Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

Shevchenko, Radomyra; Slaoui, Meryem; Tudor, Ciprian A.

doi:10.48550/arxiv.1903.02369

Cited by 2 publications

(5 citation statements)

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“…In the extreme case where observations are only available at one point t in time (and assuming ϑ 1 = ϑ 0 = 0 as well as X 0 = 0) Cialenco and Huang [7] showed that V sp (t) is asymptotically normal with 1/ √ M -rate of convergence. An analogous result has been proved by Shevchenko et al [28] for the wave equation. Proposition 2.1 reveals that V sp (t) is in fact a rescaled realized quadratic variation of the Itô diffusion y → X t (y).…”

Section: Central Limit Theorems For Realized Quadratic Variationssupporting

confidence: 77%

“…The general case will be necessary to verify asymptotic normality of the M-estimator in Section 4. It is worth noting that Proposition 3.1 reveals a quite elementary proof strategy to verify several central limit theorems in [3,7,28,29] instead of advanced techniques from Malliavin calculus or mixing theory. Remark 3.2.…”

Section: Central Limit Theorems For Realized Quadratic Variationsmentioning

confidence: 99%

“…Assuming X is observed on a discrete grid (t i , y k ) i=0,...,N,k=0,...,M ⊂ [0, T ] × [0, 1], approximate maximum likelihood estimators have been first investigated by Markussen [22] for T → ∞. For various linear SPDEs central limit theorems for method of moment type estimators based on realized quadratic variations have been studied by Torres et al [29], Cialenco and Huang [7], Bibinger and Trabs [3,2], Chong [4,5], Shevchenko et al [28], as well as Kaino and Uchida [18]. However, all these works only give partial answers to the estimation problem.…”

Section: Introductionmentioning

confidence: 99%

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Parameter estimation for SPDEs based on discrete observations in time and space

Hildebrandt¹,

Trabs²

2019

Preprint

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Parameter estimation for a parabolic linear stochastic partial differential equation in one space dimension is studied observing the solution field on a discrete grid in a fixed bounded domain. Considering an infill asymptotic regime in both coordinates, we prove central limit theorems for realized quadratic variations based on temporal and spatial increments as well as on double increments in time and space. Resulting method of moments estimators for the diffusivity and the volatility parameter inherit the asymptotic normality and can be constructed robustly with respect to the sampling frequencies in time and space. Upper and lower bounds reveal that in general the optimal convergence rate for joint estimation of the parameters is slower than the usual parametric rate. The theoretical results are illustrated in a numerical example.

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Section: Central Limit Theorems For Realized Quadratic Variationssupporting

confidence: 77%

Section: Central Limit Theorems For Realized Quadratic Variationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parameter estimation for SPDEs based on discrete observations in time and space

Hildebrandt¹,

Trabs²

2019

Preprint

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“…and Fubini's theorem as well as (43) show that there exists K > 0 such that for non-random initial conditions ξ = x ∈ E, we have…”

Section: Technical Lemmas For the Nonparametric Estimator Of Fmentioning

confidence: 99%

“…To estimate the parameters σ 2 and ϑ based on fully discrete observations, we extend the realized quadratic variation based methods developed in Hildebrandt and Trabs [26] and Bibinger and Trabs [7] to the semilinear framework (1). Similar methods have also been applied to various linear SPDE models, e.g., in [11,14,34,43,45]. Specifically, assuming f ∈ C 1 (R) in the model (1), we show that the asymptotic properties of the realized quadratic variations based on space, time and the double increments D ik := X ti+1 (y k+1 ) − X ti+1 (y k ) − X ti (y k+1 ) + X ti (y k ) derived for the case f ≡ 0 are robust with respect to a nonlinear perturbation of the equation.…”

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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Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

Cited by 2 publications

References 0 publications

Parameter estimation for SPDEs based on discrete observations in time and space

Parameter estimation for SPDEs based on discrete observations in time and space

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

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