Testing the rank of the spot covariance matrix of a multidimensional It\^o semi-martingale

Abstract

This work develops a statistical test for the maximal rank of the deterministic instantaneous (or spot) covariance matrix of a continuous-time $\mathbb{R}^d$-valued It\^o semi-martingale $X(t)$ using high-frequency observations with a particular focus on the impact of an adapted drift. We explicitly account for the presence of an adapted drift process, which, as our results demonstrate, cannot be neglected, by introducing a re-centred covariance estimator instead of relying solely on a second moment estimator. Building on this estimator, we test the null hypothesis that the rank of the spot covariance matrix is at most $r<d$ for all $t$ against local alternatives in which the $(r+1)$th eigenvalue is greater than some vanishing signal detection rate. Critical values are derived in a non-asymptotic framework and can be significantly affected by a potential drift. However, the power analysis establishes asymptotic consistency for separation rates, which depend on the H\"older regularity of both the drift and the spot covariance matrix, as well as on a potential spectral gap $\underline{\lambda}_r \geq 0$ under the null hypothesis. Simulation results indicate that the covariance-based test achieves higher power across a wider range of alternatives compared to classical second moment-based procedures.