Abstract
In this paper, we are interested in the rate of convergence for the central limit theorem of the maximum likelihood estimator of the drift coefficient for a stochastic partial differential equation based on continuous time observations of the Fourier coefficients ui(t),i=1,…,N of the solution, over some finite interval of time [0,T]. We provide explicit upper bounds for the Wasserstein distance for the rate of convergence when N→∞ and/or T→∞. In the case when T is fixed and N→∞, the upper bounds obtained in our results are more efficient than those of the Kolmogorov distance given by the relevant papers of Mishra and Prakasa Rao, and Kim and Park.
Highlights
Coefficient of a Stochastic PartialConsider the process {u(t, x ), 0 < x < 1, 0 ≤ t ≤ T } defined on a probability space (Ω, F, P) as a solution to the stochastic partial differential equationDifferential Equation
The goal of this paper is to provide Berry–Esseen bounds in Wasserstein distance for the Maximum Likelihood Estimator (MLE) θbN,T when N → ∞ and/or T → ∞
We are interested in the rate of convergence for the central limit theorem of the maximum likelihood estimator of the drift coefficient for a stochastic partial differential equation based on continuous time observations of the Fourier coefficients ui (t), i = 1, . . . , N of the solution, over some finite interval of time [0, T ]
Summary
Consider the process {u(t, x ), 0 < x < 1, 0 ≤ t ≤ T } defined on a probability space (Ω, F , P) as a solution to the stochastic partial differential equation. This problem has been solved by Kim and Park in [4], where they improved the bound in (5) to that converging to zero when N → ∞ and T fixed, by using techniques based the combination Malliavin calculus and Stein’s method They proved, in the case when f = 0, that, for sufficiently large N, there exists a constant Cθ,T depending on θ and T such that φ N (θ ) θbN,T − θ ≤ z − P( Z ≤ z) ≤ Cθ,T , sup P where the normalizing factor φ N (θ ) is given by φ N (θ ) = 2θ. We included a lemma that plays an important role in the proof of Theorem 1
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