Double Merging of the Phase Space for Stochastic Differential Equations with Small Additions in Poisson Approximation Co
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DOUBLE MERGING OF THE PHASE SPACE FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH SMALL ADDITIONS IN POISSON APPROXIMATION CONDITIONS I. V. Samoilenko1† and A. V. Nikitin1‡
UDC 519.21+62
Abstract. Double merging of phase space for the stochastic evolutionary system is performed. The case is considered where system’s perturbations are determined by the impulse process at the Poisson approximation scheme. The limiting process under such conditions has two components: deterministic shift and Poisson jump addition. Keywords: stochastic evolutionary system, double merging of phase space, Poisson approximation scheme. INTRODUCTION In many cases, difficulties emerge during the analysis of complex systems, which mean substantial complication of system’s phase space. This often makes practically impossible to visualize the model. An important problem of the modern systems theory is development of mathematically substantiated methods of creating simplified models whose analysis does not involve considerable difficulties and whose respective characteristics can be taken as respective characteristics of real models. The ideas of analyzing the properties of complex systems based on the analysis of properties of their parts with further passage to the general system underlie many methods of systems analysis. Korolyuk and Turbin [1, 7] were the first to propose and describe the algorithm of phase merging of system states. Analysis of a merged system is considerably simplified; however, at the same time, in case of successful decomposition of the phase space, main characteristics of the simplified system can fairly present characteristics of the original one. In turn, closeness of the real and merged systems means closeness of the global characteristics, which are defined on increasing time intervals. The possibility of creating the $ hierarchy of merged systems S$ , S$ , K is an important property of phase merging algorithms. Random evolution in the form of a differential equation with stochastic terms is used to describe a wide class of natural processes in many scientific fields. Analysis of the behavior of such evolutionary systems in a random medium is an extremely important case. These systems are analyzed in many studies by A. V. Skorokhod, M. I. Gikhman, M. M. Bogolyubov, et al. A detailed bibliography can be found, for example, in Korolyuk’s monographs [2, 3, 4]. Special attention should be paid to [5, 6, 8, 9], where the approaches applied in this paper, in particular to the analysis of asymptotic properties of evolutionary systems under Poisson approximation are established. In the present paper, we will analyze the case where system’s perturbations are defined by a jump process in the Poisson approximation scheme. First of all, we will consider double merging of the state space of these evolutionary models. 1
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2019, pp. 108–116. Original article submitted M
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