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The paper considers the basic model of competitive interactions and its use for the analysis and description of social processes. The peculiarity of the model is that it describes the interaction of several competing actors, while actors can vary the strategy of their actions, in particular, form coalitions to jointly counter a common enemy. As a result of modeling, various modes of competitive interaction were identified, their classification was conducted, and their features were described. In the course of the study, the attention is paid to the so-called “rough” (according to A.A. Andronov) cases of the implementation of competitive interaction, which until now have rarely been considered in the scientific literature, but are quite common in real life. Using a basic mathematical model, the conditions for the implementation of various modes of competitive interactions are considered, the conditions for the transition from one mode to another are determined, examples of the implementation of these modes in the economy, social and political life are given. It is shown that with a relatively low level of competition, which is non-antagonistic in nature, competition can lead to an increase in the activity of interacting actors and to overall economic growth. Moreover, in the presence of expanding resource opportunities (as long as such opportunities remain), this growth may have a hyperbolic character. With a decrease in resource capabilities and increased competition, there is a transition to an oscillatory mode, when weaker actors unite to jointly counteract stronger ones. With a further decrease in resource opportunities and increased competition, there is a transition to the formation of stable hierarchical structures. At the same time, the model shows that at a certain moment there is a loss of stability, the system becomes “rough” according to A.A. Andronov and sensitive to fluctuations in parameter changes. As a result, the existing hierarchies may collapse and be replaced by new ones. With a further increase in the intensity of competition, the actor-leader completely suppresses his opponents and establishes monopolism. Examples from economic, social, and political life are given, illustrating the patterns identified on the basis of modeling using the basic model of competition. The obtained results can be used in the analysis, modeling and forecasting of socioeconomic and political processes.

The paper is devoted to the study of relationships between discrete and continuous models financial processes and their probabilistic characteristics. First, a connection is established between the price processes of stocks, hedging portfolio and options in the models conditioned by binomial perturbations and their limit perturbations of the Brownian motion type. Secondly, analogues in the coefficients of stochastic equations with various random processes, continuous and jumpwise, and in the coefficients corresponding deterministic equations for their probabilistic characteristics. Statement of the results on the connections and finding analogies, obtained in this paper, led to the need for an adequate presentation of preliminary information and results from financial mathematics, as well as descriptions of related objects of stochastic analysis. In this paper, partially new and known results are presented in an accessible form for those who are not specialists in financial mathematics and stochastic analysis, and for whom these results are important from the point of view of applications. Specifically, the following sections are presented.

• In one- and n-period binomial models, it is proposed a unified approach to determining on the probability space a risk-neutral measure with which the discounted option price becomes a martingale. The resulting martingale formula for the option price is suitable for numerical simulation. In the following sections, the risk-neutral measures approach is applied to study financial processes in continuous-time models.

• In continuous time, models of the price of shares, hedging portfolios and options are considered in the form of stochastic equations with the Ito integral over Brownian motion and over a compensated Poisson process. The study of the properties of these processes in this section is based on one of the central objects of stochastic analysis — the Ito formula. Special attention is given to the methods of its application.

• The famous Black – Scholes formula is presented, which gives a solution to the partial differential equation for the function $v(t, x)$, which, when $x = S (t)$ is substituted, where $S(t)$ is the stock price at the moment time $t$, gives the price of the option in the model with continuous perturbation by Brownian motion.

• The analogue of the Black – Scholes formula for the case of the model with a jump-like perturbation by the Poisson process is suggested. The derivation of this formula is based on the technique of risk-neutral measures and the independence lemma.

Technological development determines the emergence of highly detailed data in time and space, which expands the possibilities of analysis, allowing us to consider consumer decisions and the competitive behavior of enterprises in all their diversity, taking into account the context of the territory and the characteristics of time periods. Despite the promise of such studies, they are currently limited in the scientific literature. This is due to the range of problems, the solution of which is considered in this paper. The article draws attention to the complexity of the analysis of depersonalized high-frequency data and the possibility of modeling consumption changes in time and space based on them. The features of the new type of data are considered on the example of real depersonalized data received from the fiscal data operator “First OFD” (JSC “Energy Systems and Communications”). It is shown that along with the spectrum of problems inherent in high-frequency data, there are disadvantages associated with the process of generating data on the side of the sellers, which requires a wider use of data mining tools. A series of statistical tests were carried out on the data under consideration, including a Unit-Root Test, test for unobserved individual effects, test for serial correlation and for cross-sectional dependence in panels, etc. The presence of spatial autocorrelation of the data was tested using modified tests of Lagrange multipliers. The tests carried out showed the presence of a consistent correlation and spatial dependence of the data, which determine the expediency of applying the methods of panel and spatial analysis in relation to high-frequency data accumulated by fiscal operators. The constructed models made it possible to substantiate the spatial relationship of sales growth and its dependence on the day of the week. The limitation for increasing the predictive ability of the constructed models and their subsequent complication, due to the inclusion of explanatory factors, was the lack of open access statistics grouped in the required detail in time and space, which determines the relevance of the formation of high-frequency geographically structured data bases.

The mathematical estimation model for employee labor process, built on the basis of Bayesian network is presented in the article. The great attention is given to the estimation of qualitative characteristics of labor product. Usage of described model is supposed in the companies with the management employee workflows system.

This work describes the idea of the system of informational support for the scientific projects and the development of computational task tracking complex. Due to large requirements for computational experiments the problem of presentation of the information about HPC tasks becomes one of the most important. Nonstandard usage of the service desk system as a basis of the computational task tracking and support system can be the solution of this problem. Particular attention is paid to the analysis and the satisfaction of the conflicting requirements to the task tracking complex from the different user groups. Besides the web service kit used for the integration of the task tracking complex and the datacenter environment is considered. This service kit became the main interconnect between the parts of the scientific project support system and also this kit allows to reconfigure the whole system quickly and safely.