A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

Abstract

A new modification of the minimum-contrast estimator (the weighted MCE) of drift parameter in a linear stochastic evolution equation with additive fractional noise is introduced in the setting of the spectral approach (Fourier coordinates of the solution are observed). The reweighing technique, which utilizes the self-similarity property, achieves strong consistency and asymptotic normality of the estimator as number of coordinates increases and time horizon is fixed (the space consistency). In this respect, this modification outperforms the standard (non-weighted) minimum-contrast estimator. Compared to other drift estimators studied within spectral approach (eg. maximum likelihood, trajectory fitting), the weighted MCE is rather universal. It covers discrete time as well as continuous time observations and it is applicable to processes with any value of Hurst index H ∈ (0, 1). To the author's best knowledge, this is so far the first space-consistent estimator studied for H < 1/2. 2010 Mathematics Subject Classification. 60H15, 60G22, 62M09. 1 2 PAVEL KŘÍŽreader to the articles [28] for continuous-time setting, [10] for discrete-time setting and [12] for comparison with the LSE, to name just a few. These works benefit from the relation of Malliavin calculus and central limit theorems -a popular theory initiated in [23] and further developed by many authors (see e.g.[22] and references therein). These techniques were recently applied to the MCE in infinitedimensional setting in [18] and are also utilized in the present work. Another interesting work on minimum-contrast estimation is [3], where a different version of the MCE based on fundamental-martingale technique for a real-valued fOU with Hurst index H > 1/2 is studied in the frameworks of both continuous-time and discrete-time sampling. Regarding drift parameter estimation for singular realvalued fOU processes (H < 1/2), recall the paper [12], where the LSE and the MCE for continuous-time observation are studied and the work [16], which investigates discrete-time version of the LSE.A modification of the minimum-contrast estimator (the weighted MCE) is introduced in this paper. Its construction benefits from the self-similarity property and it turns the time-consistent MCE (consistent with increasing time horizon) into a space-consistent estimator (fixed time horizon and increasing number of Fourier coordinates), which provides the best attainable speed of convergence in discrete-time setting. We believe this approach is potentially applicable to other types of timeconsistent estimators, such as the LSE, and for different types of models (but still having the self-similarity property). To the author's best knowledge, this approach is new even in the basic case of parabolic diagonalizable equations with white additive noise (in space and time).

Pathwise least-squares estimator for linear SPDEs with additive fractional noise

“…Although we did not address the asymptotic normality of the pathwise LSE in this paper (to keep it concise), we conjecture that it holds. Our conjecture is based on simulation results and asymptotic similarity with the weighted MCE, whose asymptotic normality has been proven in [17].…”

“…The theory for the weighted MCE, developed in [17], covers only singleoperator equations, so it is not directly applicable for the second example (2D stochastic heat equation with coupling with surroundings). To our best knowledge, the pathwise LSE is the first efficient tool for parameter estimation in this equation (important in oceanography, for example) when the noise is generated by a cylindrical fractional Brownian motion with general Hurst parameter 0 < H < 1.…”

“…We test performance of the pathwise LSE, defined in (17), and compare it with the weighted MCE, an alternative estimator defined and studied in [17], cf. formulas (3.21) -(3.23) and Theorem 3.2 therein.…”

“…A least-squares estimator for a one-dimensional projection of the solution has been studied in long-span asymptotic regime in [22] assuming the regular case H ≥ 1/2. Drift estimation in spectral setting (Fourier modes are observed) for infinite-dimensional equations with fractional noise has been studied in [8] (MLE approach, H ≥ 1/2) and [17] (ergodic-type estimator, H ∈ (0, 1)). We are not aware of any study on LSE in spectral setting (Fourier modes are observed) for these models.…”

“…Estimators of ergodic type for more general equations driven by a finite-dimensional fBm were recently studied in [25]. Application of this ergodic approach in the spectral setting for infinite-dimensional equations corresponds to the weighted minimum-contrast estimator (weighted MCE) introduced in [17]. Such estimator is plausible for stationary processes, but fails for processes at far-from-stationary state and is difficult to generalize to more complex equations, because (in contrast to LSE) it requires ergodicity and precise knowledge of the limiting covariance operator.…”