Stabilization of company’s income modeled by a system of discrete stochastic equations
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The paper deals with a system of difference equations where the coefficients depend on Markov chains. The functional equations for a particular density and the moment equations for the system are derived and used in the investigation of mode stability of company’s income. An application of the results is illustrated by two models.
KeywordsMarkov Chain Difference Equation Probability Space Moment Equation Stochastic Dynamic System
As early as the beginning of the 20th century, it was discovered that, even in sequences of equally distributed random variables, a marginal distribution may quite naturally occur other than normal. Most of the underlying laws behind such occurrences can only be understood on the basis of the theory of Markov chains. A random process is called a Markov chain if the way it passes from one state to another depends only on the current state regardless of the states preceding it. The set of states is finite or countable. Using stochastic approach, one can investigate a number of aspects relating to a variety of phenomena in finance and economics.
The paper constructs mathematical models of complex economic systems working in conditions of uncertainty. Incomplete or distorted information, too few observations, structure changing over time, stochastic nature of the impact of the external environment, these and other factors generate conditions of uncertainty for the system. The problems encountered in creating and solving mathematical models are caused by the fact that the input-output data type contains nonlinearities and perturbations. The paper focuses on the construction of moment equations to determine the mean value of the guaranteed profit of a company. The theoretical results are applied to two models of the profit of a company.
2 Moment equations
(for definitions see, e.g., ). The space Ω is called a sample space, ℱ is the set of all possible events (a σ-algebra) that may occur to moment t, and ℙ is some probability measure on Ω. Such a random space plays a fundamental role in the construction of models in economics, finance etc.
, is a transition matrix and .
and assume that there exist inverse matrices .
The state m-dimensional column vector-function , , is called a solution of system (1) within the meaning of a strong solution if it satisfies (1) with initial condition (2) .
Our task is to derive the moment equations of system (1) to be used for determining the mode stability of the income of a company.
We define the moments of the first and second order of a solution , , of (1) before deriving the moment equations.
is called moment of the first order for a solution , , of (1). The values , , are called particular moments of the first order.
is called moment of the second order for a solution , , of (1). The values , , are called particular moments of the second order.
3 Model problem 1
The stochastic equation (11) describes the graph of the income of a company with the initial value of income . Here the inhomogeneity represents the conditions in which a company works. For example, the value , , means the transition from one state of company activities, say, a crisis, to another state , say, the post-crisis situation.
number of employees,
and so on.
By using an iterative method, after a finite number of steps, the mean value of the income can be obtained. This is approximately 4.33 with a variance of 1.11.
Remark 1 The processes described by system (11) can be controlled introducing a control function .
which minimizes the quality criterion (15) with respect to equation (11), is called the optimal control.
We denote , , , . If we use the method developed in , we obtain the following.
with known matrices , .
The theorem gives the necessary conditions for an optimal solution of system (14) making it possible to transform the problem of synthesis of the optimal control to the problem of determining the matrices in system (17).
4 Model problem 2
where o is an m-dimensional zero vector describing the behavior of the random value , which stands for the company income at a moment n.
From the first equation, it is easy to see that , which means that the company will be left without the expected net profit in the above conditions.
5 Concluding remarks
While the theoretical results as such are original and valuable too, the present models confirm the practical importance of the methods devised to study discrete systems with random parameters. The construction of moment equations is a classical method and the results shown imply that its use in investigating different types of equations brings elegant results in modeling problems in the theory of finance.
The first author is supported by the Operational Programme Research and Development for Innovations, No. CZ.1.05/2.1.00/03.0097 (AdMaS). The third author is supported by the Grant KEGA č. 004ŽU-4/2014 of the Slovak Grant Agency.
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