Parameter Estimation for Controlled Semilinear Stochastic Systems: Identifiability and Consistency

Gołdys, Beniamin; Maslowski, Bohdan

doi:10.1006/jmva.2001.1989

Cited by 6 publications

(3 citation statements)

References 23 publications

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“…He also investigated the asymptotic properties of the maximum likelihood estimator and the Bayes estimator of the drift parameter in stochastic equations driven by fractional Brownian motion, see [23] and [24]. Goldys and Maslowski considered in [7] a controlled stochastic semilinear equation with the drift depending on the unknown parameter and with the Wiener process as the driving process. They showed that the maximum likelihood estimator is consistent for a class of bounded predictable controls.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity and Parameter Estimates for Infinite-Dimensional Fractional Ornstein-Uhlenbeck Process

Maslowski¹,

Pospı́šil

2007

Appl Math Optim

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Abstract. Existence and ergodicity of a strictly stationary solution for linear stochastic evolution equations driven by cylindrical fractional Brownian motion are proved. Ergodic behaviour of non-stationary infinite-dimensional fractional Ornstein-Uhlenbeck processes is also studied. Based on these results, strong consistency of suitably defined families of parameter estimators is shown. The general results are applied to linear parabolic and hyperbolic equations perturbed by a fractional noise.

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Section: Introductionmentioning

confidence: 99%

Ergodicity and Parameter Estimates for Infinite-Dimensional Fractional Ornstein-Uhlenbeck Process

Maslowski¹,

Pospı́šil

2007

Appl Math Optim

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“…To the authors' knowledge, the only work on estimation of the nonlinearity in semilinear SPDEs was conducted by Goldys and Maslowski [22] who studied a parametric problem, assuming a full observation (X t ) t≤T of a controlled SPDE as T → ∞. Somewhat related, Pasemann et al [40] studied estimation of the diffusivity parameter when the nonlinearity is only known up to a finite-dimensional nuisance parameter by using a joint maximum likelihood approach for spectral observations.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations

Hildebrandt¹,

Trabs²

2021

Preprint

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Nonparametric estimation for semilinear SPDEs, namely stochastic reaction-diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, based on a precise analysis of the Hölder regularity of the solution process and its nonlinear component, we show that the asymptotic properties of diffusivity and volatility estimators derived from realized quadratic variations in the linear setup generalize to the semilinear SPDE. In particular, we obtain a rate-optimal joint estimator of the two parameters. Secondly, we derive a nonparametric estimator for the reaction function specifying the underlying equation. The estimate is chosen from a finite-dimensional function space based on a simple least squares criterion. Oracle inequalities with respect to both the empirical and usual L 2 -risk provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection.

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“…[Kut04] and references therein. We will omit this analysis here, and for more details specific to SPDEs, we refer the reader to [LR17,CX15] for the spectral approach, to [KL85,Log84] for non-spectral approach, and to [GM02] for controlled SPDEs. As already mentioned, with the idea of approximating a singular model by regular models, in the spectral approach, we will take the large number of Fourier modes asymptotics, N → ∞, which will be one of the main focuses of this paper; see Section 2.…”

Section: Introductionmentioning

confidence: 99%

Statistical inference for SPDEs: an overview

Cialenco

2018

Stat Inference Stoch Process

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The aim of this work is to give an overview of the recent developments in the area of statistical inference for parabolic stochastic partial differential equations. Significant part of the paper is devoted to the spectral approach, which is the most studied sampling scheme under which the observations are done in the Fourier space over some finite time interval. We also discuss into details the practically important case of discrete sampling of the solution. Other relevant methodologies and some open problems are briefly discussed over the course of the manuscript.

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Parameter Estimation for Controlled Semilinear Stochastic Systems: Identifiability and Consistency

Cited by 6 publications

References 23 publications

Ergodicity and Parameter Estimates for Infinite-Dimensional Fractional Ornstein-Uhlenbeck Process

Ergodicity and Parameter Estimates for Infinite-Dimensional Fractional Ornstein-Uhlenbeck Process

Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations

Statistical inference for SPDEs: an overview

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