UDC 517.934

ON THE STABILITY OF DIFFERENCE STOCHASTIC SYSTEMS UNDER CONSTANTLY ACTING PERTURBATIONS

T. AKHMETKALIEV

In the article, the theory of stability in the first approximation and under constantly acting perturbations, known for differential equations [1—13], is extended to stochastic difference systems subjected to the action of a Markov random influence. The investigation is based on Lyapunov’s second method, modernized for the problems under consideration.

§ 1. Consider the system described by the stochastic vector difference equation

\[ x_{k+1}-x_k=[A(\eta_{k+1})x_k+Q(x_k,\eta_{k+1})]h, \tag{1.1} \]

where \(x\) is the \(n\)-dimensional vector of phase coordinates of the system \(\{x^{(1)},\ldots,x^{(n)}\}\); \(h\) is the step of the equation; \(k\) is the step number; \(\eta\in Y\{\eta^{(1)},\ldots,\eta^{(r)},\ldots\}\) is a homogeneous Markov chain with a countable number of states \(\eta^{(j)}\), where the probability \(p_{ij}=p_{ij}(h)\) of transition of the values \(\eta^{(i)}\) to \(\eta^{(j)}\) in one step satisfies the condition [14]

\[ p_{ij}= \begin{cases} \alpha_{ij}h, & i\ne j,\\ 1-\alpha_{ij}h, & i=j \end{cases} \left( \alpha_{ij}=\mathrm{const},\quad \alpha_{jj}=\sum_{j\ne i}\alpha_{ij} \right); \]

\(A(\eta)\) is the matrix \(\{a_{im}(\eta)\}_{1}^{n}\), whose elements \(a_{im}(\eta)\) are bounded functions of \(\eta\).

Let the domain of definition of system (1.1) be the space

\[ -\infty<x^{(j)}<+\infty,\quad \eta\in Y,\quad k\ge k_0\ge0. \tag{1.2} \]

We shall assume that the nonlinear functions \(Q^{(i)}(x,\eta)\) in system (1.1) are continuous in the domain (1.2) and satisfy Lipschitz conditions with respect to the variables \(x^{(s)}\) for all \(\eta\).

We shall also assume that \(Q^{(i)}(0,\eta)=0\) and, consequently, for \(Q^{(i)}(x,\eta)\) the inequalities

\[ |Q^{(i)}(x,\eta)|\le \gamma\|x\|, \tag{1.3} \]

hold, where \(\gamma=\mathrm{const}>0\);

\[ \|x\|=\left(\sum_{i=1}^{n}(x^{(i)})^2\right)^{1/2}. \]

Along with system (1.1), consider its first approximation

\[ x_{k+1}-x_k=A(\eta_{k+1})x_kh. \tag{1.4} \]

We shall denote by \(P[d_1/d_2]\) the conditional probability; by \(M[\psi/Y]\) we shall denote the conditional mathematical expectation of the random variable \(\psi\) [15]; and by \(\{x_k(x_{k_0},\eta_{k_0}),\eta_k(\eta_{k_0})\}\) the random solution of system (1.1).

By a random solution we shall mean a Markov random vector-function whose realizations \(\{x_{k,\omega}(x_{k_0},\eta_{k_0}),\eta_{k,\omega}(\eta_{k_0})\}\) satisfy system (1.1) for the corresponding realizations \(\eta_{k,\omega}\).

Let us formulate definitions generalizing to the stochastic system of difference equations (1.1) the concept of Lyapunov stability.

