论文标题
在小扩散率下半线性SPDE中反应项的估计
Estimation for the reaction term in semi-linear SPDEs under small diffusivity
论文作者
论文摘要
我们考虑在随机热量中的非线性反应项的估计，或在半线性随机偏微分方程（SPDE）中估计。通过研究较小的扩散水平来实现一致的推断，这在应用中是现实的。我们的主要结果是参数估计器的估计误差的中心限制定理，可以从中构建置信区间。通过建立局部渐近正态性来证明统计效率。估计方法扩展到时间和空间的局部观察结果，这允许对时间和空间变化的反应强度的参数估计。此外，可以处理时间和空间中的离散观察结果。统计分析需要从随机分析（例如Malliavin微积分）的高级工具，用于SPDES，无限维高斯庞加莱的不平等和SPDES的规律性结果。
We consider the estimation of a non-linear reaction term in the stochastic heat or more generally in a semi-linear stochastic partial differential equation (SPDE). Consistent inference is achieved by studying a small diffusivity level, which is realistic in applications. Our main result is a central limit theorem for the estimation error of a parametric estimator, from which confidence intervals can be constructed. Statistical efficiency is demonstrated by establishing local asymptotic normality. The estimation method is extended to local observations in time and space, which allows for non-parametric estimation of a reaction intensity varying in time and space. Furthermore, discrete observations in time and space can be handled. The statistical analysis requires advanced tools from stochastic analysis like Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincaré inequality and regularity results for SPDEs in $L^p$-interpolation spaces.
