Estimation of the Hurst and diffusion parameters in fractional stochastic heat equation

Avetisian, D.; Ralchenko, K.

doi:10.1090/tpms/1145

Authors: D. A. Avetisian and K. V. Ralchenko
Journal: Theor. Probability and Math. Statist. 104 (2021), 61-76
MSC (2020): Primary 60G22, 60H15, 62F10, 62F12
DOI: https://doi.org/10.1090/tpms/1145
Published electronically: September 24, 2021
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper deals with a one-dimensional stochastic heat equation driven by a space-only fractional Brownian noise. We construct strongly consistent estimators of two unknown parameters, namely, the diffusion parameter $\sigma$ and the Hurst parameter $H\in (0,1)$, based on the discrete-time observations of a solution. We also prove joint asymptotic normality of the estimators in the case $H\in \left (0,\frac 34\right )$.

References

Shin Ichi Aihara and Arunabha Bagchi, Stochastic hyperbolic dynamics for infinite-dimensional forward rates and option pricing, Math. Finance 15 (2005), no. 1, 27–47. MR 2116795, DOI 10.1111/j.0960-1627.2005.00209.x
S. I. Aihara and A. Bagchi, Parameter estimation of parabolic type factor model and empirical study of US treasury bonds, System modeling and optimization, IFIP Int. Fed. Inf. Process., vol. 199, Springer, New York, 2006, pp. 207–217. MR 2249335, DOI 10.1007/0-387-33006-2_{1}9
Miguel A. Arcones, Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors, Ann. Probab. 22 (1994), no. 4, 2242–2274. MR 1331224
Diana Avetisian and Kostiantyn Ralchenko, Ergodic properties of the solution to a fractional stochastic heat equation, with an application to diffusion parameter estimation, Mod. Stoch. Theory Appl. 7 (2020), no. 3, 339–356. MR 4159153
D. A. Avetisian and G. M. Shevchenko, Estimation of diffusion parameter for stochastic heat equation with white noise, Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics & Mathematics (2018), no. 3, 9–16.
Markus Bibinger and Mathias 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] ©2019, pp. 69–84. MR 4043170
Markus Bibinger and Mathias Trabs, Volatility estimation for stochastic PDEs using high-frequency observations, Stochastic Process. Appl. 130 (2020), no. 5, 3005–3052. MR 4080737, DOI 10.1016/j.spa.2019.09.002
Peter Breuer and Péter Major, Central limit theorems for nonlinear functionals of Gaussian fields, J. Multivariate Anal. 13 (1983), no. 3, 425–441. MR 716933, DOI 10.1016/0047-259X(83)90019-2
Baohua Chen and Jinqiao Duan, Stochastic quantification of missing mechanisms in dynamical systems, Recent development in stochastic dynamics and stochastic analysis, Interdiscip. Math. Sci., vol. 8, World Sci. Publ., Hackensack, NJ, 2010, pp. 67–76. MR 2807813, DOI 10.1142/9789814277266_{0}004
Carsten Chong, High-frequency analysis of parabolic stochastic PDEs, Ann. Statist. 48 (2020), no. 2, 1143–1167. MR 4102691, DOI 10.1214/19-AOS1841
Pao-Liu Chow, Stochastic partial differential equations, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2295103
Igor Cialenco, Francisco Delgado-Vences, and Hyun-Jung Kim, Drift estimation for discretely sampled SPDEs, Stoch. Partial Differ. Equ. Anal. Comput. 8 (2020), no. 4, 895–920. MR 4174073, DOI 10.1007/s40072-019-00164-4
Igor Cialenco and Yicong Huang, A note on parameter estimation for discretely sampled SPDEs, Stoch. Dyn. 20 (2020), no. 3, 2050016, 28. MR 4101083, DOI 10.1142/S0219493720500161
I. Cialenco and H.-J. Kim, Parameter estimation for discretely sampled stochastic heat equation driven by space-only noise revised, 2020.
