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It is substantiated that the determination of any characteristics of random variables can be reduced to computer calculations using the corresponding numerically determined integral laws of their distribution, and therefore the implementation of the stage of "equalization" of histograms with classical functions, which are proposed in classical mathematical statistics to approximate the random variable distribution laws, becomes unnecessary. A method of synthesis of equivalent models of the random variables distribution laws is proposed, in which instead of the procedure of "equalization" of histograms, a much simpler procedure of their approximation to numerically given integral distribution functions is implemented, which, after interpolation and differentiation, are transformed into equivalent models of differential laws of distribution in the event that the researcher has these laws for further studies are needed in a differential form. It is shown that the use of equivalent models of the random variable distribution laws when calculating their numerical characteristics gives results that are closer to the characteristics of the implementations of random variables generating these laws compared to the results that occur after the "equalization" of histograms by known classical functions using the Pirson procedure The effectiveness of the proposed method is demonstrated on the example of building a database suitable for identifying the laws of the distribution of grades of students who receive these grades while studying in a small-format group. The starting conditions for this example are taken from a specific previous publication of the authors, in which they created a database of student evaluations in a group of 10 people during their study: "Methodology and organization of scientific research in the field of information technologies." [ABSTRACT FROM AUTHOR]

When managing production or commodity stocks, two main questions arise: when to replenish the stock and what should be its optimal size. The purpose of this study is to build a probabilistic model, which can be proposed as a new inventory model, with the help of which the relationships between the period factors between the purchase of parts and their shelf life, which affect inventory management, are established. Research methods are based on the approach using continuous distribution laws. The size of the reserve of parts is calculated depending on the established risk factor. Using the statistical method, point estimates were found for the studied parameters: average and root mean square deviation. A histogram of relative frequencies between the dates of two consecutive purchases is constructed. Critical areas for the studied parameters are illustrated. The value of the difference in days between the purchase of parts and the amount of the purchase of parts, which correspond to the normal laws of the distribution of random variables with the appropriate parameters, as well as the critical values of the need for parts in the production process, were calculated. The size of the parts reserve was found, which corresponds to the normal distribution law, depending on the established risk factor. For different values of this coefficient, the value of the difference in days between purchases of parts, the amount of purchases and the reserve of parts, which correspond to the distributions of random values, as well as the critical value of the need for parts in the production process to avoid production downtime are given. Using the central limit theorem, it is shown that the purchase volume of parts and the volume of used parts are normally distributed. Taking into account the degree of uncertainty associated with the structure of demand and the time of use of stocks at the enterprise, the authors chose probabilistic models that make it possible to flexibly change simulated demand and take this into account in forecasting. The research concluded that the probabilistic approach is the basis for forecasting inventory management at the enterprise, which takes into account the risks associated with determining the optimal need for raw materials at the enterprise. [ABSTRACT FROM AUTHOR]

The problems of financial assets portfolio beta-coefficient diversification and evaluation play an important role among aspects of risk management in the financial institutions activities. The paper is dedicated to statistical properties of the sample estimator of the financial assets minimum Value-at-Risk portfolio beta-coefficient. We assume that the weights of the portfolio benchmark are constant and the vector of portfolio assets returns is multivariate normally distributed. We find an analytical expression for calculating the beta-coefficient if the parameters of the portfolio assets returns distribution vector are known. In practice, these parameters are usually unknown, so we make use of distribution parameters sample estimators. By replacing the unknown parameters in the expression for calculating the beta-coefficient with their sample estimators, we obtain a sample estimator of the beta-coefficient, which is a random variable. We provide an asymptotic distribution of this sample estimator. We investigate the rate of empirical distributions convergence to asymptotic ones using simulation modeling with 100,000 repetitions. We take the corresponding sample estimators as the precise value of the distribution parameters of the portfolio assets returns vector. These estimators are obtained from the data on daily stock prices of companies included in the Dow Jones list for the period from 01.07.2020 to 11.01.2022. Six portfolios with different dimensions ({5, 10, 15, 20, 25, 30} of the first companies from the Dow Jones list sorted in alphabetically order) are considered. We select the equally weighted portfolio for the benchmark portfolio. It is noted that the beta-coefficient sample estimator is significantly biased and the bias increases with the number of assets in portfolio. We propose a corrected estimator whose bias is significantly smaller and does not depend on the number of assets in portfolio. It is concluded that the obtained asymptotic results can be used in practice for modeling the behavior of the sample estimator of the beta-coefficient in the presence of moderate dimension historical data samples (500-1000 observations). [ABSTRACT FROM AUTHOR]

It is studied the new approaches for grounding and making optimal investment decisions under uncertainty and risk with the use of economic-mathematical methods. The volume of future earnings and the amount of investment for their implementation are compared for the efficiency evaluation of investment projects. Evaluating the solution implementation effectiveness is suggested to perform using the «probability tree» and the meanings of numerical characteristics of the studied random variable. Taking into account the use of the discount rate value and the future payments (receipts) value, and hence the possible scenarios of projects for different economic situations and their corresponding productive probabilities, present values of future earnings are calculated by using the built-in NPV program EXCEL. The mathematical modeling of the optimization decision-making process with taking into account the stochastic conditions of their implementation and risks not to receive predictable results is developed and practically implemented. It is highlighted that the construction of the decision tree allows managers to monitor the projects performance and to reanalyze projects in each point with the flow of the new information. [ABSTRACT FROM AUTHOR]

