Bibliography

Surveys/books

 I. Cialenco. Statistical inference for SPDEs: an overview. Statistical Inference for Stochastic Processes, 21(2):309–329, 2018. arXiv:1712.05445

S. V. Lototsky and B. L. Rozovsky. Stochastic partial differential equations. Universitext, Springer, 2017. [Chapter 6, Parameter estimation for diagonal SPDEs]

S. V. Lototsky. Statistical inference for stochastic parabolic equations: a spectral approach. Publ. Mat., 53(1):3-45, 2009. 

Papers

S. Gaudlitz, Non-parametric estimation of the reaction term in semi-linear SPDEs with spatial ergodicity, Preprint arXiv:2307.05457, 2023 

Y. Tonaki, Y. Kaino and  M. Uchida, Parametric estimation for linear parabolic SPDEs in two space dimensions based on temporal and spatial increments, Preprint arXiv:2304.09441, 2023 

J. Janák and M. Reiß, Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements, Preprint arXiv:2303.00074, 2023

Y. Tonaki, Y. Kaino, and  M. Uchida,Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise, Preprint arXiv:2206.10363, 2022.

J.  Gamain and C. A. Tudor, Exact variation and drift parameter estimation for the nonlinear fractional stochastic heat equation. Jpn J Stat Data Sci 6, 381–406, 2023. 

S. Gaudlitz, M. Reiß, Estimation for the reaction term in semi-linear SPDEs under small diffusivity, Bernoulli, 29, 3033 - 3058, 2023. arXiv

Y. Tonaki, Y. Kaino and M. Uchida, Parameter estimation for linear parabolic SPDEs in two space dimensions based on high frequency data, Scand J Statist, 1–22, 2023. arXiv

M. Bibinger, and P. Bossert, Efficient parameter estimation for parabolic SPDEs based on a log-linear model for realized volatilities. Jpn J Stat Data Sci, 6, 407–429, 2023. arXiv

F. E. Benth, D. Schroers, and A. E. D. Veraart, A feasible central limit theorem for realised covariation of SPDEs in the context of functional data, Preprint arXiv:2205.03927, 2022

P. Kříž, and J. Šnupárková, Pathwise least-squares estimator for linear SPDEs with additive fractional noise, Electronic Journal of Statistics, 16, 1561-1594, 2022. arXiv

C. A. Tudor, Stochastic Partial Differential Equations With Additive Gaussian Noise-Analysis And Inference, World Scientific, 2022

O. Lang, P. J.  van Leeuwen, D. Crisan and R. Potthast, Bayesian inference for fluid dynamics: A case study for the stochastic rotating shallow water model, Frontiers in Applied Mathematics and Statistics, 8, 2022 arXiv

R. Shevchenko. On quadratic variations of the fractional-white wave equation. Preprint arXiv:2111.13659, 2021.

V. R. Martinez, Convergence analysis of a viscosity parameter recovery algorithm for the 2D Navier–Stokes equations, Nonlinearity, 35, 2241, 2022 arXiv

O. Assaad, J. Gamain and C. A. Tudor, Quadratic variation and drift parameter estimation for the stochastic wave equation with space-time white noise, Stochastics and Dynamics, 22, 2240014, 2022

M. Kovács, Lang A. and A. Petersson. Approximation of SPDE covariance operators by finite elements: A semigroup approach. Preprint arXiv:2107.10109, 2021.

G. Pang and W. Cao. Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations. Fractal and Fractional 5, no. 3 ,129, 2021.

R. Dhoyer and C. A. Tudor, Non-central limit theorem for the spatial average of the solution to the wave equation with Rosenblatt noise, Theory of Probability and Mathematical Statistics,106, 105-119, 2022

O. Assaad and C. A. Tudor, Pathwise analysis and parameter estimation for the stochastic Burgers equation, Bulletin des Sciences Mathématiques, 170, 102995, 2021.

F. Delgado-Vences, and J. J. Pavon-Español. Statistical inference for a stochastic wave equation with Malliavin calculus, Stochastic Analysis and Applications, 41, 447-473, 2023.  arXiv:2104.04176, 2021.

I. Cialenco, H.-J. Kim, G. Pasemann. Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach. Preprint arXiv:2103.04211, 2021.

F. Hildebrandt, M. Trabs. Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations. Stochastic Processes and their Applications, 162, 171-217, 2023. arXiv:2102.13415. 

O. Assaad, Obayda, and C. A.  Tudor. Pathwise analysis and parameter estimation for the stochastic Burgers equation. Bulletin des Sciences Mathématiques 170: 102995, 2021.

