arXiv subjects

Pavel Kriz

Publications and source records attributed to Pavel Kriz.

3 recordsLinked to original sources

arXiv2025

Inference for SDEs driven by Hermite processes

In the paper, we address parametric and non-parametric estimation for nonlinear stochastic differential equations with additive Hermite noise with possibly nonlinear scaling. We assume that a single trajectory of the solution is observed discretely and we propose estimators of the Hurst parameter and the Hermite order of the driving process as well as of the average noise intensity and noise intensity function. The estimators are based on the weighted quadratic variation whose properties are used, in particular, to prove weak consistency of the proposed estimators under in-fill asymptotics.

math.ST↗

arXiv2019

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

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 $H \in (0,1)$. To the author's best knowledge, this is so far the first space-consistent estimator studied for $H < 1/2$.

math.PR↗

arXiv2019

Central Limit Theorems and Minimum-Contrast Estimators for Linear Stochastic Evolution Equations

Central limit theorems and asymptotic properties of the minimum-contrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular ($H < \frac{1}{2})$ and regular ($H > \frac{1}{2})$ types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter $H < \frac{3}{4}$) is the so-called $4^{th}$ moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.

math.PR↗