Penalized estimation methods for diffusion processes and dependent data 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 the least squares approximation of the quasi-likelihood with adaptive $\ell_1$ and $\ell_2$ regularization. This approach allows to enhance prediction accuracy and interpretability while effectively recovering the sparse underlying structure of the model. In the spirit of analyzing high-dimensional scenarios, we provide finite-sample guarantees for the (block-diagonal) estimator's performance by deriving high-probability non-asymptotic bounds for the $\ell_2$ estimation error. These results complement the established oracle properties in the high-frequency 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 $\ell_1$ 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.

翻译：针对扩散过程及相关数据的惩罚估计方法，因其在处理高维随机系统方面的有效性，近来受到广泛关注。本文针对高频采样方案下观测到的遍历扩散过程，提出了一种自适应弹性网估计量。该方法将拟似然的最小二乘逼近与自适应$\ell_1$和$\ell_2$正则化相结合，能够在有效恢复模型稀疏底层结构的同时，提高预测精度与可解释性。在高维场景分析框架下，我们通过推导$\ell_2$估计误差的高概率非渐近界，为（块对角）估计量的性能提供了有限样本保证。这些结果与混合收敛速率的高频渐近体系中的既定oracle性质相互补充，确保了对相关交互项的一致选择，并达到了最优收敛速率。此外，我们利用所得结果分析一步超前预测，为$\ell_1$预测误差提供了非渐近控制。通过仿真和实际数据应用评估了本方法的性能，结果表明其具有显著有效性，尤其在变量强相关场景中表现突出。