LABORATORY 16

Laboratory of Stochastic Dynamical Systems

Head of Laboratory – Dr.Sc. (Mathematics), Prof. Aleksandr Veretennikov

Tel.: (095) 299-94-15; E-mail: veretenn@iitp.ru

 

The leading researchers of the laboratory include:

Dr.Sc. (Math.)

V. Kalashnikov

Dr.

P. Kitsul

Dr.Sc. (Techn.)

R. Liptser

Dr.

A. Puhalskii

Dr.

F. Grigoriev

Dr.

A. Serebrovskii

Dr.

O. Gulinsky

Dr.

S. Lototsky

Dr.

V. Kistlerov

    

Currently R. Liptser, P. Kitsul, A. Puhalskii and S. Lototsky are working abroad remaining the staff members of Laboratory.

Directions of Activity

MAIN RESULTS

Stochastic Averaging Principle and new mixing bounds helped to investigate a stochastic approach to "sliding modes'' – engineering oriented theory which studies motions with discontinuity of forces (right hand sides of an ODE) along some lines or surfaces in the right hand side of an ordinary differential equation. Stochastic approach introduced in this work prescribes to add a small white noise to the right hand side of the equation and study the limit when the intensity of the noise tends to zero. (A. Veretennikov.)

For a specific class of models of Stochastic Averaging Principle with "slow" and "fast diffusions" the problem concerning the Large Deviation Principle for the systems with "complete dependence" was solved; the problem had been open for

about 25 years. (A. Veretennikov.)

An approach to large deviations for asymptotic problems without evident probabilistic representation behind is proposed. Some examples provided by the mean field models of quantum statistical mechanics are considered. The approach is based on the Choquet theory of capacities. Following Choquet's ideas we introduce a class of strongly sublinear functionals and show that if a sequence of properly normalized functionals of this class has a limit than it obeys the sup-property (the principle of the largest terms). A generalisation of the Choquet theorem provides us by a sup-integral or idempotent integral representation of sup-functional(the analogue of the Riesz theorem). This functional is proved to be an extreme element of a certain cone of functionals. Based on this representation we are able to prove an analogue of Gartner's theorem and the Varadhan type variational principle. (O. Gulinsky.)

A case when a component of a vector gaussian Markov process is a Markov process itself is considered. The necessary and sufficient conditions are obtained. (A. Serebrovskii.)

An application of recent mixing bounds for Markov chains provided parameter estimation in dependent (markovian) experiments. Under exponential mixing it was proved that a maximum likelihood estimator is effective in the sense of Hajek – Le Cam with any polynomial growth loss function. (A. Veretennikov.)

Two-sided quantitative stability bounds of queueing models in terms of weighted probability metrics are obtained. (V. Kalashnikov.)

New two-sided bounds of reliability of redundant systems working in the range of small time arguments are proposed. (V. Kalashnikov.)

Spatial-time model of mobile communication systems is proposed. This model takes into account spatial and time inhomogeneity of the arrival process reflecting the inhomogeneity of the information transmission process (video, audio, etc.) as well as movement of customers. (V. Kalashnikov.)

New control and synthesis results were obtained for certain classes of nonlinear moving objects. New bounds for optimal control are proposed. (F. Grigoriev.)

Semantics of functional programming languages was studied on the base of FLAC language. (V. Kistlerov.)

Teaching:

International collaboration. Fruitful collaboration is established with the probability group of the LATP CMI Universite de Provence, Marseille, France, and, in particular, with Professor Etienne Pardoux as its leader.

We also have close contacts with Universite Paris 6 (Professors Jean Jacod and Pierre Priouret); Universite du Main in France (Professor Yuri Kutoyants); Weierstrass Institute for Applied Analysis and Stochastics – WIAS, Berlin, Germany; the Univerity of Warwick, UK (Professor David Elworthy); Mathematical Institute of the University of Copenhagen; University of Trier (Professor Dieter Baum); University of Wuerzburg (Professor Elart von Collani), and some others.

Other activities. V. Kalashnikov is an Associate Editor in the journals: “Queueing Systems. Theory and Applications”; “Stochastic Models”; “Pattern Recognition and Image Analysis”; “Fundamental and Applied Mathematics”.

Conferences: V. Kalashnikov was a member of the Organizing Committee of the International Symposium "Defining the Science Stochastics" (4-6 October, Germany), under support of the Bernoulli statistical Society.

V. Kalashnikov was an invited speaker at the following international meetings:

 

Grants from:

A. Veretennikov.

 

Publications in 2000

  1. Veretennikov A. Yu. On polynomial mixing for SDEs with a gradient-type drift // Teoriya Veroyatn. i ee Primenen. 2000. V. 45(1). P. 163-166.

  2. Milstein G. N., Veretennikov A. Yu. On deterministic and stochastic sliding modes via small diffusion approximation // Markov Processes and Related Fields. 2000. V. 6. No. 3. P. 371-395.

  3. Veretennikov A. Yu. On large deviations for SDEs with small diffusion and averaging // Stochastic Processes and their Applications. 2000. V. 89. No. 1. 2000. P. 69-79.

  4. Веретенников А. Ю. Параметрическое и непараметрическое оценивание для цепей Маркова. М.: Изд-во ЗПИ. МГУ. 2000.

  5. Pardoux E., Veretennikov A. Yu. On regularity of an invariant density of a Markov chain in a parameter // Russian Math. Dokl. 2000.

  6. Cai J., Kalashnikov V.V. NWU property of a class of random sums // Journal of Applied Probability. 2000. V. 37. P. 283-289.

  7. Enikeeva F., Kalashnikov V.V., Rusaityte D. Continuity estimates for ruin probabilities // Scandinavian Actuarial Journal. 2000. № 2.

  8. Kalashnikov V.V. Quantification in stochastics and the stability concept // Proc. of the Millenial Symposium "Defining the Science Stochastics" (October 4–6). Würzburg, Germany. 2000. P. 37-72.

  9. Kalashnikov V.V., Norberg R. On the sensitivity of premiums and reserves to changes in valuation elements // Laboratory of Actuarial Mathematics, University of Copenhagen. Working paper. 2000.

  10. Kalashnikov V.V., Konstantinides D. Ruin under interest force and subexponential claims: A simple treatment // Insurance: Math. & Econ. 2000. No. 27. P. 145-149.

  11. Калашников В., Носовский Г., Фоменко А. Астрономический анализ хронологии. М.: Деловой экспресс. 2000.

  12. Bon J-L., Kalashnikov V.V. Some estimates of geometric sums // Proc. of the 2nd Internatinal Conference on Mathematical Methods in Reliability. Bordeaux. 4-7 June. P. 45-49.

  13. Григорьев Ф.Н., Кузнецов Н.А. Оптимальное по быстродействию управление в одной задаче // Автоматики и телемеханика. 2000. № 8. С. 11-25.

  14. Pardoux E., Veretennikov A.Yu. On Poisson equation and diffusion approхimation, 1 // Annals of Probability (to appear).

  15. Veretennikov A.Yu. On KPP equations via large deviations: Larson's result // Proc. 8 Vilnius Conf. on Prob. Th. and Math. Stat. (to appear).

  16. Kitsul P.I. , Liptser R.Sh. and Serebrovski A.P. When a Component of a Vector Gaussian Markov Process is a Markov Process Itself // Systems & Control Letters (to appear).

  17. Gulinsky O.V. The Principle of the Largest Terms and Large Deviations for a Class of Nonlinear Functionals with Applications to Some Model of Quantum Statistical Mechanics (to appear).