Abstract
Via Malliavin calculus, we analyze the limit behavior in distribution of the spatial wavelet variation for the solution to the stochastic linear wave equation with fractional Gaussian noise in time and white noise in space. We propose a wavelet-type estimator for the Hurst parameter of the this solution and we study its asymptotic properties.
Highlights
IntroductionThe parameter estimation for stochastic (partial) differential equations constitutes a topic of wide interest (see, among many others, the monographs or surveys [8, 14] or [20])
The parameter estimation for stochastic differential equations constitutes a topic of wide interest
The statistical inference for stochastic models driven by fractional Brownian motion and related processes became a popular topic, due to the developments of the stochastic calculus for fractional processes
Summary
The parameter estimation for stochastic (partial) differential equations constitutes a topic of wide interest (see, among many others, the monographs or surveys [8, 14] or [20]). We will consider the linear stochastic wave equation (2.1) driven by a fractional-white Gaussian noise (i.e. a Gaussian noise that behaves as a fractional Brownian motion in time and as a white noise in space) and we construct and analyze statistical estimators for the Hurst index of the solution, based on the discrete observations of the solution in space and time. By analyzing the asymptotic behavior of the wavelet variation VN (t, a) as N → ∞, we are able to construct, via a log–log regression of the empirical variance onto several scales, an estimator for the Hurst parameter of the solution to (2.1) and to analyze its asymptotic behavior.
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