Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2369510.22363/2658-4670-2020-28-1-35-48Modeling and SimulationМатематическое моделированиеResearch ArticleSimulation of non-stationary event flow with a nested stationary componentМоделирование нестационарного потока событий с вложенным стационарным компонентомPleshakovRuslan V.ПлешаковРуслан Владимирович

PhD student

аспирант

ruslanplkv@gmail.comKeldysh Institute of Applied MathematicsИнститут прикладной математики им. М. В. Келдыша РАН15122020281VOL 28, NO1 (2020)ТОМ 28, №1 (2020)354809052020Copyright ©; 2020, Pleshakov R.V.Copyright ©; 2020, Плешаков Р.В.2020Pleshakov R.V.Плешаков Р.В.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/23695

A method for constructing an ensemble of time series trajectories with a nonstationary flow of events and a non-stationary empirical distribution of the values of the observed random variable is described. We consider a special model that is similar in properties to some real processes, such as changes in the price of a financial instrument on the exchange. It is assumed that a random process is represented as an attachment of two processes - stationary and non-stationary. That is, the length of a series of elements in the sequence of the most likely event (the most likely price change in the sequence of transactions) forms a non-stationary time series, and the length of a series of other events is a stationary random process. It is considered that the flow of events is non-stationary Poisson process. A software package that solves the problem of modeling an ensemble of trajectories of an observed random variable is described. Both the values of a random variable and the time of occurrence of the event are modeled. An example of practical application of the model is given.

В статье описан метод построения ансамбля траекторий временных рядов с нестационарным потоком событий и нестационарным эмпирическим распределением значений наблюдаемой случайной величины. Мы рассматриваем специальную модель, которая похожа по свойствам на некоторые реальные процессы, такие как изменения цены финансового инструмента на бирже. Предполагается, что случайный процесс представлен как совокупность двух процессов - стационарного и нестационарного. То есть длина ряда элементов в последовательности наиболее вероятного события (например, наиболее вероятное изменение цены в последовательности транзакций) образует нестационарный временной ряд, а длина ряда других событий является стационарным случайным процессом. Считается, что поток событий является нестационарным пуассоновским процессом. В работе описан программный комплекс, решающий задачу моделирования ансамбля траекторий наблюдаемой случайной величины. Моделируются как значения случайной величины, так и время возникновения события. Приведён пример практического применения модели.

non-stationary time seriesnon-stationary flow of eventsmodeling of an ensemble trajectoriesнестационарные временные рядынестационарный поток событиймоделирование ансамблевых траекторийA. D. Bosov and Y. N. Orlov, “Kinetic and hydrodynamic approach to the non-stationary time series forecasting on the base of Fokker-Planck equation [Kinetiko-gidrodinamicheskiy podkhod k prognozirovaniyu nestatsionarnykh vremennykh ryadov na osnove uravneniya FokkeraPlanka],” in Proceedings of MIPT [Trudy MFTI], 4. 2012, vol. 4, pp. 134- 140, in Russian.Y. N. Orlov and S. L. Fedorov, “Modeling and statistical analysis of functionals set on samples from a non-stationary time series,” in Preprints of IPM im. M. V. Keldysh, 43. 2014, in Russian.Y. N. Orlov, Kinetic methods for studying non-stationary time seriesy [Kineticheskiye metody issledovaniya nestatsionarnykh vremennykh ryadov]. Moscow: MIPT, 2014, in Russian.Y. N. Orlov and S. L. Fedorov, Methods of numerical modeling of nonstationary random walk processes [Metody chislennogo modelirovaniya protsessov nestatsionarnogo sluchaynogo bluzhdaniya]. Moscow: MIPT, 2016, in Russian.Y. N. Orlov and K. P. Osminin, “Sample distribution function construction for non-stationary time-series forecasting [Postroyeniye vyborochnoy funktsii raspredeleniya dlya prognozirovaniya nestatsionarnogo vremennogo ryada],” Mathematical modeling, no. 9, pp. 23-33, 2008, in Russian.D. S. Kirillov, O. V. Korob, N. A. Mitin, Y. N. Orlov, and R. V. Pleshakov, “On the stationary distributions of the Hurst indicator for the non-stationary marked time series [Raspredeleniya pokazatelya Hurst nestatsionarnogo markirovannogo vremennogo ryada],” in Preprints of IPM im. M. V. Keldysh, 11. 2013, in Russian.M. H. Numan Elsheikh, D. O. Ogun, Y. N. Orlov, R. V. Pleshakov, and V. Z. Sakbaev, “Averaging of random semigroups and quantization [Usredneniye sluchaynykh polugrupp i neodnoznachnost’ kvantovaniya gamil’tonovykh sistem],” in Preprints of IPM im. M. V. Keldysh, 19. 2014, in Russian.E. Bacry, S. Delattre, M. Hoffmann, and J. F. Muzy, “Modeling microstructure noise with mutually exciting point processes,” Quantitative Finance, vol. 13, no. 1, pp. 65-77, 2013.G. Bhardwaj and N. R. Swanson, “An empirical investigation of the usefulness of ARFIMA models for predicting macroeconomic and financial time series,” Journal of Econometrics, vol. 131, pp. 539-578, 2006. DOI: 10.1016/j.jeconom.2005.01.016.P. Embrechts, T. Liniger, and L. Lin, “Multivariate Hawkes Processes: an Application to Financial Data,” Journal of Applied Probability, vol. 48A, pp. 367-378, 2011. DOI: 10.1239/jap/1318940477.N. S. Kremer and B. A. Putko, Econometrica [Ekonometrika]. Moscow: UNITY-DANA, 2005, in Russian.D. E. Bestens, V. M. van der Berth, and D. Wood, Neural networks and financial markets: decision-making in trading operations [Neyronnyye seti i finansovyye rynki: prinyatiye resheniy v torgovykh operatsiyakh]. Moscow: TVP, 1998, in Russian.A. Zeifman, A. Korotysheva, K. Kiseleva, V. Korolev, and S. Shorgin, “On the bounds of the rate of convergence and stability for some queueing models [Ob otsenkakh skorosti skhodimosti i ustoychivosti dlya nekotorykh modeley massovogo obsluzhivaniya],” Informatics and Applications, vol. 8, no. 3, pp. 19-27, 2014, in Russian. DOI: 10.14357/ 19922264140303.V. I. Khimenko, “Scatterplots in Analysis of Random Streams of Events [Diagrammy rasseyaniya v analize sluchaynykh potokov sobytiy],” Informatsionno-upravlyayushchiye sistemy, no. 4, pp. 85-93, 2016, in Russian. DOI: 10.15217/issn1684-8853.2016.4.85.M. V. Zhitlukhin, A. A. Muravlyov, and A. N. Shiryaev, “On confidence intervals for Brownian motion changepoint times,” Russian Mathematical Surveys, vol. 71, no. 1, pp. 159-160, 2016. DOI: 10.1070/RM9702.R. V. Pleshakov, “NSTS Software package for modeling non-stationary non-equidistant time series,” 2018.