Про один з методiв побудови моделi строго φ-субгауссового узагальненого дробового броунiвського руху

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = exp{|x|} − |x| − 1, x ∈ R. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment. Pages of the article in the issue: 18 - 25 Language of the article: UkrainianIn the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = exp{|x|} − |x| − 1, x ∈ R. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.  Pages of the article in the issue: 18 - 25 Language of the article: Ukrainia

Про оцінку ймовірності переповнення буферу для мереж зв’язку

In recent years, a large number of research of telecommunications traffic have been conducted. It was found that traffic has a number of specific properties that distinguish it from ordinary traffic. Namely: it has the properties of self-similarity, multifractality, long-term dependence and distribution of the amount of load coming from one source. At present, many other models of traffic with self-similarity properties and so on have been built in other researched works on this topic. Such models are investigated in this paper, which considers traffic in telecommunications networks, the probability of overflow traffic buffer. Statistical models are built to analyze traffic in telecommunications networks, in particular to research the probability of buffer overflow for communication networks. The article presents the results of the analysis of processes in telecommunication networks, in particular traffic; research of possibilities of representation of real processes in the form of random processes on the basis of use of statistical simulation model; the necessary mathematical and statistical models are selected and analyzed; software-implemented models using the Matlab environment; visual graphs for comparison of the received data are given; the analysis of the received models is carried out. Pages of the article in the issue: 64 - 69 Language of the article: UkrainianУ даній статті досліджується трафік у телекомунікаційних мережах, ймовірність переповнення буфера трафіку. Для цього у роботі проаналізовано процеси у телекомунікаційних мережах, зокрема залежності трафіку; проведено дослідження можливостей представлення реальних процесів у вигляді випадкових процесів на основі використання статистичного імітаційного моделювання; підібрано та проаналізовано необхідні математичні, статистичні моделі; програмно реалізовано дані моделі за допомогою середовища Matlab; побудовано необхідні графіки для порівняння отриманих даних; проведено аналіз отриманих моделей

Мiжнародна наукова конференцiя “Сучасна стохастика: теорiя та застосування. V” (MSTA-V). 1-4 червня 2021

Pages of the article in the issue: 9 Language of the article: UkrainianPages of the article in the issue: 9 Language of the article: Ukrainia

Sample continuity with probability one for the estimator of impulse response function

The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output crosscorrelogram is taken as an estimator of the response function. The conditions on sample continuousness with probability one for impulse response function are investigated.Key words: impulse response function, linear time-invariant system (LTI), Gaussian process, crosscorrelogram, sample continuity.Pages of the article in the issue: 96 - 102Language of the article: Ukrainia

Convergence rate for the estimation of impulse response function in the space of continuous functions

The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of the distribution of the supremum of the estimation error is found that gives a convergence rate of estimator to impulse response function.Key words: impulse response function, linear time-invariant system (LTI), Gaussian process, crosscorrelogram.Pages of the article in the issue: 30 - 36Language of the article: Ukrainia

On the convergence rate for the estimation of impulse response function in the space Lp(T)

The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of the distribution of the estimation error is found that gives a convergence rate of estimator to impulse response function in the space Lp(T).Key words: impulse response function, linear time-invariant system (LTI), Gaussian process, cross-correlogram. Pages of the article in the issue: 36-41Language of the article: UkrainianThe problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of  the distribution of the estimation error is found that gives a convergence rate of estimator to impulse response function in the space Lp(T).Key words: impulse response function, linear time-invariant system (LTI), Gaussian process, crosscorrelogram.Pages of the article in the issue: 36 - 41Language of the article: Ukrainia

Analytical properties of sample paths of some stochastic processes

The study of the analytical properties of random processes and their functionals, without a doubt, was and remains the relevant topic of the theory of random processes. The first result from which the study of the local properties of random processes began is Kolmogorov’s theorem on sample continuity with probability one. The classic result for Gaussian random processes is Dudley’s theorem. This paper is devoted to the study of local properties of sample paths of random processes that can be represented as a sum of squares of Gaussian random processes. Such processes are called square-Gaussian. We investigate the sufficient conditions of sample continuity with probability 1 for square-Gaussian processes based on the convergence of entropy Dudley type integrals. The estimation of the distribution of the continuity module is studied for square-Gaussian random processes. It is considered in detail an example with an estimator (correlogram) of the covariance function of a Gaussian stationary random process. The conditions on continuity of correlogram’s trajectories with probability one are found and the distribution of the continuity module is also estimated.Key words: Gaussian process, square-Gaussian process, correlogram, sample continuity.Pages of the article in the issue: 11 - 15Language of the article: Ukrainian

