Modeling of applied problems by stochastic systems and their analysis using the moment equations
© Diblík et al.; licensee Springer 2013
Received: 1 March 2013
Accepted: 8 May 2013
Published: 29 May 2013
The paper deals with systems of linear differential equations with coefficients depending on the Markov process. Equations for particular density and the moment equations for given systems are derived and used in the investigation of solvability of initial problems and stability. Results are illustrated by examples.
MSC:34K50, 60H10, 60H30, 65C30.
Most of the notable achievements in theoretical economics in the last fifty years were related to finances. The first Nobel Memorial Prize in Economic Sciences was awarded in 1969 jointly to Frisch and Tinbergen ‘for having developed and applied dynamic models for the analysis of economic processes’. Many of their works were devoted to the development of mathematical methods to the analysis of economic processes, including mathematical modeling of financial processes . The second Nobel Memorial Prize in Economic Sciences was awarded in 1970 to Samuelson ‘for the scientific work through which he has developed static and dynamic economic theory and actively contributed to raising the level of analysis in economic science’. In his works he studied the role of expectations in the theory of finance.
The growing importance of finance theory in economics is linked to two trends: still wider use of mathematics in the modeling of economic processes, and using the results of theoretical economics in practice. The both trends have a close relationship to finance. Mathematical modeling assumes the exact determination of the parameters - they usually are expressed in the finances; application of the theory in practice assumes description of the cash flows and the risk of using models.
Significant development of the theory of finance, which includes the theory of corporate finance and the theory of investment, occurred in the twentieth century. Until then, the theory of finance was developed as a theory of state finance, in the twentieth century became the theory of capital markets. The amount of significant works on the theory of finance were written in the years 1950 to 2000. Bachelier, the founder of the modern theory of finance, has merit that the theory of finance received a mathematical basics. He anticipated many of the ideas of the twentieth century in his works: the relationship between random and diffusion processes, Markov processes, the theory of Brownian motion and much more than today lies not only in the investment theory. One of the first models of the offers loan funds was built in the early twentieth century by Fisher. Equations to balance between savings and investments (known as IS-LM model, and Mandella-Fleming model) are the basis of the modern macroeconomics. Its authors Hicks and Mundell are Nobel Prize winners. Mundell, in addition, created the theory of optimum currency areas, which allows to call him the father of the euro. In the theory of financial investment, there is no concept that would be such widely verified and so little credible as ‘efficient markets’. The so-called efficient market hypothesis performs a primary function - to justify the use of probabilistic calculation in the analysis of capital markets. But if markets are ‘nonlinear stochastic dynamical systems’, the use of standard statistical analysis can lead to erroneous results, especially if they are based on the model of random walks.
One of the methods that permit to examine the stability of stochastic systems is a traditional method of Lyapunov functions, which was developed, for example, in the works by Barbashin , Hasminski , Valeev , Zubov  and others.
Investigating the mean stability or mean square stability of solutions of differential equations with random coefficients depending on Markov process is a current problem. The theory of Markov processes was studied in the works by Chung , Davis , Dynkin [8, 9], Kolmogorov , Lèvy , Skorohkod  and others. The use of the theory of Markov processes to the study of various economic processes can be found in the works by Elliot, Kopp , Malliaris, Brock  and Williams .
Dynamic systems considered in the present paper belong to the class of the so-called systems with random states. The works by Artem’ev , Katz, Krasovskii  and others are dedicated to such systems.
We offer a new approach to simulation by creating algorithms for the construction of moment equations and their quantification. The origin of the theory of moment equations and their use in the examination of the stability can be found in the works by Valeev  and his scientific school (e.g., ).
In the present paper we derive the functional equations for particular density functions and the moment equations for the system which are used in the investigation of solvability and mean square stability. There is shown the application of the results to solve various problems of practice.
2 Statement of the problem
with the transition matrix .
Definition 1 The m-dimensional random vector function , the components of which are random variables is called a solution of the initial value problem (1), (2) if satisfies (1) and initial condition (2) in the meaning of strong solution (defined in ) of the initial Cauchy problem.
Our task is to obtain a reliable and simple method for investigating the stability of solutions of this class of systems. To solve this task, we present below the method of moment equations. On a series of examples, we demonstrate that the method is effective and useful.
are called moments of the first or second order of the random variable x respectively. The values and , , are called particular moments of the first or second order respectively.
The in Definition 2 denotes an m-dimensional Euclidian space, functions , are the particular density functions of the random variable x.
Several different stability statements are possible. We here recall mean square stability definition, which is based on that given in .
3 Moment equations for the linear differential equations
Before the initial value problem (1), (2) formulated in the previous section will be investigated, a simpler problem will be studied. First we derive the moment equations in the scalar case of system (1), that is, if instead of the system there is an equation. In the first part of this section, the linear homogenous differential equation, the coefficient of which depends on a random Markov process, with two states only is considered. In the second part, the moment equations are derived for nonhomogenous linear differential equations with q possible states of a random process, on which the coefficients depend.
3.1 Homogenous linear differential equations
Finally, multiplying equations (11), (12) by , and integrating them by parts from −∞ to ∞, in accordance with Definition 2, a system of linear differential equations with constant coefficients (7) can be obtained. □
If we put the obtained expressions into the first equation of (13), then under assumption , we get the first equation of system (7). In the same way, using the second equation of (13), the second equation of system (7) can be constructed.
where a is independent of a random variable . This case corresponds to the moment equations of the zeroth order, i.e., if .
3.2 Nonhomogenous linear differential equation
to system (15).
The system of moment equations (16) can be derived from the last system for particular probability density functions by using the same modifications as in the proof of Theorem 1. □
4 Moment equations for the linear differential system
Now we come back to the initial problem (1), (2) that we have formulated in Section 2. We also suppose that the matrix A and vector B depend on a random Markov process with q possible states, the probabilities of which (3) satisfy the system of linear differential equations (4).
Finally, multiplying equation (21) by x and integrating it by parts on the Euclidean space , in accordance with Definition 2, we obtain a system of linear equations for the particular moments of the first order in the form (18). The particular moments of the second order satisfy the matrix system of differential equations (19) which we get in the same way. The difference is that (21) is multiplied by the matrix , next it is integrated over the Euclidean space . □
The following examples illustrate the use of moment equations for the investigation of stability.
It is easy to see that the real parts of all eigenvalues are negative or equal to zero. Therefore, the solutions of the system of equations (22) are stable in the mean square.
The first author was supported by the Grant No. P201/11/0768 of the Grant Agency of the Czech Republic (GA CR). The third and fourth authors were supported by the Grant No. 081ŽU-4/2011 of the Grant Agency of the Slovak Republic (KEGA).
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