Let (\mathrm{\Omega},\mathcal{F},F,\mathbb{P}) be a filtered probability space (stochastic basis) consisting of a probability space (\mathrm{\Omega},\mathcal{F},\mathbb{P}) and a filtration

F=\{{\mathcal{F}}_{t},\mathrm{\forall}t\ge 0\}\subset \mathcal{F}

(for definitions see, *e.g.*, [9]). 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 (\mathrm{\Omega},\mathcal{F},\mathbb{P}), we consider an initial problem formulated for the stochastic dynamic system with random coefficients in the form

{x}_{n+1}=A({\xi}_{n+1},{\xi}_{n}){x}_{n}+B({\xi}_{n+1},{\xi}_{n}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,

(1)

{x}_{0}=\phi (\omega ),

(2)

where *A* is an m\times m matrix with random elements, *B* is an *m*-dimensional column vector with random elements, {\xi}_{n} is the Markov chain of a finite number of states {\theta}_{1},{\theta}_{2},\dots ,{\theta}_{q} with probabilities {p}_{k}(n)=P\{{\xi}_{n}={\theta}_{k}\}, k=1,2,\dots ,q, n=1,2,\dots , that satisfy the system of difference equations

p(n+1)=\mathrm{\Pi}p(n),

(3)

p(n)={({p}_{1}(n),{p}_{2}(n),\dots ,{p}_{k}(n))}^{T}, \mathrm{\Pi}={({\pi}_{ks})}_{k,s=1}^{q} is a q\times q transition matrix and \phi :\mathrm{\Omega}\to {\mathbb{R}}^{m}.

If the random variable {\xi}_{n+1} is in state {\theta}_{k}, k=1,2,\dots ,q, and the random variable {\xi}_{n} is in state {\theta}_{s}, s=1,2,\dots ,q, we denote

{A}_{ks}=A({\theta}_{k},{\theta}_{s}),\phantom{\rule{2em}{0ex}}{B}_{ks}=B({\theta}_{k},{\theta}_{s}),\phantom{\rule{1em}{0ex}}k,s=1,2,\dots ,q,

and assume that there exist inverse matrices {A}_{ks}^{-1}.

The state *m*-dimensional column vector-function {x}_{n}, n=1,2,\dots , is called a solution of system (1) within the meaning of a strong solution if it satisfies (1) with initial condition (2) [10].

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 {x}_{n}, n=1,2,\dots , of (1) before deriving the moment equations.

In the sequel, {\mathbb{E}}_{m} denotes an *m*-dimensional Euclidean space, *m*-dimensional row vector-functions {f}_{k}(n,z), n=1,2,\dots , k=1,2,\dots ,q, z\in {\mathbb{E}}_{m} are the particular probability density functions of {x}_{n}, n=1,2,\dots , determined by the formula (see in [11])

{\int}_{{\mathbb{E}}_{m}}{f}_{k}(n,z)\phantom{\rule{0.2em}{0ex}}dz=P\{{x}_{n}\in {\mathbb{E}}_{m},{\xi}_{n}={\theta}_{k}\},

(4)

and they satisfy the following equations:

{f}_{k}(n+1,z)=\sum _{s=1}^{q}{\pi}_{ks}{f}_{s}(n,{A}_{ks}^{-1}(z-{B}_{ks}))det{A}_{ks}^{-1}.

(5)

**Definition 1** The vector function

{E}^{(1)}\{{x}_{n}\}=\sum _{k=1}^{q}{E}_{k}^{(1)}\{{x}_{n}\},

where

{E}_{k}^{(1)}\{{x}_{n}\}={\int}_{{\mathbb{E}}_{m}}z{f}_{k}(n,z)\phantom{\rule{0.2em}{0ex}}dz,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,q,

(6)

is called moment of the first order for a solution {x}_{n}, n=1,2,\dots , of (1). The values {E}_{k}^{(1)}\{{x}_{n}\}, k=1,2,\dots ,q, are called particular moments of the first order.

**Definition 2** The matrix function

{E}^{(2)}\{{x}_{n}\}=\sum _{k=1}^{q}{E}_{k}^{(2)}\{{x}_{n}\},

where

{E}_{k}^{(2)}\{{x}_{n}\}={\int}_{{\mathbb{E}}_{m}}z{z}^{\ast}{f}_{k}(n,z)\phantom{\rule{0.2em}{0ex}}dz,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,q,

(7)

is called moment of the second order for a solution {x}_{n}, n=1,2,\dots , of (1). The values {E}_{k}^{(2)}\{{x}_{n}\}, k=1,2,\dots ,q, are called particular moments of the second order.

**Theorem 1** *Systems of moment equations of the first or second orders for a solution* {x}_{n}, n=1,2,\dots , *of* (1) *are of the form*

{E}_{k}^{(1)}\{{x}_{n+1}\}=\sum _{s=1}^{q}{\pi}_{ks}({A}_{ks}{E}_{k}^{(1)}\{{x}_{n}\}+{B}_{ks}{p}_{s}(n)),

(8)

{E}_{k}^{(2)}\{{x}_{n+1}\}=\sum _{s=1}^{q}{\pi}_{ks}({A}_{ks}{E}_{k}^{(2)}\{{x}_{n}\}{A}_{ks}^{\ast}+{A}_{ks}{E}_{s}^{(1)}{B}_{ks}^{\ast}+{B}_{ks}{E}_{s}^{(1)}{A}_{ks}^{\ast}+{B}_{ks}{B}_{ks}^{\ast}{p}_{s}(n)),

(9)

*respectively*.

*Proof* Multiplying equation (5) by *z* and integrating them on the Euclidean space {\mathbb{E}}_{m}, we obtain the system

{\int}_{{\mathbb{E}}_{m}}z{f}_{k}(n+1,z)\phantom{\rule{0.2em}{0ex}}dz=\sum _{s=1}^{q}{\pi}_{ks}{\int}_{{\mathbb{E}}_{m}}z{f}_{s}(n,{A}_{ks}^{-1}(z-{B}_{ks}))det{A}_{ks}^{-1}\phantom{\rule{0.2em}{0ex}}dz.

(10)

Using the substitution {Y}_{ks}={A}_{ks}^{-1}z, integrating by parts, in regard to {\int}_{{\mathbb{E}}_{m}}{f}_{k}(n+1,z)\phantom{\rule{0.2em}{0ex}}dz={p}_{k}(n), we get, as in the proof of Theorem 2 in [2], 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 z{z}^{\ast} and integrated by parts on the Euclidean space {\mathbb{E}}_{m}. □