrete observations of the integrated variance process and extends their applicability to general nonlinear functionals of the latent fractional Brownian motion—i.e., nonlinear composite processes. The work establishes, for the first time, the consistency and almost-sure convergence rate of such estimators under generalized rSV models, thereby overcoming prior restrictions to linear or narrowly specified functionals. Methodologically, the analysis integrates tools from functional analysis of fractional Brownian motion, path regularity theory, and an enhanced law of large numbers. The theoretical results provide a rigorous foundation for empirical estimation of $H$ in practical models such as fractional SV (fSV), while also yielding computationally efficient algorithms.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In [8], 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 fractional stochastic volatility model.

Problem

• Estimating Hurst parameter for rough volatility models
• Extending convergence to nonlinear function cases
• Proving almost sure convergence rate generally

Innovation

• Scale-invariant estimator for Hurst parameter
• Extends convergence to nonlinear functions
• Applies to rough stochastic volatility models