The Volatility Index (VIX) is a real-time index that has been used as the first measure to quantify market expectations for volatility, which affects the financial market as a main actor of the overall economy that is sensitive to the environmental and social aspects of investors and companies. The VIX is calculated using option prices for the S&P 500 Index (SPX) and is expressed as a percentage. Taking into account that VIX only shows the implicit volatility of the S&P 500 for the next 30 days, the authors develop a model for a near-optimal state trying to avoid uncertainty and insufficient accuracy. The researchers are trying to make a contribution to the theory of socially responsible portfolio management. The developed approach allows potential investments to make decisions regarding such important topics as ethical investing, performance analysis, as well as sustainable investment strategies. The approach of this research allows to use deep probabilistic convolutional neural networks based on conditional variance as a linear function of errors with the aim of estimating and predicting the VIX. For this purpose, the use of technical indicators and economic indexes such as Chicago Board Options Exchange (CBOE) VIX and S&P 500 is considered. The results of estimating and predicting the VIX with the proposed method indicate high precision and create a certainty in modeling to achieve the goals.

Globalization has given a powerful impetus to the development of international commercial activity and logistics management systems taking full advantage of cross-border networking. The solution lies at the intersection of information technologies, technical means of machine-to-machine (M2M) interaction, mobile high-speed networks, geolocation, cloud services, and a number of international standards. The current trend towards creating digital logistics platforms has set a number of serious challenges for developers. The most important requirement is the condition of sustainability of the obtained solutions with respect to disturbances in the conditions of logistics activities caused not only by market uncertainty but also by a whole set of unfavorable factors accompanying the transportation process. Within the framework of the presented research, the problem of obtaining the conditions for the stability of solutions obtained on the basis of mathematical models is set. At the same time, the processes of transferring not only discrete but also continuous material flows through complex structured networks are taken into account. This study contains the results of the analysis of the stability of solutions of differential systems of various types that simulate the transfer processes in network media. Initial boundary value problems for evolutionary equations and differential-difference systems are relevant in logistics, both for the discrete transportation of a wide range of goods and for the quasi-continuous transportation of, for example, liquid hydrocarbons. The criterion for the work of a logistics operator is the integral functional. For the mathematical description of the transport process of continuous and discrete media, a wide class of integrable functions are used, which adequately describe the transport of media with a complex internal rheological structure.

The development of modern logistics systems requires state-of-the-art solutions for simultaneously achieving sustainable development goals and ambitious business targets. On this issue, the challenge of implementing digital technologies in social life, in particular in smart cities, deserves special attention. The diversified application of Industry 4.0 doctrine and digital shadow penetration to all fields of socioeconomic systems highlights the gaps in the design, control, and efficiency assessment of digital tools in the logistics of smart cities. Another challenge concerns the need to consider environmental, social, and governance (ESG) principles amid the deployment of harmonic digital tools within urban territories. All these issues require a complex methodological approach toward understanding the role of IT in the modern economy through an ESG prism. The article contains a contemporary literature review on the related topic and the conceptual framework of city logistics digitalization under ESG perspectives and constraints. The mathematical model proposed by the researchers enables a multidimensional design of digital solution applications within smart city logistics performance. The designed discrete stochastic model is eligible for scaling toward and the further development of variables other than the key ones mentioned above. The mathematical formalization of the proposed model considers the distribution of the limited budgeting of administrative branches within city logistics, highlighting the research relevance in connection with the ESG principles.

The paper presents the results of the developed algorithms aimed at optimizing management decision-making by the administration of megalopolises. A mathematical model is obtained within the concept of digital economy. The regulatory action of dispositive decisions is aimed at business entities whose activities are externalized while consuming energy resources. Since any resources are used unevenly throughout the year, the authors apply the methods of the theory of optimal decisions. The criterion is the functional reflection of the balance between the maximum profit, the comfort of living conditions, and the environmental conditions. The results obtained make it possible to take administrative decisions in an optimal way, which reduces the negative effects of externalities and results in the most efficient use of energy resources.

Nowadays, digitizing the data stream is enabling the evolution of logistics processes. The data-driven network logistics processes are evaluated by leading economic indicators. The purpose of this study is to further develop the algorithmic foundations of economic and mathematical modeling of logistics networks based on advanced technologies that the digitalization process provides. Methods of mathematical modeling of flows of various resources in logistic networks, the structure of which is presented in the form of a graph, are used in the work. The economic benefit lies in the implementation of the planning concept based on leading indicators. As a result, using a set of formalisms, a mathematical model has been developed, which will make it possible to search for optimal solutions based on the criterion of economic efficiency. The results obtained will find application not only in transport problems. Network logistics includes the transportation of liquid and gaseous materials through pipelines, energy through electrical networks. A universal mathematical model will allow the results to be applied in many industries and economic activities when it is required to make economically sound decisions based on a stream of digital data coming in real time.

The new kind of practical issues was born due to modern trend on globalization and transition to the concept of industry 4.0. These are the processes in commercial networks representing the movement of goods and finance, which have specific economic indicators; migration of populations and labor resources; transference of one-parameter continuous media (diffusion effect). The issues with higher level of complexity occur during the research of multi-phase environments dynamics in network-like media and mathematical modeling of such processes. The authors divide these two problems and offer the formalized description for the laminar processes and dynamic convective processes with feedback. There was developed an approach to solve the problems of optimal control for these processes. The research is based on methods for analyzing systems of partial differential equations with distributed parameters on a graph or network-like domain.