Abstract

In many cases, it is difficult to find a solution to a system of difference equations with random structure in a closed form. Thus, a random process, which is the solution to such a system, can be described in another way, for example, by its moments. In this paper, we consider systems of linear difference equations whose coefficients depend on a random Markov or semi-Markov chain with jumps. The moment equations are derived for such a system when the random structure is determined by a Markov chain with jumps. As an example, three processes: Threats to security in cyberspace, radiocarbon dating, and stability of the foreign currency exchange market are modelled by systems of difference equations with random parameters that depend on a semi-Markov or Markov process. The moment equations are used to obtain the conditions under which the processes are stable.

Highlights

  • Many real processes in both research and practice can be well modelled by dynamical systems with random parameters

  • The moment equations for systems of linear and nonlinear differential and difference equations with a random structure determined by a Markov or semi-Markov process are derived in [20,21,22]

  • In [9], moment equations are obtained for systems of difference equations if the random structure is determined by a semi-Markov chain with jumps

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Summary

Introduction

Many real processes in both research and practice can be well modelled by dynamical systems with random parameters. The moment equations for systems of linear and nonlinear differential and difference equations with a random structure determined by a Markov or semi-Markov process are derived in [20,21,22]. In [9], moment equations are obtained for systems of difference equations if the random structure is determined by a semi-Markov chain with jumps. We deal with the derivation of moment equations for systems of difference equations with random coefficients if these coefficients depend on a Markov chain with jumps. S is the state space of all random variables for which there exists a squared first-order moment In such a probability space, we consider a non-stationary system of linear difference equations.

Preliminary Remarks
Markov Chain ξ is Markovian
Markov Chain ξ is Semi-Markovian
Moment Equations for Difference Systems with Random Jumps
Model Problems
Stability of Foreign Currency Exchange Market
Radiocarbon Dating Modelled by a System with Markov Parameters
Conclusions
Full Text
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