Conference Agenda

In this talk, we explore the parametric statistical inference of the drift function in high-dimensional diffusion processes. Specifically, we examine the statistical performance of the Lasso estimator when the process is observed discretely and under a sparsity condition over the parameter. In particular, we present error bounds for the estimator, and we provide an in-depth analysis of the contribution of the discretization error, clarifying the impact of the sampling effects in a high-dimensional regime.

This is based on joint work with Chiara Amorino and Mark Podolskij.

Our work deals with the estimation of the instantaneous or spot covariance matrix of a continuous semi-martingale X(t) using high-frequency observations. It has been shown that estimating and testing the rank of a covariance matrix leads to fast rates that are affected by a potential drift. Therefore, a potential drift term cannot be neglected and a modified estimator has to be considered. For this reason, we propose a re-centered covariance estimator instead of a second moment estimator. Our aim is to infer the maximal rank of the spot covariance matrix of the multidimensional process X(t). Consequently, we consider the null hypothesis that the rank of the spot covariance matrix is at most r for all t, and compute a corresponding critical value.

This talk is based on joint work with Markus Reiß.

Penalized estimation methods for diffusion processes have recently gained significant attention due to their effectiveness in handling high-dimensional stochastic systems. In this work, we introduce an adaptive Elastic-Net estimator for ergodic diffusion processes observed under high-frequency sampling schemes. Our method combines adaptive $\ell_1$ and $\ell_2$ regularization, enhancing prediction accuracy and interpretability while effectively recovering the sparse underlying structure of the model.

We provide finite-sample guarantees for the estimator's performance by deriving high-probability non-asymptotic bounds for the $\ell_2$ estimation error. These results complement the established oracle properties in an asymptotic regime with mixed convergence rates, ensuring consistent selection of the relevant interactions and achieving optimal rates of convergence. Furthermore, we utilize our results to analyze one-step-ahead predictions, offering non-asymptotic control over the prediction error.

The performance of our method is evaluated through simulations and real data applications, demonstrating its effectiveness, particularly in scenarios with strongly correlated variables.

We consider the class of stationary-increment harmonizable stable processes $X=\{X(t):t\in\mathbb{R}\}$ defined by \begin{align*} X(t)=Re\left(\int_\mathbb{R}\left(e^{itx}-1\right)\Psi(x)M_\alpha(dx)\right),\quad t\in\mathbb{R}, \end{align*} where $M_\alpha$ is an isotropic complex symmetric $\alpha$-stable ($S\alpha S$) random measure with Lebesgue control measure. This class contains real harmonizable fractional stable motions, which are a generalization of the harmonizable representation of fractional Brownian motions to the stable regime, when $\Psi(x) = \vert x\vert^{-H-1/\alpha}$, $x\in\mathbb{R}$. We give conditions for the integrability of the path of $X$ with respect to a finite, absolutely continuous measure, and show that the convolution with a suitable measure yields a real stationary harmonizable $S\alpha S$ process with finite control measure. Such a process admits a LePage type series representation consisting of sine and cosine functions with random amplitudes and frequencies, which can be estimated consistently using the periodogram. Combined with kernel density estimation, this allows us to construct consistent estimators for the index of stability $\alpha$ as well as the kernel function $\Psi$ in the integral representation of $X$ (up to a constant factor). For real harmonizable fractional stable motions consistent estimators for the index of stability $\alpha$ and its Hurst parameter $H$ are given, which are computed directly from the periodogram frequency estimates.