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## Вариация мартингалов со значениями в вероятностных мерах и повторяющиеся игры с неполной информацией

In the book deals with modern methods and models of socio-economic forecasting, the most frequently used in practice. Essential part of economic decisions aimed at obtaining results in the future, so to make the right management decisions need to be reliable socio-economic forecasting, which is impossible without knowledge of methods and models. In connection with this prerequisite training of highly economist and manager is to study their discipline "methods of social and economic forecasting." The second volume contains the basic textbook methods and models used today in the socio-economic forecasting. Consistently provides methods and models short, medium and long-term forecasting how simple models of trends and using factor models. In some groups are identified and methods of forecasting models of evolutionary processes of socio-economic dynamics. The textbook is designed for students of undergraduate academic, but may be useful to undergraduates, postgraduates and doctoral students, as well as practitioners dealing with the forecasting of socio-economic processes.

Variational Inference is a powerful tool in the Bayesian modeling toolkit, however, its effectiveness is determined by the expressivity of the utilized variational distributions in terms of their ability to match the true posterior distribution. In turn, the expressivity of the variational family is largely limited by the requirement of having a tractable density function. To overcome this roadblock, we introduce a new family of variational upper bounds on a marginal log-density in the case of hierarchical models (also known as latent variable models). We then derive a family of increasingly tighter variational lower bounds on the otherwise intractable standard evidence lower bound for hierarchical variational distributions, enabling the use of more expressive approximate posteriors. We show that previously known methods, such as Hierarchical Variational Models, Semi-Implicit Variational Inference and Doubly Semi-Implicit Variational Inference can be seen as special cases of the proposed approach, and empirically demonstrate superior performance of the proposed method in a set of experiments.

There are a lot of models of bankruptcies prediction, which differ in methods of modeling and in set of factors. These methods are mainly belong to 5 groups: the classic statistical methods, regression analysis, the method of discriminant analysis, logit-analysis methods, methods of fuzzy sets and neural network methods. Combinations of these methods also can take place. The last three groups of methods are currently being developed especially quickly. As for the choice of factors bankruptcy, prevails heuristics. There is no formal methodology for selection and comparison groups of economic indicators to build the model of bankruptcies, as well as effective methods for data preprocessing. In this paper we propose an original method for the choice of indicators, followed by the construction of neural network model diagnostic of bankruptcies based on Bayesian approach.

**Introduction.** There are a lot of models of bankruptcies prediction, which differ in methods of modeling and in set of factors. These methods are mainly belong to 5 groups: the classic statistical methods, regression analysis, the method of discriminant analysis, logit-analysis methods, methods of fuzzy sets and neural network methods. Combinations of these methods also can take place. The last three groups of methods are currently being developed especially quickly. As for the choice of factors bankruptcy, prevails heuristics. There is no formal methodology for selection and comparison groups of economic indicators to build the model of bankruptcies, as well as effective methods for data preprocessing. In this paper we propose an original method for the choice of indicators, followed by the construction of neural network model diagnostic of bankruptcies based on Bayesian approach.

**Methods. **The study applied the methods of mathematical statistics, correlation analysis, neural network modeling, methods of knowledge representation in intelligent information technologies.

**Results. **The developed concept of formalization of choice and comparative evaluation of a system of indicators has created the preconditions for the development of effective neural network model of bankruptcies.

**Discussion.** The proposed concept was tested for the construction sector of the economy. However, the authors believe that the generality of the approach, the concept and the method may be useful in other industries for a wide range of economic problems, such as the formation of the loan portfolio, an external audit or evaluation of the financial condition of the company.

The paper considers a game-theoretical model of bidding with asymmetric information. One player has the inside information on the liquidation price of risky asset. The model is formalized with the repeated game with incomplete information on the side of uninformed player. We consider the case of external stopping of the game at the random moment. Insider's expected profit in the game of random duration if she applies the strategy optimal in infinite-stage game is obtained. This result allows to calculate the loss of insider in case of sudden disclosure of his private information.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l2 space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.

For a class of optimal control problems and Hamiltonian systems generated by these problems in the space *l *2, we prove the existence of extremals with a countable number of switchings on a finite time interval. The optimal synthesis that we construct in the space *l *2 forms a fiber bundle with piecewise smooth two-dimensional fibers consisting of extremals with a countable number of switchings over an infinite-dimensional basis of singular extremals.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

In this paper, we construct a new distribution corresponding to a real noble gas as well as the equation of state for it.