Modelling a Disintegrating Economy by Markov Processes

V.A. Dikarev, A.E. Veprik, and V.I. Zabelin
Kharkov Technical University of Radioelectronics, Ukraine

Abstract

Possibility of control of single economic structure disintegration to not connected or weakly connected economies, when the main parameters of them are equal to any pre-assigned values at the moments, near or equal to the moment of disintegration is investigated in this report. Disintegration of economies is studied by means of finite non-homogeneous Markov processes with continious time. Disinegration is assumed to occur during finite time interval . The analysis is made by Kolmogorov equations with infinitesimal matrix . The state j, (j=1,...,n) of studied process is assumed to be one of the possible states of investigated economy. It can be described by main economic indices. Elements of matric describe interaction economic indices dynamics.

Let and be diagonal blocks of matrix . They correspond to disintegrating economy. Two cases are considered: when elements of not belonging to and tend to zero when or are small in the vicinity of . These are cases of complete break and reduction of economical contacts. Components of vector p(s) are probabilities of process being in state j at moment s. Let some set of values , j=1,...,n, , , determines state probabilities of (state of disinegrating economy) at the moment of stabilization. It means, that tend to when s tend to , , j=1,...,n or when s is enough close to and is small. Let both conditions be accomplished provided by any initial distribution at . In mentioned cases moment will be named as a point of strict stabilization or point of rough stabilization ( -stabilization). Cases , , for strict and rough stabilizations are investigated. In all cases matrices and were determined to provide stabilization at moment :

Realization of stabilization processes in economy causes great expenses. Matrix , stabilizing the process at moment , can de determined not in unique way. Expenses for stabilization can be estimated by functional, describing expenses for building of matrix . Expenses for tend to zero. The strict stabilization is impossible if expenses are limited with the exception of some special cases.

Then the problem of rough stabilization on finite time interval [a,b] is investigated. Let , , be continuous functions defined on [a,b]. It was determined, that there is continuous on [a,b] matrix , which describe the process state probabilities of which satisfay to the following conditions:

Computer models of non-homogeneous Markov process were designed on base of Kolmogorov equations. These models were created for systems with numbers of states from 2 till 300. Some computer modelling experiments were carried out. We have found, that under certain conditions the process of stabilization or rough stabilization was taking place at certain moments. Also it was confirmed, that the nature of splitting of these processes depends on sort of decomposition of their infinitesimal matrices to non-interacting or almost non-interacting blocks. Computer modelling experiments on rough stabilization for finite time interval were also carried out.