Electronic Journal of Statistics

Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise

Marwa Khalil and Ciprian A. Tudor

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We compute the covariance function of the solution to the linear stochastic wave equation with fractional noise in time and white noise in space. We apply our findings to analyze the correlation structure of this Gaussian process and to study the asymptotic behavior in distribution of its spatial quadratic variation. As an application, we construct a consistent estimator for the Hurst parameter.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 3639-3672.

Received: October 2017
First available in Project Euclid: 31 October 2018

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60H05: Stochastic integrals 60G18: Self-similar processes

Fractional Brownian motion stochastic wave equation quadratic variation Stein-Malliavin calculus central limit theorem almost sure central limit theorem Hurst parameter estimation

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Khalil, Marwa; Tudor, Ciprian A. Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise. Electron. J. Statist. 12 (2018), no. 2, 3639--3672. doi:10.1214/18-EJS1488. https://projecteuclid.org/euclid.ejs/1540951344

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  • [1] Balan, R. M. and Tudor, C. A. (2010)., The stochastic wave equation with Fractional colored noise: a random field approach. Stoch. Process. Appl. 120, 2468-2494.
  • [2] Bercu, B. and I. Nourdin, I. and Taqqu, M. (2010)., Almost sure central limit theorems on the Wiener space. Stochastic Process. Appl. 120 (9), 1607-1628.
  • [3] Biermé, H. and Bonami, A. and Nourdin, I. and Peccati, G. (2012)., Optimal Berry-Essén rates on the Wiener space. ALEA 9 (2), 473-500.
  • [4] Breuer, P. and Major, P. (1983)., Central limit theorems for non-linear functionals of Gaussian fields. J. Mult. Anal. 13, 425-441.
  • [5] Cabana, E. (1970)., The vibrating string forced by white noise. Z. Wahrscheinlichkeits theorie Verw Geb. 15, 111-130.
  • [6] Clarke De la Cerda, J and Tudor, C. A. (2014)., Hitting probabilities for the stochastic wave equation with fractional colored noise. Rv. Mat. Iberoam. 30 (2), 685-709.
  • [7] Coeurjolly, J.F. (2001)., Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 30, 199-227.
  • [8] Dalang, R. C. (1999)., Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDE. Electr. J. Probab. 4 (6), 1-29.
  • [9] Dalang, R. C. (1962)., The stochastic wave equation, A minicourse on stochastic partial differential equations, 2nd edn. Springer-Berlin.
  • [10] Dalang, R. C. and Sanz-Solé, M. (2009)., Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three. Memoirs of the American Mathematical Society 199, 1-70.
  • [11] Dudley, R. M. (1967)., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290-330.
  • [12] Fernique, X. (1975)., Régularité des trajectoires des fonctions aléatoires gaussiennes. Ecole d’été de Probabilités de Saint-Flour IV-1974, Lecture Notes in Math., 480, Springer.
  • [13] Folland, G. B. (1975)., Introduction to Partial Differential Equations. Princeton Univ. Press.
  • [14] Ibragimov, I. A. and Lifshits, M. A. (2000)., On the convergence of generalized moments in almost sure central limit theorem. Theory Probab. Appl. 44 (2), 254-272.
  • [15] Khalil, M. and Tudor, C. A and Zili, M. (2017)., Spatial variation for the solution to the stochastic linear wave equation driven by additive space-time white noise. Stochastics and Dynamics 18 (5).
  • [16] Lévêque, O. (2001)., Hyperbolic stochastic partial differential equations driven by boundary noises. Thèse, Vol. 2452, EPFL, Lausanne.
  • [17] Lévy, P. (1937)., Théorie de l’addition des variables aléatoires. Gauthiers-Villars.
  • [18] Mocioalca, O. and Viens, F. G. (2004)., Skorohod integration and stochastic calculus beyond the fractional Brownian scale. J. Funct. Anal. 222 (2), 385-434.
  • [19] Neufcourt, L. and Viens, F. G. (2016)., A third-moment theorem and precise asymptotics for variations of stationary Gaussian sequences. ALEA 13 (1), 239-264.
  • [20] Nourdin, I. and Peccatti, G. (2012)., Normal Approximations with Malliavin Calculus from Stein’s Method to Universality. Cambridge University Press, Cambridge.
  • [21] Nourdin, I. and Peccati, G. (2015)., The optimal fourth moment theorem. Proc. Amer. Math. Soc. 143 (7), 3123-3133.
  • [22] Nualart, D. (2006)., The Malliavin calculus and related topics. Probability and its Applications. Second edition. Springer-Verlag, Berlin.
  • [23] Pipiras, V. and Taqqu, M. S. (2000)., Integration questions related to fractional Brownian motion. Prob. Theory Relat. Fields. 118, 251-291.
  • [24] Quer-Sardanyons, L. and Tindel, S. (2007)., The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stoch. Proc. Appl. 117, 1448-1472.
  • [25] Tindel, S. and Tudor, C. A. and Viens, F. G. (2004)., Sharp Gaussian regularity on the circle and application to the fractional stochastic heat equation. J. Funct. Anal. 217 (2), 280-313.
  • [26] Treves, F. (1975)., Basic Linear Partial Differential Equations. Academic Press.
  • [27] Torres, S. and Tudor, C. A. and Viens, F. G. (2014)., Quadratic variations for the fractional-colored stochastic heat equation. 19, 76.
  • [28] Tudor, C. A (2013)., Analysis of variations for self-similar processes. A Stochastic Calculus Approach. Springer, Cham.
  • [29] Tudor, C. A and Viens, F. G (2009)., Variations and estimators for the selfsimilarity parameter through Malliavin calculus. Annals of Probability. 37 (6), 2093-2134.
  • [30] Viens, F. G. and Vizcarra, A. (2007)., Supremum concentration inequality and modulus of continuity for sub-$n$th-chaos processes. Journal of Functional Analysis 248 (1), 1-26.