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, we derive a nonparametric estimator for the reaction function of the underlying equation. The estimate is chosen from a finite-dimensional function space based on a least squares criterion. Oracle inequalities provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection. Secondly, we show that the asymptotic properties of realized quadratic variation based estimators for the diffusivity and volatility carry over from linear SPDEs. In particular, we obtain a rate-optimal joint estimator of the two parameters. The result relies on our precise analysis of the H\"older regularity of the solution process and its nonlinear component, which may be of its own interest. Both steps of the calibration can be carried out simultaneously without prior knowledge of the parameters.
翻译:对半线性SPDEs的非参数估计,即一个空间维度中的随机反应反扩散方程式,进行了研究。我们考虑在时间和空间的离散网格上对溶液场的观测,同时在两个坐标上填充充静态。首先,我们为基础方程式的反应功能得出一个非对称估计值。根据一个最小平方标准从一个有限维功能空间中选择这一估计值。Oracle不平等为估量器提供了达到通常的非对称融合率的条件。通过模型选择提供了适应性。第二,我们表明,基于分性和波动的测算器的已实现的二次变异的偏差性特性从线性SPDEs上传过来。特别是,我们获得了两个参数的速率-最佳联合估计器。结果取决于我们对溶液过程及其非线性组成部分的H\“老性规律性”及其非线性成分的精确分析,这可能符合其自身的利益。两个校准步骤可以在不事先了解参数的情况下同时进行。