In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is -- with a high probability -- the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three-five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV -- and also RV -- of full rank.

翻译：本文针对高维连续Itô半鞅的二次变差估计问题，提出了一种惩罚实现方差估计量。我们将线性回归中的正则化思想推广至连续时间高频场景下的协方差估计。研究表明，在核范数惩罚下，PRV可通过软阈值化实现方差矩阵的特征值来计算，从而促进奇异值的稀疏性（即解的低秩性）。我们证明该估计量在至多对数因子范围内达到极小极大最优性。通过推导集中不等式，我们发现PRV的秩以高概率等于QV非可忽略特征值的数量。此外，我们还对瞬时方差进行了相应的非渐近分析。为选择收缩参数，我们提出了一种直观的数据驱动子抽样方法。理论结果通过模拟研究和实证应用得到验证。PRV在股票市场中检测到约三至五个因子，且在金融市场动荡时期出现显著的秩下降现象。这与大多数标准资产定价模型相一致：驱动股票横截面收益的有限系统性因子受到异质性误差扰动，使得QV（及RV）呈现满秩特性。