Let be a filtered probability space (stochastic basis) consisting of a probability space and a filtration
(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.
On the probability space , we consider an initial problem formulated for the stochastic dynamic system with random coefficients in the form
where A is an matrix with random elements, B is an m-dimensional column vector with random elements, is the Markov chain of a finite number of states with probabilities , , , that satisfy the system of difference equations
, is a transition matrix and .
If the random variable is in state , , and the random variable is in state , , we denote
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.
In the sequel, denotes an m-dimensional Euclidean space, m-dimensional row vector-functions , , , are the particular probability density functions of , , determined by the formula (see in )
and they satisfy the following equations:
Definition 1 The vector function
is called moment of the first order for a solution , , of (1). The values , , are called particular moments of the first order.
Definition 2 The matrix function
is called moment of the second order for a solution , , of (1). The values , , are called particular moments of the second order.
Theorem 1 Systems of moment equations of the first or second orders for a solution , , of (1) are of the form
Proof Multiplying equation (5) by z and integrating them on the Euclidean space , we obtain the system
Using the substitution , integrating by parts, in regard to , we get, as in the proof of Theorem 2 in , systems of moment equations (8).
In the same way, the system of moment equations (9) can be derived. This means that equation (5) is multiplied by and integrated by parts on the Euclidean space . □