We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

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关于离散采样 SPDE 参数估计的注释

当数据在时间或空间分量上离散采样时，我们考虑一维随机热方程的参数估计问题。我们证明，拉普拉斯算子旁边的实值参数（漂移）和噪声前面的常数参数（波动率）可以在令人惊讶的最小信息下一致地估计。也就是说，在固定时间和离散空间网格上，或在固定空间点和有限间隔的离散时间实例上观察解就足够了，假设网格大小变为零。所提出的估计量具有相同的形式和渐近属性，而不管域的性质——有界域或整个空间。估计量的推导及其渐近性质的证明是基于对一些相关随机过程的幂变化的计算。我们使用 Malliavin 演算的元素来建立有界域情况下的渐近正态性属性。我们还讨论了漂移和波动系数的联合估计问题。我们以一些数值实验结束，这些实验说明了所获得的理论结果。