Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

TL;DR

This paper studies the behavior of generalized $k$-variations for solutions to a fractional wave equation driven by Gaussian noise, establishing CLTs, convergence rates, and applying these results to estimate the Hurst parameter.

Contribution

It introduces a new analysis of generalized $k$-variations for fractional wave equations and develops consistent estimators for the Hurst parameter using Malliavin calculus.

Findings

CLT holds for $k$-variations when $p> H+1/4$

Convergence rates are estimated via Stein-Malliavin calculus

Several consistent estimators for the Hurst index are proposed and validated

Abstract

We analyze the generalized $k$-variations for the solution to the wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian with Hurst parameter $H>\frac{1}{2}$ in time and which is white in space. The $k$-variations are defined along {\it filters} of any order $p\geq 1$ and of any length. We show that the sequence of generalized $k$-variation satisfies a Central Limit Theorem when $p> H+\frac{1}{4}$ and we estimate the rate of convergence for it via the Stein-Malliavin calculus. The results are applied to the estimation of the Hurst index. We construct several consistent estimators for $H$ and these estimators are analyzed theoretically and numerically.