F. Espen Benth, D. Schroers, and A. E.D. Veraart, A weak law of large numbers for realised covariation in a Hilbert space setting, Stochastic  Processes  and  their  Applications  145, pp.  241–268, 2022. arXiv

R. Altmeyer, T. Bretschneider, J. Janák, M. Reiß. Parameter Estimation in an SPDE Model for Cell Repolarisation. Preprint arxiv:2010.06340, 2020.

L. Sharrock and N. Kantas. Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation. Preprint arXiv:2009.08693, 2020.

Y. Kaino,  and M. Uchida. Adaptive estimator for a parabolic linear SPDE with a small noise, Jpn J Stat Data Sci 4, 513–541, 2021.  arXiv:2008.05353

C. Chong, R.C. Dalang. Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs. Preprint arXiv:2006.15817, 2020.

D. Avetisian and K. Ralchenko. Ergodic properties of the solution to a fractional stochastic heat equation, with an application to diffusion parameter estimation.  Modern Stochastics: Theory and Applications 7, no. 3, 339-356, 2020.

R. Altmeyer, I. Cialenco, G. Pasemann. Parameter estimation for semilinear SPDEs from local measurements. Preprint arXiv:2004.14728, 2020.

S. Reich, and P. Rozdeba. Posterior contraction rates for non-parametric state and drift estimation. Preprint arXiv:2003.09219, 2020

I. Cialenco, H.-J. Kim. Parameter estimation for discretely sampled stochastic heat equation driven by space-only noise. Stochastic Processes and their Applications 143, 1-30, 2022. arXiv 

G. Pasemann, S. Flemming, S. Alonso, C. Beta, W. Stannat. Diffusivity Estimation for Activator-Inhibitor Models: Theory and Application to Intracellular Dynamics of the Actin Cytoskeleton. Preprint arXiv:2005.09421, 2020. 

F. Hildebrandt, M. Trabs. Parameter estimation for SPDEs based on discrete observations in time and space. Electronic Journal of Statistics, 15(1), 2716-2776, 2021. arXiv 

Z. M. Khalil and C. Tudor, Estimation of the drift parameter for the fractional stochastic heat equation via power variation, Modern Stochastics: Theory and Applications 6(2019), no. 4, 397-417, DOI 10.15559/19-VMSTA141.  

Y. Kaino,  and M. Uchida. Parametric estimation for a parabolic linear SPDE model based on discrete observations. Journal of Statistical Planning and Inference, 211, 190-220, 2021. arXiv

C. Chong. High-frequency analysis of parabolic stochastic PDEs with multiplicative noise: Part I. Preprint arXiv:1908.04145, 2019.

I. Cialenco, F. Delgado-Vences, and H.-J. Kim. Drift estimation for discretely sampled SPDEs. Stochastics and Partial Differential Equations: Analysis and Computations 1-26, 2020.  arXiv

I. Cialenco, H.-J. Kim, and S. V. Lototsky. Statistical analysis of some evolution equations driven by space-only noise. Stat. Inference Stoch. Process. 23(1):83-103, 2020. arXiv

R. Shevchenko, M. Slaoui, and C. A. Tudor. Generalized k-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus. Journal of Statistical Planning and Inference, 207, 155-180, 2020. arXiv.

P. Kříž. A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations. Stochastics and Dynamics, 20(03), p.2050019, 2019. arXiv.

P. Kříž and B. Maslowski. Central limit theorems and minimum-contrast estimators for linear stochastic evolution equations. Stochastics, 1-32, 2019. arXiv

R. Altmeyer and M. Reiß. Nonparametric estimation for linear SPDEs from local measurements. Ann. Appl. Probab. 31 (1) 1 - 38, 2021.  arXiv

G. Pasemann and W. Stannat. Drift estimation for stochastic reaction-diffusion systems. Electronic Journal of Statistics 14(1):547-579,  2020.  arXiv

M. Bibinger and M. Trabs. On central limit theorems for power variations of the solution to the stochastic heat equation. In Workshop on Stochastic Models, Statistics and their Application (pp. 69-84). Springer, Cham, 2019. arXiv

C. Chong. High-frequency analysis of parabolic stochastic PDEs. Ann. Statist., 48(2):1143-67, 2020. arXiv

Z. Cheng, I. Cialenco, and R. Gong. Bayesian estimations for diagonalizable bilinear SPDEs. Stochastic Process. Appl., 130(2):845-77, 2020. arXiv

I. Cialenco. Statistical inference for SPDEs: an overview. Stat. Inference Stoch. Process., 21(2):309–329, 2018.

M. Bibinger and M. Trabs. Volatility estimation for stochastic PDEs using high-frequency observations. Stochastic Process. Appl., 130(5):3005-3052, 2020. arXiv

I. Cialenco and Y. Huang. A note on parameter estimation for discretely sampled SPDEs. Stoch. Dyn., 20(3):2050016, 2020. arXiv.

