Drift estimation for discretely sampled SPDEs
Drift estimation for discretely sampled SPDEs
复制标题
离散采样 SPDE 的漂移估计
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Hyun
中科院分区：
文献类型：
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作者：
Igor Cialenco;Francisco Delgado;Hyun
研究了加性时空噪声驱动下分数阶随机热方程漂移系数的极大似然估计的渐近性质。我们考虑传统的随机偏微分方程统计实验时，测量是在谱域中进行的，与现有的文献相比，我们研究的最大似然（型）估计（MLE）的渐近性质时，傅立叶模式的数量和时间都趋于无穷大。在本文的第一部分中，我们考虑通常的设置连续时间观测的解决方案的傅里叶系数，并表明，MLE是一致的，渐近正常的和最佳的均方意义。在本文的第二部分中，我们研究的自然时间离散化的MLE，通过假设的第一个N傅立叶模式测量M时间网格点，均匀分布在时间间隔[0，T ]。我们提供了一个严格的渐近分析的建议估计时，$$N
N → ∞和/或T，M的八次幂
八箭头infty $$ T，M → ∞。我们建立了N，M和T的增长率的充分条件，保证这些估计的一致性和渐近正态性。
The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consider the usual setup of continuous time observations of the Fourier coefficients of the solutions, and show that the MLE is consistent, asymptotically normal and optimal in the mean-square sense. In the second part of the paper we investigate the natural time discretization of the MLE, by assuming that the first N Fourier modes are measured at M time grid points, uniformly spaced over the time interval [0, T ]. We provide a rigorous asymptotic analysis of the proposed estimators when $$N
ightarrow infty $$ N → ∞ and/or $$T,M
ightarrow infty $$ T , M → ∞ . We establish sufficient conditions on the growth rates of N , M and T , that guarantee consistency and asymptotic normality of these estimators.