Generalized k-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus
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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>1/2 in time and which is white in space. The k-variations are defined along filters of any order p≥ 1 and of any length. We show that the sequence of generalized k-variation satisfies a Central Limit Theorem when p> H+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.
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