Proposal Description

IPS “Statistical Estimation in Diffusion and Fractional Diffusion Models and High-Dimensional Analysis, with Applications” is devoted to extremely important statistical problems: modeling phenomena in physics, technology, finance, economics and assessing their parameters. On the one hand, the fact is that modern technological processes and the functioning of financial markets are much more complicated than 50 or even 20 years ago. Neural networks also create a very modern branch of statistics and its applications. On the other hand, these processes are characterized by high performance, and this rapid development leads to the fact that the internal characteristics of the models are inhibited, that is, in a sense, they have memory. Therefore, these processes need to be modeled using so-called fractional diffusion models, and then it is necessary to evaluate their parameters. Fractional diffusion models can be based on fractional Brownian motion and other Gaussian processes with memory, including Gaussian-Volterra processes, bi-fractional, tempered fractional Brownian motions, related mixed fractional processes and their generalizations. The latter models are more complicated but more flexible. Compared to standard diffusion models, fractional models have more parameters, including fractionality in one form or another, so new methods for estimating parameters when working with such models are needed.

Thus, our section represents something like a bridge between serious mathematical and statistical methods and numerous applications of the models under consideration. Distinguished statisticians from Finland, Japan, Sweden, and Ukraine will deliver invited lectures during the conference, providing insights into various aspects of statistical estimation in diffusion and fractional diffusion models.

Taras Bodnar (Stockholm University) explores the high-dimensional asymptotic properties of the Moore-Penrose inverse and the ridge-type inverse of the sample covariance matrix. The findings contribute to the construction of improved shrinkage estimators of the precision matrix and the weights of the global minimum variance portfolio.

Masaaki Fukasawa (Osaka University) focuses on the Local Asymptotic Normality property for fractional Ornstein-Uhlenbeck processes under high-frequency observations. A key role is played by a novel sharp error bound for the cumulants of Gaussian quadratic forms. The study demonstrates the asymptotic efficiency of the maximum likelihood sequence of estimators.

Yuliya Mishura (Taras Shevchenko National University of Kyiv) constructs asymptotic properties of trajectories and consistent statistical estimators for the Hurst index, volatility coefficient, and drift parameter for Bessel processes driven by fractional Brownian motion or the mixture of standard and fractional Brownian motions. The continuity of the fractional Bessel process is proven. This model is suitable for stochastic volatility and many other applications.

Kostiantyn Ralchenko (Taras Shevchenko National University of Kyiv) focuses on two types of tempered fractional Brownian motions. Least-square estimators for the unknown drift parameters within Vasicek models, driven by these processes, are constructed.

Tommi Sottinen (University of Vaasa) investigates the Langevin equation with stationary-increment Gaussian noise, demonstrating strong consistency and asymptotic normality with Berry–Esseen bound of the second-moment estimator of the mean reversion parameter. Conditions and results are presented in terms of the variance function of the noise. Continuous and discrete observations are considered, with examples including fractional and bifractional Ornstein–Uhlenbeck processes.