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High-frequency analysis of parabolic stochastic PDEs. (English) Zbl 1450.62122

This paper deals with the estimation of the noise process in an observed process defined through a parabolic stochastic partial differential equation. The author proposes a realistic mathematical model regarding the various applications that this process has in various disciplines. Let’s see a brief introduction to paper math. Be \((\Omega,F,P)\) a probability space and be \((F_{t})_{t\in\mathbb R}\) a filtration on \((\Omega,F,P)\), that is: \(F_{s}\subset F_{t}\subset F\) for \(s\leq t\), and \(F_{s}=\cap_{s\leq t}F_{t}\) for \(s\in\mathbb R\). For each \((t,x)\in\mathbb R\times\mathbb R^{d}\) \((d\geq1)\) be \(Y(t,x):\Omega\to\mathbb R\), \(\sigma(t,x):\Omega\to\mathbb R\) and \(W(t,x):\Omega\to\mathbb R\) random variables such that the below parabolic stochastic PDE is satisfied \[ \partial_{t}Y(t,x)=(\kappa/2)\Delta Y(t,x)-\lambda Y(t,x)+\sigma(t,x)W(t,x),\tag{1} \] where \(W=(W(t,x))_{(t,x)\in\mathbb R\times\mathbb R^{d}}\) is a Gaussian noise and \(\Delta\) is the Laplacian operator on the spatial coordinates of the process \(Y=(Y(t,x))_{(t,x)\in\mathbb R\times\mathbb R^{d}}\). In practice, \(Y\) is the observed process, \(\kappa>0\), \(\lambda>0\) and the process \(\sigma\) are estimated or some of them are considered known. This paper proposes a consistent estimator and asymptotic confidence limits for the process \(\sigma\) called stochastic volatility process, not only in the context of finance. Here it is said that, if \[ \sup_{(t,x)\in\mathbb R\times\mathbb R^{d}} E\left[\sigma^2(t,x)\right] < \infty, \] then (1) has a solution given by \[ Y(t,x)=\int_{\mathbb R^{d}}\int_{-\infty}^{t}G(t-s,x-y)\sigma(s,y)\cdot W(ds,dy), \] where \[ G(t,x)=(2\pi\kappa t)^{-d/2} \exp(-\Vert x\Vert^2/(2\kappa t)-\lambda t) \cdot 𝟙_{(0,+\infty)}(t). \] It is assumed that the process \(Y\) is observed in a finite number of \(\mathbb R^{d}\) and in a much larger number of times \(t=\Delta_{n},2\Delta_{n},\dots,[(T/(\Delta_{n}))]\Delta_{n}\) within the interval \([0,T]\), \(T<\infty\) (\([a]\) is the integer part of \(a\in\mathbb R\)). The proposed scheme of observations, of low spatial resolution and high frequency in time, is a realistic approach in many applications.
The authors study properties of the solution to Equation (1) for \(\Delta_{n}\to 0\). This requires prior knowledge of two concepts: “uniform convergence in probability over compacts”, and “stable functional convergence in law with respect to uniform topology”. These concepts and basic results related to them can be studied in the following two books that are in the bibliography of this paper: [Y. Aït-Sahalia and J. Jacod, High-frequency financial econometrics. Princeton, NJ: Princeton University Press (2014; Zbl 1298.91018)] and [J. Jacod and P. Protter, Discretization of processes. Berlin: Springer (2012; Zbl 1259.60004)]. Thorough knowledge of these two texts is essential for reading this paper. In several remarks, the author comments on the need for certain suppositions and relaxed approaches in order to extend the results stated in the paper to other models. In this paper, the author does not give the proof of any of the several results that he enunciates by means of lemmas and theorems. He only gives various comments on them and he says that the proofs are in another “Supplement” paper that is not provided by the author free of charge. No examples of application of the results to real or simulated situations are given either. It is hard to understand this work without good knowledge, typical of a mathematician, about stochastic differential equations.

