I. A. Ibragimov, R. Z. Khas'minskii, “Estimation problems for coefficients of stochastic partial differential equations. Part I”, Teor. Veroyatnost. i Primenen., 43:3 (1998), 417–438; Theory Probab. Appl., 43:3 (1999), 370

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Teoriya Veroyatnostei i ee Primeneniya, 1998, Volume 43, Issue 3, Pages 417–438
DOI: https://doi.org/10.4213/tvp1550 (Mi tvp1550)

This article is cited in 23 scientific papers (total in 24 papers)

Estimation problems for coefficients of stochastic partial differential equations. Part I

I. A. Ibragimov^a, R. Z. Khas'minskii^b

^a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
^b Wayne State University, Detroit, USA

Full-text PDF (1059 kB) Citations (24)

DOI: https://doi.org/10.4213/tvp1550

Abstract: This paper considers the problem of estimating functional parameters

a_{k} (t, x)

$a_k(t,x)$ ,

f (t, x)

$f(t,x)$ by observing a solution

u_{ε} (t, x)

$u_{\varepsilon}(t,x)$ of a stochastic partial differential equation

$du_{\varepsilon}(t)=\sum_{|k|\le2p}a_kD_x^ku_{\varepsilon}+f\,dt+\varepsilon\,dw(t),$
where

$w(t)$ is a Wiener process. The asymptotic statement of the problem is considered when the noise level

$\varepsilon\to0$ . In the first part of the work we determine what is considered the statistics of the problem and investigate the problem of estimating

$f$ .

Keywords: keywords inverse problems, stochastic partial differential equations, statistical estimating, nonparametric estimating problems.

Received: 09.12.1997

English version:
Theory of Probability and its Applications, 1999, Volume 43, Issue 3, Pages 370–387
DOI: https://doi.org/10.1137/S0040585X97976982

Bibliographic databases:

Language: Russian

Citation: I. A. Ibragimov, R. Z. Khas'minskii, “Estimation problems for coefficients of stochastic partial differential equations. Part I”, Teor. Veroyatnost. i Primenen., 43:3 (1998), 417–438; Theory Probab. Appl., 43:3 (1999), 370–387

Citation in format AMSBIB

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\pages 417--438

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\zmath{https://zbmath.org/?q=an:0983.62051}

\transl

\jour Theory Probab. Appl.

\yr 1999

\vol 43

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\pages 370--387

\crossref{https://doi.org/10.1137/S0040585X97976982}

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Linking options:

https://www.mathnet.ru/eng/tvp1550

https://doi.org/10.4213/tvp1550

https://www.mathnet.ru/eng/tvp/v43/i3/p417

This publication is cited in the following 24 articles:

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A. A. Borovkov, Al. V. Bulinski, A. M. Vershik, D. Zaporozhets, A. S. Holevo, A. N. Shiryaev, “Ildar Abdullovich Ibragimov (on his ninetieth birthday)”, Russian Math. Surveys, 78:3 (2023), 573–583
Sascha Gaudlitz, Markus Reiß, “Estimation for the reaction term in semi-linear SPDEs under small diffusivity”, Bernoulli, 29:4 (2023)
Tomás Caraballo, Tran Bao Ngoc, Tran Ngoc Thach, Nguyen Huy Tuan, “On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion”, Stoch. Dyn., 22:02 (2022)
Yuan G., “Inverse Problems For Stochastic Parabolic Equations With Additive Noise”, J. Inverse Ill-Posed Probl., 29:1 (2021), 93–108
Yan L., “Global Uniqueness of An Inverse Problem For Stochastic Degenerate Wave Equation With Three Unknowns”, Math. Meth. Appl. Sci., 44:17 (2021), 12545–12558
Nguyen Huy Tuan, Caraballo T., Tran Ngoc Thach, “On Terminal Value Problems For Bi-Parabolic Equations Driven By Wiener Process and Fractional Brownian Motions”, Asymptotic Anal., 123:3-4 (2021), 335–366
Jaya P. N. Bishwal, “Bernstein-von Mises Theorem and small noise asymptotics of Bayes estimators for parabolic stochastic partial differential equations”, Theory Stoch. Process., 23(39):1 (2018), 6–17
Igor Cialenco, “Statistical inference for SPDEs: an overview”, Stat Inference Stoch Process, 21:2 (2018), 309
Yuan G., “Conditional stability in determination of initial data for stochastic parabolic equations”, Inverse Probl., 33:3 (2017), 035014
Yuan G., “Determination of two unknowns simultaneously for stochastic Euler?Bernoulli beam equations”, J. Math. Anal. Appl., 450:1 (2017), 137–151
Lototsky S., Rozovsky B., “Stochastic Partial Differential Equations”, Stochastic Partial Differential Equations, Universitext, Springer, 2017, 1–508
Yuan G., “Determination of Two Kinds of Sources Simultaneously For a Stochastic Wave Equation”, Inverse Probl., 31:8 (2015), 085003
Lu Q., Zhang X., “Global Uniqueness For An Inverse Stochastic Hyperbolic Problem With Three Unknowns”, Commun. Pure Appl. Math., 68:6 (2015), 948–963
Effective Dynamics of Stochastic Partial Differential Equations, 2014, 257
Lu Q., “Carleman Estimate for Stochastic Parabolic Equations and Inverse Stochastic Parabolic Problems”, Inverse Probl., 28:4 (2012), 045008
Liu W., Lototsky S.V., “Parameter estimation in hyperbolic multichannel models”, Asymptot Anal, 68:4 (2010), 223–248
Lototsky S.V., “Statistical Inference for Stochastic Parabolic Equations: a Spectral Approach”, Publicacions Matematiques, 53:1 (2009), 3–45
Igor Cialenco, Sergey V. Lototsky, “Parameter estimation in diagonalizable bilinear stochastic parabolic equations”, Stat Inference Stoch Process, 12:3 (2009), 203
Lototsky S.V., “Optimal filtering of stochastic parabolic equations”, Recent Developments in Stochastic Analysis and Related Topics, 2004, 330–353

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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