Igor Cialenco, Hyun-Jung Kim, and Sergey V. Lototsky, Statistical analysis of some evolution equations driven by space-only noise, Stat. Inference Stoch. Process. 23 (2020), no. 1, 83–103. MR 4072253, DOI 10.1007/s11203-019-09205-0
Rama Cont, Modeling term structure dynamics: an infinite dimensional approach, Int. J. Theor. Appl. Finance 8 (2005), no. 3, 357–380. MR 2144706, DOI 10.1142/S0219024905003049
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR 3236753, DOI 10.1017/CBO9781107295513
Donald A. Dawson, Qualitative behavior of geostochastic systems, Stochastic Process. Appl. 10 (1980), no. 1, 1–31. MR 581688, DOI 10.1016/0304-4149(80)90002-2
S. S. De, Stochastic model of population growth and spread, Bull. Math. Biol. 49 (1987), no. 1, 1–11. MR 891484, DOI 10.1016/S0092-8240(87)80032-0
R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussian fields, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27–52. MR 550122, DOI 10.1007/BF00535673
Massimiliano Gubinelli, Antoine Lejay, and Samy Tindel, Young integrals and SPDEs, Potential Anal. 25 (2006), no. 4, 307–326. MR 2255351, DOI 10.1007/s11118-006-9013-5
Yaozhong Hu and David Nualart, Stochastic heat equation driven by fractional noise and local time, Probab. Theory Related Fields 143 (2009), no. 1-2, 285–328. MR 2449130, DOI 10.1007/s00440-007-0127-5
L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika 12 (1918), no. 1/2, 134–139.
Yusuke Kaino and Masayuki Uchida, Parametric estimation for a parabolic linear SPDE model based on discrete observations, J. Statist. Plann. Inference 211 (2021), 190–220. MR 4153870, DOI 10.1016/j.jspi.2020.05.004
Sergey V. Lototsky and Boris L. Rozovskii, Stochastic partial differential equations driven by purely spatial noise, SIAM J. Math. Anal. 41 (2009), no. 4, 1295–1322. MR 2540267, DOI 10.1137/070698440
Zeina Mahdi Khalil and Ciprian Tudor, Estimation of the drift parameter for the fractional stochastic heat equation via power variation, Mod. Stoch. Theory Appl. 6 (2019), no. 4, 397–417. MR 4047392
P. Major, Non-central limit theorem for non-linear functionals of vector valued gaussian stationary random fields, 2019.
Bohdan Maslowski and David Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal. 202 (2003), no. 1, 277–305. MR 1994773, DOI 10.1016/S0022-1236(02)00065-4
Yu. Mishura, K. Ralchenko, and G. Shevchenko, Existence and uniqueness of mild solution to stochastic heat equation with white and fractional noises, Teor. Ĭmovīr. Mat. Stat. 98 (2018), 142–162 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 98 (2019), 149–170. MR 3824684, DOI 10.1090/tpms/1068
L. Piterbarg and B. Rozovskii, Maximum likelihood estimators in the equations of physical oceanography, Stochastic modelling in physical oceanography, Progr. Probab., vol. 39, Birkhäuser Boston, Boston, MA, 1996, pp. 397–421. MR 1383880, DOI 10.1007/978-1-4612-2430-3_{1}5
Kostiantyn Ralchenko and Georgiy Shevchenko, Existence and uniqueness of mild solution to fractional stochastic heat equation, Mod. Stoch. Theory Appl. 6 (2019), no. 1, 57–79. MR 3935427, DOI 10.15559/18-vmsta122
Boris L. Rozovsky and Sergey V. Lototsky, Stochastic evolution systems, Probability Theory and Stochastic Modelling, vol. 89, Springer, Cham, 2018. Linear theory and applications to non-linear filtering; Second edition of [ MR1135324]. MR 3839316, DOI 10.1007/978-3-319-94893-5
Murad S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 53–83. MR 550123, DOI 10.1007/BF00535674
S. Tindel, C. A. Tudor, and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields 127 (2003), no. 2, 186–204. MR 2013981, DOI 10.1007/s00440-003-0282-2
John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI 10.1007/BFb0074920