Periodic Reporting for period 4 - STAMFORD (Statistical Methods For High Dimensional Diffusions)
Reporting period: 2023-07-01 to 2025-02-28
The project STAMFORD addresses numerous mathematical problems in high dimensional analysis of diffusion models. In modern applications such as finance, physics or biology just to name a few, high dimensional diffusions play a prominent role, and their statistical analysis is of immense importance.
Our project aims at developing statistical techniques for these models, which combine elements of theoretical statistics and advanced probability theory. We are mostly focusing on high dimensional procedures for estimation of the drift and volatility functions of a diffusion processes. The theoretical results are expected to shed light on large particle system models in sciences.
Our project aims at developing statistical techniques for these models, which combine elements of theoretical statistics and advanced probability theory. We are mostly focusing on high dimensional procedures for estimation of the drift and volatility functions of a diffusion processes. The theoretical results are expected to shed light on large particle system models in sciences.
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.
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.
In the article "Optimal Estimation of Some Random Quantities of a Lévy Process", we present a comprehensive theoretical study on the optimal estimation of key random characteristics of Lévy processes, including the maximum, minimum, local time, and occupation time measure. Unlike earlier contributions in this area, our work is the first to rigorously address the question of estimator optimality. As such, it goes beyond the current state of the art. While the framework is tailored to Lévy processes, the underlying methodology offers ideas that may prove valuable in broader settings.
In the papers “On Lasso Estimation for the Drift Function in Diffusion Models” and "Consistent Support Recovery for High-Dimensional Diffusions", we investigate the high-dimensional estimation of the drift function—an essential milestone of the proposal. We develop a comprehensive theoretical framework for the estimation error and demonstrate the support recovery property of the adaptive Lasso estimator. Our results provide a theoretical foundation for analyzing complex high-dimensional diffusion models, with applications in fields such as finance, physics, and biology.
In a series of follow-up works—“Semiparametric Estimation of McKean–Vlasov SDEs”, “Polynomial Rates via Deconvolution for Nonparametric Estimation in McKean–Vlasov SDEs”, and “On Nonparametric Estimation of the Interaction Function in Particle System Models”—we introduce novel techniques for estimating the interaction function, a central object in models of interacting particle systems. Despite the growing relevance of McKean–Vlasov SDEs, the statistical inference theory for these models remains underdeveloped. Our contributions address this gap by advancing estimation strategies, analyzing rate efficiency, and tackling inverse problems inherent to such systems.
The work “Quantitative and Stable Limits of High-Frequency Statistics of Lévy Processes: A Stein’s Method Approach” develops a new Stein-type framework tailored to non-Gaussian statistics that exhibit asymptotic mixed normality. This methodology enables a refined understanding of quantitative limit theorems in non-Gaussian settings and offers promising applications in both probability theory and statistical inference.
Finally, the article “Estimation of Mixed Fractional Stable Processes Using High-Frequency Data” focuses on statistical inference for superpositions of fractional stable motions. These processes have recently gained prominence within the rough volatility paradigm in mathematical finance. Our research contributes rigorous results on identifiability and convergence rates, providing a theoretical foundation for the analysis and estimation of mixed fractional stable models.
In the papers “On Lasso Estimation for the Drift Function in Diffusion Models” and "Consistent Support Recovery for High-Dimensional Diffusions", we investigate the high-dimensional estimation of the drift function—an essential milestone of the proposal. We develop a comprehensive theoretical framework for the estimation error and demonstrate the support recovery property of the adaptive Lasso estimator. Our results provide a theoretical foundation for analyzing complex high-dimensional diffusion models, with applications in fields such as finance, physics, and biology.
In a series of follow-up works—“Semiparametric Estimation of McKean–Vlasov SDEs”, “Polynomial Rates via Deconvolution for Nonparametric Estimation in McKean–Vlasov SDEs”, and “On Nonparametric Estimation of the Interaction Function in Particle System Models”—we introduce novel techniques for estimating the interaction function, a central object in models of interacting particle systems. Despite the growing relevance of McKean–Vlasov SDEs, the statistical inference theory for these models remains underdeveloped. Our contributions address this gap by advancing estimation strategies, analyzing rate efficiency, and tackling inverse problems inherent to such systems.
The work “Quantitative and Stable Limits of High-Frequency Statistics of Lévy Processes: A Stein’s Method Approach” develops a new Stein-type framework tailored to non-Gaussian statistics that exhibit asymptotic mixed normality. This methodology enables a refined understanding of quantitative limit theorems in non-Gaussian settings and offers promising applications in both probability theory and statistical inference.
Finally, the article “Estimation of Mixed Fractional Stable Processes Using High-Frequency Data” focuses on statistical inference for superpositions of fractional stable motions. These processes have recently gained prominence within the rough volatility paradigm in mathematical finance. Our research contributes rigorous results on identifiability and convergence rates, providing a theoretical foundation for the analysis and estimation of mixed fractional stable models.