The development of software for simulation of random processes with a given accuracy and reliability

Today, the theory of random processes and time series prediction is widely used in various fields of science, not only in natural fields. That is why one of the urgent problems is to build a mathematical model of a random process and study its properties. Numerical modeling tasks become especially important due to the powerful capabilities of computer technology, which allows you to create software modeling tools and predict the behavior of a random process. There are different methods of modeling random processes and fields. In some works related to the modeling of random processes, the issues of accuracy and reliability have not been studied. In [1, 2, 3] for various stochastic processes and fields this problem was investigated. In this paper the question of accuracy and reliability of the constructed model is considered. This means that we first build the model and then test it using some adequacy tests with known accuracy and reliability. We also find the estimators of the model parameters using methods of moments. All theoretical results are applied to develop software for model construction of stochastic processes.Key words: simulation, accuracy, reliability, estimation.Pages of the article in the issue: 83 - 87Language of the article: Ukrainia

Моделювання технічних резервів страхової компанії

In the modern rapidly evolving society, the science and the business are facing new needs and challenges constantly. The insurance industry and its mathematical foundation, the actuarial science, are not exceptions. Currently, the greatest challenge that the insurance system has to cope with is the issue of the new international financial standard that affects the calculation of reserves among other things. So far, insurers have mainly used common classical deterministic methods. However, the new standard emphasizes the necessity of the realistic prognosis that is best achieved with stochastic modelling tools since deterministic models do not represent the uncertainty and the random nature of future possible losses. This article considers the advantage of using stochastic modelling for reserve calculation in comparison to the deterministic approach.The article consists of five sections.In the first section, we briefly present the technique that lies in the basis of technical reserves calculation.The second section is devoted to such deterministic methods of reserve calculation as the Bornhuetter-Ferguson method and the chain-ladder method.In the third section, we consider modifications of two stochastic models – the Mack method and the bootstrapping technique.The fourth section considers the adjustment of reserves for the time value of money and inflation.In the fifth section, the results of modelling in the programming language R are presented.Key words: reserve calculation, Mack method, bootstrapping, Bornhuetter-Ferguson method, chainladder method.Pages of the article in the issue: 46 - 51Language of the article: EnglishУ сучасному суспільстві, що розвивається швидкими темпами, наука та бізнес постійно стикаються з новими потребами та викликами. Страхова галузь та її теоретична основа, актуарна математика, не є винятками. У даний час найважливішим питанням, над яким працює страхова система, є підготовка до нового міжнародного фінансового стандарту, положення якого, зокрема, впливають на розрахунок резервів. Досі страховики в основному застосовували поширені класичні детерміністичні методи. Однак новий стандарт підкреслює необхідність реалістичного прогнозу, який найкраще досягається за допомогою інструментів стохастичного моделювання, оскільки детерміністичні моделі не враховують невизначеність та випадкову природу майбутніх можливих втрат. У цій статті розглядається перевага використання стохастичного моделювання для розрахунку резервів порівняно з детерміністичним підходом.Стаття складається з п'яти розділів.У першому розділі коротко описується методика, яка лежить в основі розрахунку технічних резервів.Другий розділ присвячений таким детерміністичним методам розрахунку резервів, як метод Борнхуеттера-Фергюсона та метод ланцюгових сходів.У третьому розділі ми розглянемо модифікації двох стохастичних моделей – методу Мака та бутстрепу.У четвертому розділі розглядається коригування резервів з врахуванням вартості грошей у часі та інфляції.У п’ятому розділі представлені результати моделювання мовою програмування R

Professor Yu.V. Kozachenko (01.12.1940 - 05.05.2020) - prominent scientist and teacher

Pages of the article in the issue: 9 - 29Language of the article: Ukrainia