MSC:

62M40 Random fields; image analysis
62G20 Asymptotic properties of nonparametric inference
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F15 Strong limit theorems
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References:

[1] Aït-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184-222. · Zbl 1155.62057 · doi:10.1214/07-AOS568
[2] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37 2202-2244. · Zbl 1173.62060 · doi:10.1214/08-AOS640
[3] Aït-Sahalia, Y. and Jacod, J. (2014). High-Frequency Financial Econometrics. Princeton Univ. Press, Princeton, NJ. · Zbl 1298.91018
[4] Barndorff-Nielsen, O. E., Corcuera, J. M. and Podolskij, M. (2011). Multipower variation for Brownian semistationary processes. Bernoulli 17 1159-1194. · Zbl 1244.60039 · doi:10.3150/10-BEJ316
[5] Barndorff-Nielsen, O. E., Corcuera, J. M. and Podolskij, M. (2013). Limit theorems for functionals of higher order differences of Brownian semi-stationary processes. In Prokhorov and Contemporary Probability Theory (A. N. Shiryaev, S. R. S. Varadhan and E. L. Presman, eds.). Springer Proc. Math. Stat. 33 69-96. Springer, Heidelberg. · Zbl 1273.60021
[6] Barndorff-Nielsen, O. E. and Graversen, S. E. (2011). Volatility determination in an ambit process setting. J. Appl. Probab. 48A 263-275. · Zbl 1251.60043 · doi:10.1239/jap/1318940470
[7] Barndorff-Nielsen, O. E., Pakkanen, M. S. and Schmiegel, J. (2014). Assessing relative volatility/intermittency/energy dissipation. Electron. J. Stat. 8 1996-2021. · Zbl 1302.60115 · doi:10.1214/14-EJS942
[8] Barndorff-Nielsen, O. E. and Schmiegel, J. (2004). Lévy-based spatial-temporal modelling, with applications to turbulence. Russian Math. Surveys 59 65-90. · Zbl 1062.60039 · doi:10.1070/RM2004v059n01ABEH000701
[9] Bibinger, M. and Trabs, M. (2018). Volatility estimation for stochastic PDEs using high-frequency observations. Available at arXiv:1710.03519 [math.ST]. · Zbl 1462.60082 · doi:10.1016/j.spa.2019.09.002
[10] Bibinger, M. and Trabs, M. (2019). On central limit theorems for power variations of the solution to the stochastic heat equation. Available at arXiv:1901.01026 [math.ST]. · Zbl 1434.60064
[11] Brockwell, P. J. (2009). An overview of asset-price models. In Handbook on Financial Time Series (T. Mikosch, J.-P. Kreiß, R. A. Davis and T. G. Andersen, eds.) 403-419. Springer, Berlin. · Zbl 1178.91063
[12] Brockwell, P. J. and Matsuda, Y. (2017). Continuous auto-regressive moving average random fields on \(\Bbb{R}^n \). J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 833-857. · Zbl 1411.62277 · doi:10.1111/rssb.12197
[13] Brouste, A. and Fukasawa, M. (2018). Local asymptotic normality property for fractional Gaussian noise under high-frequency observations. Ann. Statist. 46 2045-2061. · Zbl 1411.62045 · doi:10.1214/17-AOS1611
[14] Chong, C. (2020). Supplement to “High-frequency analysis of parabolic stochastic PDEs.” https://doi.org/10.1214/19-AOS1841SUPP.