I. Cialenco, R. Gong, and Y. Huang. Trajectory fitting estimators for SPDEs driven by additive noise. Stat. Inference Stoch. Process., 21(1):1–19, 2018. arXiv

J. Janák. Parameter estimation for stochastic wave equation based on observation window. Acta Appl. Math., 172(2), 2021. arXiv

J. Janák. Parameter estimation for stochastic partial differential equations of second order. Appl. Math. Optim., 82:353-397, 2020. arXiv

I. Cialenco and L. Xu. Hypothesis testing for stochastic PDEs driven by additive noise. Stochastic Process. Appl., 125(3):819-866, 2015. arXiv

D. Crisan, Y. Otobe and S. Peszat (2015). Inverse problems for stochastic transport equations. Inv. Probl., 31(1), 015005.

I. Cialenco and L. Xu. A note on error estimation for hypothesis testing problems for some linear SPDEs. Stoch. Partial Differ. Equ. Anal. Comput., 2(3):408-431, 2014. arXiv

B. Maslowski and C. Tudor. Drift parameter estimation for infinite-dimensional fractional Ornstein–Uhlenbeck process. Bulletin Des Sciences Mathématiques, 137(7), 880–901. http://doi.org/10.1016/j.bulsci.2013.04.008, 2013.

A. Gupta and M. Khammash. Unbiased estimation of parameter sensitivities for stochastic chemical reaction networks. SIAM Journal on Scientific Computing 35, no. 6, 2598-2620, 2012. DOI: 10.1137/120898747

I. Cialenco and N. Glatt-Holtz. Parameter estimation for the stochastically perturbed Navier-Stokes equations. Stochastic Process. Appl., 121(4):701-724, 2011.

I. Cialenco. Parameter estimation for SPDEs with multiplicative fractional noise. Stoch. Dyn., 10(4):561-576, 2010.

I. Cialenco and S. V. Lototsky. Parameter estimation in diagonalizable bilinear stochastic parabolic equations. Stat. Inference Stoch. Process., 12(3):203-219, 2009.

S. V. Lototsky. Statistical inference for stochastic parabolic equations: a spectral approach. Publ. Mat., 53(1):3-45, 2009.

I. Cialenco, S. V. Lototsky, and J. Pospíšil. Asymptotic properties of the maximum likelihood estimator for stochastic parabolic equations with additive fractional Brownian motion. Stoch. Dyn., 9(2):169-185, 2009.

B. Maslowski and J. Pospíšil. Ergodicity and parameter estimates for infinite-dimensional fractional Ornstein-Uhlenbeck process. Appl. Math. Optim., 57(3):401-429, 2008.

J. Pospíšil and R. Tribe. Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise. Stoch. Anal. Appl., 25(3):593-611, 2007.

J. Pospíšil. On parameter estimates in stochastic evolution equations driven by fractional Brownian motion. PhD thesis, University of West Bohemia, Plzen, 2005.

B. L. S. Prakasa Rao. Parameter estimation for some stochastic partial differential equations driven by infinite dimensional fractional Brownian motion. Theory Stoch. Process., 10(3-4):116-125, 2004.

S. V. Lototsky. Optimal  filtering of stochastic parabolic equations. In S. Albeverio, Z.-M. Ma, M. Rockner edts, Recent developments in stochastic analysis and related topics, pages 330-353. World Sci.  Publ., Hackensack, NJ, 2004.

B. L. S. Prakasa Rao. Estimation for some stochastic partial differential equations based on discrete observations. II. Calcutta Statist. Assoc. Bull., 54(215-216):129-141, 2003.

B. Markussen. Likelihood inference for a discretely observed stochastic partial differential equation. Bernoulli, 9(5):745-762, 2003.

S. V. Lototsky. Parameter estimation for stochastic parabolic equations: asymptotic properties of a two-dimensional projection-based estimator. Stat. Inference Stoch. Process., 6(1):65-87, 2003. 

B. Goldys and B. Maslowski. Parameter estimation for controlled semilinear stochastic systems: identifiable and consistency. J. Multivariate Anal., 80(2):322-343, 2002.

J. P. N. Bishwal. The Bernstein-von Mises theorem and spectral asymptotics of Bayes estimators for parabolic SPDEs. J. Aust. Math. Soc., 72(2):287-298, 2002.

B. L. S. Prakasa Rao. Nonparametric inference for a class of stochastic partial differential equations based on discrete observations. Sankhy a Ser. A, 64(1):1-15, 2002.

I. A. Ibragimov and R. Z. Khasminskii. Estimation problems for coefficients of stochastic partial differential equations. Part III. Theory Probab. Appl., 45(2), 210-232, 2001.

