On quadratic variations for the fractional-white wave equation
Author:
Radomyra Shevchenko
Journal:
Theor. Probability and Math. Statist. 108 (2023), 185-207
MSC (2020):
Primary 60G15, 60G22, 60H15, 62F12; Secondary 62M30, 60F05
DOI:
https://doi.org/10.1090/tpms/1192
Published electronically:
May 2, 2023
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Additional Information
Abstract: This paper studies the behaviour of quadratic variations of a stochastic wave equation driven by a noise that is white in space and fractional in time. Complementing the analysis of quadratic variations in the space component carried out in [Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise, Electron. J. Stat. 12 (2018), no. 2, 3639–3672] and [Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180], it focuses on the time component of the solution process. For different values of the Hurst parameter a central and a noncentral limit theorems are proved, allowing to construct parameter estimators and compare them to the findings in the space-dependent case. Finally, rectangular quadratic variations in the white noise case are studied and a central limit theorem is demonstrated.
References
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- Radomyra Shevchenko, Meryem Slaoui, and C. A. Tudor, Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180. MR 4066122, DOI 10.1016/j.jspi.2019.10.008
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References
- R. Altmeyer, T. Bretschneider, J. Janák, and M. Reiß, SIAM/ASA J. Uncertain. Quantif. 10 (2022), no. 1, 179–199. MR 4376300
- S. Bai and M. S. Taqqu, Behavior of the generalized Rosenblatt process at extreme critical exponent values, Ann. Probab. 45 (2017), no. 2, 1278–1324. MR 3630299
- M. Bibinger and M. Trabs, On central limit theorems for power variations of the solution to the stochastic heat equation, Stochastic models, statistics and their applications, Springer Proc. Math. Stat., vol. 294, Springer, Cham, 2019, pp. 69–84. MR 4043170
- C. Chong and R. C. Dalang, Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs, Bernoulli (in press), https://doi.org/10.48550/arXiv.2006.15817.
- I. Cialenco, Statistical inference for SPDEs: an overview, Stat. Inference Stoch. Process. 21 (2018), no. 2, 309–329. MR 3824970
- J. Clarke de la Cerda and C. A. Tudor, Hitting times for the stochastic wave equation with fractional colored noise, Rev. Mat. Iberoam. 30 (2014), no. 2, 685–709. MR 3231213
- F. Hildebrandt and M. Trabs, Parameter estimation for SPDEs based on discrete observations in time and space, Electron. J. Stat. 15 (2021), no. 1, 2716–2776. MR 4280160
- M. Khalil and C. A. Tudor, Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise, Electron. J. Stat. 12 (2018), no. 2, 3639–3672. MR 3870508
- M. Khalil, C. A. Tudor, and M. Zili, Spatial variation for the solution to the stochastic linear wave equation driven by additive space-time white noise, Stoch. Dyn. 18 (2018), no. 5, 1850036, 20. MR 3853264
- D. Nualart, The Malliavin calculus and related topics, second ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
- G. Pasemann, S. Flemming, S. Alonso, C. Beta, and W. Stannat, Diffusivity Estimation for Activator-Inhibitor Models: Theory and Application to Intracellular Dynamics of the Actin Cytoskeleton, J. Nonlinear Sci. 31 (2021), no. 3, 59. MR 4255684
- G. Peccati and C. A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, Séminaire de Probabilités XXXVIII, Lecture Notes in Math., vol. 1857, Springer, Berlin, 2005, pp. 247–262. MR 2126978
- R. Shevchenko, M. Slaoui, and C. A. Tudor, Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180. MR 4066122
- C. A. Tudor, Analysis of variations for self-similar processes, Probability and its Applications (New York), Springer, Cham, 2013, A stochastic calculus approach. MR 3112799
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Additional Information
Radomyra Shevchenko
Affiliation:
Fakultät für Mathematik, LSIV, TU Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany
Email:
radomyra.shevchenko@tu-dortmund.de
Keywords:
Hurst parameter estimation,
fractional Brownian motion,
stochastic wave equation,
quadratic variations,
Stein–Malliavin calculus
Received by editor(s):
June 30, 2021
Accepted for publication:
January 23, 2022
Published electronically:
May 2, 2023
Additional Notes:
The support of the German Research Foundation (DFG) via SFB 823 is gratefully acknowledged.
Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv
