On the research activity of the Laboratory
The classical statistics dealt mainly with the random structures depending on the relatively small number of parameters. These restrictions were caused by the small volume of statistical data, computational complexity etc.The situation has changed dramatically over the last years. It happened due to the fast progress of the computer and information technologies. First, very large databases became available in different application problems (such as biometrics, econometrics, medicine). Second, totally new methods of statistical analysis have been developed and many are being developed. In particular, it is happening in the case of the sparse data. The social, demographic and economics processes are so complex, that it is not principally possible to collect the full information about them. This information will be inevitably fractured. Besides, the parametrization of these processes is either impossible in principle or cannot be practically done.All the theoretical models in the financial mathematics and economics are based on the Stochastic Differential Equations (SDE’s): different versions of the Black-Sholes model, Cramer-Lundberg type equations in the insurance business etc. The numerical solution of these equations must be based on the discretization (Euler, Mil’shtein methods). The fundamental meaning take the questions why and how this discretization is good, whether it adequately models the asymptotic properties of SDE’s and how to estimate the error following from the discretization. In the case of non-degenerated elliptic and parabolic PDE’s the discussion of these questions is well covered in the literature. However,many mechanical, physical and financial models had e very essential deterministic component, so the diffusion processes (SDE’s) are degenerated. Here the hypo-elliptic theory proposed by Hörmander and developed by Malliavin in the probabilistic setting is very helpful. The discretization problem of the hypo-elliptic equations is very important and remains essentially open.
The research in the Laboratory has several directions:
- statistical analysis of the complex structures in the case of sparse data: algorithms, estimation of the errors etc.- the non-parametric methods in econometrics, biometrics, demography
- the construction and the study of the stationary in space and time models in population dynamics and demography. Problem of stability of these models with respect to random fluctuations of the environment
- asymptotic analysis and statistical algorithms for the mixed distributions with applications to the time series (stock prices, returns, options)
- the approximation of the non-degenerated diffusion processes by Markov chains. Local limit theorems, errors estimates based on the different modifications of the parametric technique
- Brownian motions on the Riemannian manifolds and their approximation by the “master processes”
- Special features of the Brownian motions on the nilpotent and solvable Lee groups. The approximation of these Brownian processes. Application to the European and Asian options
- Study of the degenerated diffusion processes and of their approximation by the random walks. Local and quasi local limit theorems.
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