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Home » Issues » Archive of Issues » 2015 - 4 issue » pp. 501-516

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Russia Perm
Year
2015
Volume
25
Issue
4
Pages
501-516
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Section Mathematics
Title Stochastic differential equations with random delays in the form of discrete Markov chains
Author(-s) Poloskov I.E.a
Affiliations Perm State National Research Universitya
Abstract The paper provides an overview of the problems that lead to a necessity for analyzing models of linear and nonlinear dynamic systems in the form of stochastic differential equations with random delays of various types as well as some well-known methods for solving these problems. In addition, the author proposes some new approaches to the approximate analysis of linear and nonlinear stochastic dynamic systems. Changes of delays in these systems are governed by discrete Markov chains with continuous time. The proposed techniques for the analysis of systems are based on a combination of the classical steps method, an extension of the state space of a stochastic system under examination, and the method of statistical modeling (Monte Carlo). In this case the techniques allow to simplify the task and to transfer the source equations to systems of stochastic differential equations without delay. Moreover, for the case of linear systems the author has obtained a closed sequence of systems with increasing dimensions of ordinary differential equations satisfied by the functions of conditional expectations and covariances for the state vector. The above scheme is demonstrated by the example of a second-order stochastic system. Changes of the delay in this system are controlled by the Markov chain with five states. All calculations and graphics were performed in the environment of the mathematical package Mathematica by means of a program written in the source language of the package.
Keywords stochastic dynamic system, random delay, modeling, state vector, transition process
UDC 519.21, 004.94
MSC 65C30, 60H35, 68U20
DOI 10.20537/vm150407
Received 20 August 2015
Language Russian
Citation Poloskov I.E. Stochastic differential equations with random delays in the form of discrete Markov chains, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 501-516.
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