Bipolar behavior of submodular, law-invariant capacities

In the case of a submodular, law-invariant capacity, we provide an entirely elementary proof of a result of Marinacci [M. Marinacci, Upper probabilities and additivity, Sankhyā Ser. A 61 1999, no. 3, 358–361]. As a corollary, we also show that the anticore of a continuous submodular, law-invariant nonatomic capacity has a dichotomous nature: either it is one-dimensional or it is infinite-dimensional. The results have implications for the use of such capacities in financial and economic applications.

Brauer's Fixed Point Theorem

The primary goal of the paper is to deliver a simple proof of equivalence between Brouwer’s fixed point theorem and the existence of equilibrium in a simple exchange model with monotonic consumers. To achieve this end, we discuss some equivalent formulations of Brouwer’s theorem and prove additional ones, that are ’approximating’ in character or seem to be better suited for economic applications than the standard results.

A Full Description of ω-Limit Sets of Cournot Maps Having Non-Empty Interior and Some Economic Applications

Given a continuous Cournot map F(x,y)=(f2(y),f1(x)) defined from I2=[0,1]×[0,1] into itself, we give a full description of its ω-limit sets with non-empty interior. Additionally, we present some partial results for the empty interior case. The distribution of the ω-limits with non-empty interior gives information about the dynamics and the possible outputs of each firm in a Cournot model. We present some economic models to illustrate, with examples, the type of ω-limits that appear.

Substantive-methodological concept of the formation of a special course on Markov processes applications

The concept of the formation of a special course on economic applications of Markov processes for students of economic universities is proposed. Methodical recommendations are given for teaching such a discipline, introducing students to the applied aspects of the mathematical theory of Markov processes. On the basis of many years of generalized experience in teaching the discipline in REU, the structure of the course construction, its relationship with other disciplines is described. The goals and objectives of the discipline are formulated. A detailed description of the topics under consideration is given and concrete examples of characteristic problems having both a theoretical and a computational focus are proposed. An algorithm for the implementation and control of students' independent work is proposed. Samples of individual analytical work are shown. Criteria for assessing the knowledge of students are formulated, a list of theoretical questions of the final certification is given. The practical orientation of illustrative examples even with serious theoretical foundations allows students to be interested in a wide range of areas of study. The developed approach allows expanding the professional and general cultural competencies of students aimed at the application of mathematical methods of analysis in economic applications, and also forms a scientific worldview and helps establish causal relationships in applied research. The practice of teaching a discipline in REU in the form of an elective discipline has shown its relevance by students and has consistently demonstrated interest in the tasks of this direction. Computer support for training significantly complements the methods of independent work of students, diversifies it, develops the skills of modern work with data, saves time and makes you interested in quickly getting results.