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🗀 Archive 2010-2015

Department of Mathematical Modelling

- In exchange model, we found necessary and sufficient conditions for markets cleaning under given ownership and patterns of consumer demand structure. The theorem is proven describing the structure of ownership vectors in which for a given demand structure the complete cleaning of markets takes place. The equilibrium price vector is specified that provides such a cleaning. The theorem is proven, giving conditions of degeneracy multiplicity for equilibrium price vector. The concept of equivalent distribution of property is introduced, necessary and sufficient conditions are found for the existence of equilibrium price vector, and the structure of aggregate supply is described. A number of theorems is proven determining the aggregate vector of customer satisfaction levels. An adapted aggregated description of the economic system is proposed on the basis of input-output balance and is applied to the analysis of Great Britain, Germany, Russia and Greece economies.
*N.S. Gonchar, W.H. Kozyrski, A.S. Zhokhin*

- In the frame of the theory of economic equilibrium, the mechanism of the recession phenomenon is proposed. The structure of economic equilibrium states in the economic system depending on the structure of property, supply and demand is described. The theorem of one to one correspondence between the state of economy equilibrium under that total demand on the part of goods is strictly less then supply and multiplicity of degeneration of equilibrium state is proved. It is shown that in such a state of economic equilibrium the sale of surplus products does not occur. Provided that such a set of goods is critical devaluation and falling asset values take place. The criteria of such criticality is introduced. The proposed model is applied to the analysis of the Ukrainian economy on the basis of input-output tables for 2011. It is revealed that in 2011 Ukraine's economy was in a deep recession.
*M.S. Gonchar, A.S. Zhokhin, W.G. Kozyrski*

- Previously proposed model for operating of a bank whose capital evolution dynamics is described by homogeneous Markov chain is generalized on the case of randomreceipt of deposits and capital backuping to pay on external obligations and investing in risk and unrisk assets. Investigational integral equations for bankruptcy probability were investigated. Strategies of the favourable investing in risk and unrisk assets are described under which a bank having initial can operate for a long time with a small baqnkruptcy probability.
*N.S. Gonchar* - The optional decomposition theorem significant in financial theory is generalized on the case of arbitrary structure of martingale in relation to the family of equivalent measures. Application is given to calculate the hedging cost on incomplete markets.
*N.S. Gonchar* - In the dynamic exchange model, equilibrium states are classified in some of which the economy can be arbitrarily long not being reformed. Conditions of equilibrium states branching and degeneration multipleness are established. The greater is degeneration multipleness of equilibrium state, the more unstable is the financial system of economy. On this ground, the algorithm of recession displaying on the early stage is proposed.
*N.S. Gonchar, A. S. Zhokhin*