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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 4, Pages 501–516 (Mi vuu504)  

MATHEMATICS

Stochastic differential equations with random delays in the form of discrete Markov chains

I. E. Poloskov

Higher Mathematics Department, Perm State University, ul. Bukireva, 15, Perm, 614990, Russia

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.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-96019
Ministry of Education and Science of the Russian Federation 2014/153


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Document Type: Article
UDC: 519.21+004.94
MSC: 65C30, 60H35, 68U20
Received: 20.08.2015

Citation: I. E. Poloskov, “Stochastic differential equations with random delays in the form of discrete Markov chains”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:4 (2015), 501–516

Citation in format AMSBIB
\Bibitem{Pol15}
\by I.~E.~Poloskov
\paper Stochastic differential equations with random delays in the form of discrete Markov chains
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2015
\vol 25
\issue 4
\pages 501--516
\mathnet{http://mi.mathnet.ru/vuu504}
\elib{http://elibrary.ru/item.asp?id=25109971}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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