The path integral method is applied to investigation of time dependence of interest rate of Merton and Vasicek stochastic models. The path integral is based on Wiener integral by means of variable substitution based on stochastic equations of the models. The results were obtained for term structure of interest rates in Merton and Vasicek models. The path integral method is applied to the investigation of the time dependence of the interest rate in the Merton and Vasicek stochastic models. The path integral is based on the Wiener integral by means of variable substitution based on stochastic equations of the models. The results were obtained for the term structure of interest rates in the Merton and Vasicek models. As is known, the drawback of those models is that they allow for negative values of the interest rate. Hence, one should introduce boundary conditions, which limit the stochastic variable domain to positive values. During the construction of the path integral for respective mean values, one needs use a transition probability that satisfies those boundary conditions. However, the construction of such path integral for average values, which contains an exponent dependent on some functional of integration variables, is in general a very complicated task. In the work, an easier approach is used that consists in limiting the integral domain in path integrals for the Merton and Vasicek models. In particular, in path integrals the integral domain is limited using condition ⟆t t0 r(τ )dτ > 0, which is given by the Heaviside step function θ(x). It is shown that limiting the integral domain in such a way allows us to get rid of negative values of the term structure of interest rates. The path integrals for the limited integral domain are calculated using the transformation of the Heaviside step function into a complex plane. During the calculation of mean values, a normalization of the measure is applied to a selected domain. As a result, in the models where the interest rate takes only positive values, equations for the term structure of interest rates were obtained. Positive values of interest rates are visually demonstrated by means of asymptotic analysis of their time dependence. The obtained functional forms of the interest rates dependencies can be used to calibrate models with the help of respective optimization procedures, which minimize deviation of theoretical and experimental values. Expressions found in the work allow one to accomplish the above procedures using numeric methods. [ABSTRACT FROM AUTHOR]

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The paper describes spatial features identification of the population trading service development in Kharkiv region with the use of the univariate statistical analysis. Kharkiv region as one of the most developed regions of Ukraine is characterized by a high degree of monocentric regional development. The univariate statistical analysis was chosen in order to assess this degree. Univariate statistical analysis is a type of statistical analysis used to characterize the distribution of a single variable. The following parameters of descriptive statistics were selected as characteristics of the values distribution: median, mode, mean, variance, standard deviation, coefficient of variation, coefficient of skewness, and coefficient of kurtosis. Such indicators are characterized in the paper: a number of population, a number of trade enterprises, a number of restaurant facilities in the settlements of the region. The normal distribution of a random variable was taken for the ideal distribution of the indicators values. The distribution is normal if the median, mode and mean are equal to each other, the coefficients of skewness and kurtosis should equal 0. According to the results of a univariate statistical analysis, it was determined that the distribution of the above mentioned indicators don't correspond to a normal distribution, it is extremely uneven, which confirms the monocentricity of regional development in Kharkiv region. Distribution of indicators of the number of population, trade enterprises, and restaurant facilities is similar to each other, which indicates conditionality of the existing network of trade enterprises, restaurant facilities to the population in settlements. The analysis of the descriptive statistics parameters in the context of the newly formed administrative districts of Kharkiv region was carried out and it was found that any district isn't characterized by the normal distribution of the above indicators, however, the most distant from the normal distribution is Kharkiv district, the most closed are Bogodukhivsky, Chuhuivsky, Krasnohradsky districts. [ABSTRACT FROM AUTHOR]

This paper is devoted to the finding of necessary and sufficient conditions for the almost sure convergence of the random series whose terms are elements of one-dimensional zero-mean Gaussian Markov sequences. Mainly in the paper the series whose terms are elements of weakly dependent and strongly dependent Gaussian Markov sequences are considered. Criterions which provide almost sure convergence of the series whose terms are elements of weakly dependent and strongly dependent Gaussian Markov sequences respectively are the main results of this work. These criterions are formulated in terms of correlation characteristics of Gaussian Markov sequences and are of a simple kind. Obtained results also enable to find criterions for the almost sure convergence of the series whose terms are elements of weakly dependent and strongly dependent Gaussian Markov sequences with weighted coefficients. Moreover, obtained results automatically provide sufficient conditions for the almost sure convergence of a series whose terms are elements of some regressive sequences generated by independent sub-Gaussian random variables with uniformly bounded standards. [ABSTRACT FROM AUTHOR]

The extended stochastic integral with respect to a Lévy process and the corresponding Hida stochastic derivative have many applications in the stochastic analysis, in particular, in the theory of stochastic differential and integral equations. One of main lacks of these operators, if one considers them as linear operators on the space of square integrable random variables (L²), consists in their unboundedness. Moreover, the domain of the extended stochastic integral (and of the conjugated to this integral operator -- the stochastic derivative) depends on an interval of integration. As a result, there are problems with some applications of the above-mentioned operators. Therefore an important problem is a modification of definitions of the extended stochastic integral and the Hida stochastic derivative that gives us the possibility to define these operators as linear bounded (i.e., continuous) ones. In this paper, using the theory of Hilbert equipments, we introduce and study the extended stochastic integral with respect to a Lévy process and the Hida stochastic derivative as linear continuous operators on a suitable rigging of (L²). This gives a possibility to extend an area of applications of these operators. [ABSTRACT FROM AUTHOR]