[15] Cialenco, I. (2018). Statistical inference for SPDEs: An overview. Stat. Inference Stoch. Process. 21 309-329. · Zbl 1394.60067 · doi:10.1007/s11203-018-9177-9
[16] Cialenco, I. and Huang, Y. (2017). A note on parameter estimation for discretely sampled SPDEs. Available at arXiv:1710.01649 [math.PR]. · Zbl 1451.60063
[17] Cont, R. (2005). Modeling term structure dynamics: An infinite dimensional approach. Int. J. Theor. Appl. Finance 8 357-380. · Zbl 1113.91020 · doi:10.1142/S0219024905003049
[18] Corcuera, J. M., Hedevang, E., Pakkanen, M. S. and Podolskij, M. (2013). Asymptotic theory for Brownian semi-stationary processes with application to turbulence. Stochastic Process. Appl. 123 2552-2574. · Zbl 1295.60044 · doi:10.1016/j.spa.2013.03.011
[19] Corcuera, J. M., Nualart, D. and Podolskij, M. (2014). Asymptotics of weighted random sums. Commun. Appl. Ind. Math. 6 e-486, 11. · Zbl 1329.60073
[20] Corcuera, J. M., Nualart, D. and Woerner, J. H. C. (2006). Power variation of some integral fractional processes. Bernoulli 12 713-735. · Zbl 1130.60058 · doi:10.3150/bj/1155735933
[21] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge. · Zbl 0761.60052
[22] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 no. 6, 29 pp. · Zbl 0922.60056 · doi:10.1214/EJP.v4-43
[23] Dalang, R. C. and Quer-Sardanyons, L. (2011). Stochastic integrals for spde’s: A comparison. Expo. Math. 29 67-109. · Zbl 1234.60064 · doi:10.1016/j.exmath.2010.09.005
[24] Denaro, G., Valenti, D., La Cognata, A., Spagnolo, B., Bonanno, A., Basilone, G., Mazzola, S., Zgozi, S. W., Aronica, S. et al. (2013). Spatio-temporal behaviour of the deep chlorophyll maximum in Mediterranean Sea: Development of a stochastic model for picophytoplankton dynamics. Ecol. Complex. 13 21-34. · Zbl 1371.92134 · doi:10.5506/APhysPolB.44.977
[25] El Saadi, N. and Bah, A. (2007). An individual-based model for studying the aggregation behavior in phytoplankton. Ecol. Model. 204 193-212.
[26] Foondun, M. and Khoshnevisan, D. (2009). Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 548-568. · Zbl 1190.60051 · doi:10.1214/EJP.v14-614
[27] Foondun, M., Khoshnevisan, D. and Mahboubi, P. (2015). Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion. Stoch. Partial Differ. Equ. Anal. Comput. 3 133-158. · Zbl 1327.60127 · doi:10.1007/s40072-015-0045-y
[28] Fukasawa, M. and Takabatake, T. (2019). Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations. Bernoulli 25 1870-1900. · Zbl 1466.62382 · doi:10.3150/18-BEJ1039
[29] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Stat. 33 407-436. · Zbl 0882.60032 · doi:10.1016/S0246-0203(97)80099-4
[30] Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités, XXXI (J. Azéma, M. Emery and M. Yor, eds.). Lecture Notes in Math. 1655 232-246. Springer, Berlin. · Zbl 0884.60038
[31] Jacod, J., Podolskij, M. and Vetter, M. (2010). Limit theorems for moving averages of discretized processes plus noise. Ann. Statist. 38 1478-1545. · Zbl 1196.60033 · doi:10.1214/09-AOS756
[32] Jacod, J. and Protter, P. (2012). Discretization of Processes. Stochastic Modelling and Applied Probability 67. Springer, Heidelberg. · Zbl 1259.60004
[33] Jacod, J. and Todorov, V. (2014). Efficient estimation of integrated volatility in presence of infinite variation jumps. Ann. Statist. 42 1029-1069. · Zbl 1305.62146 · doi:10.1214/14-AOS1213
[34] Jones, B. J. T. (1999). The origin of scaling in the galaxy distribution. Mon. Not. R. Astron. Soc. 307 376-386.
