Under Objective A, we have made substantial progress in the high-dimensional estimation of diffusion models, fulfilling one of the key milestones of the proposal. Our work has resulted in several completed articles addressing the estimation of drift and volatility functions in high-dimensional settings. These contributions are of critical importance for the analysis of complex systems arising in fields such as finance, physics, and biology.
A significant part of our research has focused on the estimation of drift functions in high-dimensional parametric models under both continuous and discrete observation schemes. We have established convergence rates for the proposed estimators and demonstrated their support recovery properties. In parallel, we investigated the statistical estimation of the volatility matrix under low-rank constraints. These theoretical findings have been applied to large asset portfolios, yielding insightful empirical results on the intrinsic dimensionality of such models. The outcomes of this line of research have been published in leading journals, including Bernoulli, Electronic Journal of Statistics, and Statistical Inference for Stochastic Processes.
In a different direction, we have advanced the statistical analysis of McKean–Vlasov equations, a class of high-dimensional models with applications across disciplines such as physics, biology, opinion dynamics, and deep learning. Our work spans both parametric and non-parametric inference, as well as investigations into optimality of the proposed methods. We have developed novel estimators for the interaction function, which plays a central role in modeling large particle systems. These results have significant theoretical value and practical relevance. This strand of the project has led to publications in top-tier journals such as Annales de l’Institut Henri Poincaré (B), Probability Theory and Related Fields, and Stochastic Processes and Their Applications.
Within Objective B, our group has made significant contributions to the statistical and probabilistic theory of random matrices. One major line of research established novel estimators for multivariate quadratic variation of jump semimartingales, addressing both synchronous and asynchronous observation schemes. These developments have important applications in financial mathematics, where asset prices are often modeled using semimartingales.
Another key focus has been the probabilistic analysis of the spectral properties of large random matrices. This work contributes to a deeper understanding of high-dimensional stochastic systems and has implications across multiple domains. The results from these studies have been published in respected international journals, including Random Matrices: Theory and Applications and Stochastic Processes and Their Applications.
Under Objective C, we have completed a range of projects addressing various aspects of the objective. Two articles explore the optimal estimation of local times and the suprema/infima of stable Lévy processes, along with the associated weak limit theory. These results enhance our understanding of how random quantities should be estimated in practice, with a focus on optimality.
A particularly noteworthy theoretical contribution is our discovery of a new Stein-type characterization for stable convergence to mixed normal limits, which also admits a quantitative formulation. This result has potential applications that extend well beyond the scope of the STAMFORD project and may influence future developments in both probability theory and statistics.
In addition, our group has published articles on the probabilistic and statistical analysis of non-Gaussian fractional processes under infill asymptotic. The results offer new insights into the behavior of such processes and contribute to the development of robust inference methods. This body of work has appeared in leading international journals such as Annals of Statistics, Electronic Journal of Statistics and the Electronic Journal of Probability.