S. V. Lototsky and B. L Rozovskii. Parameter estimation for stochastic evolution equations with non-commuting operators. In Skorohod's Ideas in Probability Theory, V.Korolyuk, N.Portenko and H.Syta (editors), pages 271-280. Institute of Mathematics of National Academy of Sciences of Ukraine, Kiev, Ukraine, 2000.

M. Huebner and S. V. Lototsky. Asymptotic analysis of a kernel estimator for parabolic SPDE's with time-dependent coe cients. Ann. Appl. Probab., 10(4):1246-1258, 2000.

M. Huebner and S. V. Lototsky. Asymptotic analysis of the sieve estimator for a class of parabolic SPDEs. Scand. J. Statist., 27(2):353-370, 2000.

I. A. Ibragimov and R. Z. Khasminskii. Problems of estimating the coefficients of stochastic partial differential equations Part II. Theory Probab. Appl., 44(3), 469-494, 2000.

J. P. N. Bishwal. Bayes and sequential estimation in Hilbert space valued stochastic differential equations. J. Korean Statist. Soc., 28(1):93-106, 1999.

S. V. Lototsky and B. L. Rozovskii. Spectral asymptotics of some functionals arising in statistical inference for SPDEs. Stochastic Process. Appl., 79(1):69-94, 1999. 

M. Huebner. Asymptotic properties of the maximum likelihood estimator for stochastic PDEs disturbed by small noise. Stat. Inference Stoch. Process., 2(1):57-68 (2000), 1999.

I. A. Ibragimov and R. Z. Khasminskii. Problems of estimating the coefficients of stochastic partial differential equations Part I. Theory Probab. Appl., 43(3), 370–387, 1999.

B. L. S. Prakasa Rao. Bayes estimation for some stochastic partial di erential equations. J. Statist. Plann. Inference, 91(2):511-524, 2000. Prague Workshop on Perspectives in Modern Statistical Inference:  Parametrics, Semi-parametrics, Non-parametrics (1998).

S. I. Aihara. Consistency property of extended least-squares parameter estimation for stochastic diffusion equation. Systems Control Lett., 34(5):249-256, 1998.

S. I. Aihara. Identification of a discontinuous parameter in stochastic parabolic systems. Appl. Math. Optim., 37(1):43-69, 1998.

M. Huebner. A characterization of asymptotic behaviour of maximum likelihood estimators for stochastic PDE's. Math. Methods Statist., 6(4):395-415, 1998.

L. I. Piterbarg and B. L. Rozovskii. On asymptotic problems of parameter estimation in stochastic PDE's: discrete time sampling. Math. Methods Statist., 6(2):200-223, 1997. 

M. Huebner, S. V. Lototsky, and B. L. Rozovskii. Asymptotic properties of an approximate maximum likelihood estimator for stochastic PDEs. In Yu. M. Kabanov, B. L. Rozovskii, and A. N. Shiryaev edts, Statistics and control of stochastic processes (Moscow, 1995/1996), pages 139-155. World Sci. Publishing, River Edge, NJ, 1997. 

S. V. Lototsky. Problems in statistics of stochastic di erential equations. PhD thesis, University of Southern California, Los Angeles, USA, 1996.

M. Huebner and B. L. Rozovskii. On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's. Probab. Theory Related Fields, 103(2):143-163, 1995.

M. Huebner, R. Khasminskii, and B. L. Rozovskii. Two examples of parameter estimation for stochastic partial differential equations. In S.Cambanis, J. K. Ghosh, R. L. Karandikar, P. K. Sen eds, Stochastic processes: A Festschrift in Honour of G. Kallianpur, pp 149-160. Springer, New York, 1993.

M. Huebner. Parameter Estimation for SPDEs. PhD thesis, University of Southern California, Los Angeles, USA, 1993.

S. I. Aihara. Regularized maximum likelihood estimate for an in nite-dimensional parameter in stochastic parabolic systems. SIAM J. Control Optim., 30(4):745-764, 1992. 

S. I. Aihara. Parameter identification for stochastic parabolic systems. In F. Kozin, T. Ono, eds Systems and control, pp. 1-12. Mita, 1991.

S. I. Aihara and A. Bagchi. Infinite-dimensional parameter identification for stochastic parabolic systems. Statist. Probab. Lett., 8(3):279-287, 1989.

S. I. Aihara and Y. Sunahara. Identification of an in infinite-dimensional parameter for stochastic diffusion equations. SIAM J. Control Optim., 26(5):1062-1075, 1988.

W. Loges. Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space-valued stochastic differential equations. Stochastic Process. Appl., 17(2):243-263, 1984.

T. Koski and W. Loges. Asymptotic statistical inference for a stochastic heat flow problem. Statist. Probab. Lett., 3(4):185-189, 1985.