A note on parameter estimation for discretely sampled SPDEs

Abstract

The main goal of this paper is to build consistent and asymptotically normal estimators for the drift and volatility parameter of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space R and the bounded domain (0, π). First, we establish the exact regularity of the solution and its spatial derivative, which in turn, using power-variation arguments, allows building the desired estimators. Second, we propose two sets of estimators, based on sampling either the spatial derivative or the solution itself on a discrete space-time grid. Using the so-called Malliavin-Stein's method, we prove that these estimators are consistent and asymptotically normal as the spatial mesh-size vanishes. More importantly, we show that naive approximations of the derivatives appearing in the power-variation based estimators may create nontrivial biases, which we compute explicitly. We conclude with some numerical experiments that illustrate the theoretical results.

Drift estimation for discretely sampled SPDEs

“…Moreover, if 4β < d, then consistency holds true, when N → ∞, when M, T are fixed. This, in particular implies that to estimate efficiently the drift parameter it is enough to observe the Fourier modes at one instant of time -a result that agrees with recent discoveries in [CH17,BT17] where the solution is sampled in physical domain. Under some additional technical assumptions on the growth rates of N, M and T , we also prove that the proposed estimator is also asymptotically normal, with the same rate of convergence √ T N Open problems and future work.…”

On Central Limit Theorems for Power Variations of the Solution to the Stochastic Heat Equation

“…A natural problem is parameter estimation based on discrete observations of a solution of an SPDE which was first studied in [10] and which has very recently attracted considerable interest. Applying similar methods the three related independent works [6,2,4] study parabolic SPDEs including the stochastic heat equation, consider high-frequency observations in time, construct estimators using power variations of time-increments of the solution and prove central limit theorems. As we shall see below, the marginal solution process along time at a fixed spatial point is not a (semi-)martingale such that the well-established high-frequency theory for stochastic processes from [8] cannot be (directly) applied.…”

“…In view of this difficulty, different techniques are required to prove central limit theorems. Interestingly, the proof strategies in [6,2,4] are quite different. Cialenco and Huang [6] consider the realised fourth power variation for the stochastic heat equation with both an unbounded spatial domain D = R, or a bounded spatial domain D = [0, π].…”

Statistical analysis of some evolution equations driven by space-only noise

“…In contrast to the previous section, where the measurements were done in the Fourier space, here we will assume that the solution, or its spacial derivative, is observed in the physical space. It was noted in [2] that, to estimate the drift and/or volatility in a stochastic heat equation driven by a space-time white noise, it is enough to observe the solution at only one fixed time point and at some discrete spacial points from a fixed interval.…”