[35] Liu, J. and Tudor, C. A. (2016). Central limit theorem for the solution to the heat equation with moving time. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 1650005, 17. · Zbl 1335.60113 · doi:10.1142/S0219025716500053
[36] Lototsky, S. V. and Rozovsky, B. L. (2017). Stochastic Partial Differential Equations. Universitext. Springer, Cham. · Zbl 1375.60010
[37] Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel. · Zbl 0816.35001
[38] Nguyen, M. and Veraart, A. E. D. (2017). Spatio-temporal Ornstein-Uhlenbeck processes: Theory, simulation and statistical inference. Scand. J. Stat. 44 46-80. · Zbl 1394.60052 · doi:10.1111/sjos.12241
[39] Nourdin, I., Nualart, D. and Tudor, C. A. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46 1055-1079. · Zbl 1221.60031 · doi:10.1214/09-AIHP342
[40] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Springer, Berlin. · Zbl 1099.60003
[41] Nualart, D. and Pardoux, E. (1994). Markov field properties of solutions of white noise driven quasi-linear parabolic PDEs. Stoch. Stoch. Rep. 48 17-44. · Zbl 0828.60043 · doi:10.1080/17442509408833896
[42] Nualart, D. and Quer-Sardanyons, L. (2007). Existence and smoothness of the density for spatially homogeneous SPDEs. Potential Anal. 27 281-299. · Zbl 1133.60029 · doi:10.1007/s11118-007-9055-3
[43] Pakkanen, M. S. (2014). Limit theorems for power variations of ambit fields driven by white noise. Stochastic Process. Appl. 124 1942-1973. · Zbl 1319.60110 · doi:10.1016/j.spa.2014.01.005
[44] Pereira, R. M., Garban, C. and Chevillard, L. (2016). A dissipative random velocity field for fully developed fluid turbulence. J. Fluid Mech. 794 369-408. · Zbl 1445.76045 · doi:10.1017/jfm.2016.166
[45] Pham, V. S. and Chong, C. (2018). Volterra-type Ornstein-Uhlenbeck processes in space and time. Stochastic Process. Appl. 128 3082-3117. · Zbl 1405.60064 · doi:10.1016/j.spa.2017.10.012
[46] Podolskij, M. and Vetter, M. (2010). Understanding limit theorems for semimartingales: A short survey. Stat. Neerl. 64 329-351.
[47] Pospíšil, J. and Tribe, R. (2007). Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise. Stoch. Anal. Appl. 25 593-611. · Zbl 1118.60030 · doi:10.1080/07362990701282849
[48] Prakasa Rao, B. L. S. (2002). Nonparametric inference for a class of stochastic partial differential equations based on discrete observations. Sankhyā Ser. A 64 1-15. · Zbl 1192.60084
[49] Robert, R. and Vargas, V. (2008). Hydrodynamic turbulence and intermittent random fields. Comm. Math. Phys. 284 649-673. · Zbl 1157.60322 · doi:10.1007/s00220-008-0642-y
[50] Sigrist, F., Künsch, H. R. and Stahel, W. A. (2012). A dynamic nonstationary spatio-temporal model for short term prediction of precipitation. Ann. Appl. Stat. 6 1452-1477. · Zbl 1257.62121 · doi:10.1214/12-AOAS564
[51] Sigrist, F., Künsch, H. R. and Stahel, W. A. (2015). Stochastic partial differential equation based modelling of large space-time data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 3-33. · Zbl 1414.62405
[52] Swanson, J. (2007). Variations of the solution to a stochastic heat equation. Ann. Probab. 35 2122-2159. · Zbl 1135.60041 · doi:10.1214/009117907000000196
[53] Tuckwell, H. C. (2013). Stochastic partial differential equations in neurobiology: Linear and nonlinear models for spiking neurons. In Stochastic Biomathematical Models (M. Bachar, J. Batzel and S. Ditlevsen, eds.). Lecture Notes in Math. 2058 149-173. Springer, Heidelberg. · Zbl 1390.92034
[54] Walsh, J. B. (1981). A stochastic model of neural response. Adv. in Appl. Probab. 13 231-281. · Zbl 0471.60083 · doi:10.1017/S0001867800036004
[55] Walsh, J.
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