High-frequency analysis of parabolic stochastic PDEs

Parameter estimation for SPDEs based on discrete observations in time and space

“…Assuming X is observed on a discrete grid {(t i , y k )} i=0,...,N,k=0,...,M ⊂ [0, T ] × [0, 1], approximate maximum likelihood estimators have been first investigated by Markussen [28] for T → ∞. For various linear SPDEs central limit theorems for method of moment type estimators based on realized quadratic variations have been studied by Torres et al [36], Cialenco and Huang [7], Cialenco and Kim [8], Bibinger and Trabs [2,3], Chong [4,5], Shevchenko et al [35], Liu and Tudor [25], as well as Kaino and Uchida [22]. However, all these works only give partial answers to the estimation problem.…”

“…(ii) The optimal convergence rate for estimating (ϑ 2 , σ 2 ) jointly is given by 1/ √ M 3 ∧N 3/2 which is generally slower than the parametric rate 1/ √ MN. (iii) Quadratic variations based on space-time increments, see (4) below, satisfy a central limit theorem with 1/ √ MN-rate, regardless of the relation of N and M . Furthermore, they can be used to implement a joint estimator for (σ 2 , ϑ 2 ) that reaches the optimal rate 1/…”

“…Our proofs employ directly the Gaussian distribution of X which allows for an explicit covariance condition for asymptotic normality of quadratic forms of Gaussian triangular schemes. Also, our estimators could be directly generalized to a nonparametric model with time dependent coefficients, as indicated in [3,5].…”

“…Note that the solution process X to the SPDE (1) admits continuous trajectories only in one spatial dimension. In the multi-dimensional case one could consider noise processes which are more regular in space as studied by Chong [5]. Alternatively, Kriz and Maslowski [24] as well as Altmeyer and Reiß [1] generalize the spectral approach to the observation of functionals X t , K for some (localizing) kernel K. In one space dimension, a second canonical problem is to analyze equation (1) on the whole real line instead of the unit interval, as considered in, e.g., [2,7].…”

Ergodic properties of the solution to a fractional stochastic heat equation, with an application to diffusion parameter estimation

“…It was first investigated in [29]. Later, applying similar methods, parabolic SPDEs including the stochastic heat equation had been studied in [6,10,14]. In these articles estimators based on power variations of time-increments of the solution were constructed and central limit theorems were proved.…”