Definition 1.1. The unperturbed motion \(x=0\) will be called strongly asymptotically stable in the large in probability if, for any bounded domain \(\|x_{k_0}\|<H_0\) and for any numbers \(\mu_1>0\), \(0<\mu_2<1\), \(0<q<1\), one can find a bounded domain \(\|x\|<H_1\) and a number \(K>0\) such that the conditions

\[ P\left[\|x_k(x_{k_0},\eta_{k_0})\|<H_1 \ \text{for all } k\geq k_0/\|x_{k_0}\|<H_0,\ \eta_{k_0}\in Y\right]>1-\mu_2, \]

\[ P\left[\|x_k(x_{k_0},\eta_{k_0})\|<\mu_1 \ \text{for all } k>k_0+K/\|x_{k_0}\|<H_0,\ \eta_{k_0}\in Y\right]>1-q. \]

are satisfied.

Definition 1.2. The unperturbed motion \(x=0\) will be called stable in the mean if, for every positive number \(\varepsilon\), one can choose a positive number \(\delta\) such that any solution with initial data satisfying the condition \(\|x_{k_0}\|\leq \delta\), \(\eta_{k_0}\in Y\), \(k_0\geq 0\), satisfies the inequality

\[ M\left[\|x_k(x_{k_0},\eta_{k_0})\|^2/x_{k_0},\eta_{k_0}\right]<\varepsilon \quad \text{for all } k\geq k_0. \]

Definition 1.3. The unperturbed motion \(x=0\) will be called asymptotically stable in the mean if, in addition to the fulfillment of the conditions of Definition 1.2, the condition

\[ \lim_{k\to\infty} M\left[\|x_k(x_{k_0},\eta_{k_0})\|^2/x_{k_0},\eta_{k_0}\right]=0 \]

is satisfied for all solutions with initial data \(\|x_{k_0}\|<H_0\), \(\eta_{k_0}\in Y\), where \(H_0\) is some constant.

Definition 1.4. The unperturbed motion \(x=0\) will be called exponentially stable in the mean if, for any solution of system (1.1) with arbitrary initial data \(x_{k_0}\), \(\eta_{k_0}\), there exists a pair of constants \(B>0\), \(\beta>0\) such that the inequality

\[ M\left[\|x_k(x_{k_0},\eta_{k_0})\|^2/x_{k_0},\eta_{k_0}\right]\leq Be^{-\beta(k-k_0)h}\|x_{k_0}\|^2 \tag{1.5} \]

holds for all \(k\geq k_0\).

Now one can formulate a theorem corresponding to the theorems on stability in the first approximation for ordinary [1—7] and stochastic [12] systems of differential equations.

Theorem 1.1. Let the zero solution of system (1.4) satisfy inequality (1.5).

One can indicate a number \(\gamma>0\) such that the solution \(x=0\) of system (1.1) will be strongly asymptotically stable in the large in probability, and also exponentially stable in the mean, if only in the domain (1.2) the functions \(Q^{(i)}(x,\eta)\) satisfy conditions (1.3).

Proof. According to Lemma 3.2 [16], condition (1.5) is equivalent to the existence of a positive-definite quadratic form

\[ v(x,\eta)=\sum_{i,s=1}^{n} d_{is}(\eta)x^{(i)}x^{(s)}, \]
which, by virtue of system (1.4), satisfies the following estimates:
\[ c_1\|x\|^2 \leqslant v(x,\eta)\leqslant c_2\|x\|^2,\qquad \frac{\Delta M[v]}{h}\leqslant -c_3\|x\|^2 \tag{1.6} \]
(\(c_1,c_2,c_3\) are certain positive constants).

We shall show that the form \(v(x,\eta)\) can be used to study the stability of system (1.1), provided only that the functions \(Q^{(i)}(x,\eta)\) satisfy conditions (1.3). Denote the first averaged differences \(\dfrac{\Delta M[v]}{h}\) of the function \(v(x,\eta)\), by virtue of systems (1.1) and (1.4), respectively, by
\[ \left[\frac{\Delta M[v]}{h}\right]_{(1.1)},\qquad \left[\frac{\Delta M[v]}{h}\right]_{(1.4)}. \]
By virtue of system (1.1), the first difference is computed by the formula [16]
\[ \left[\frac{\Delta M[v]}{h}\right]_{(1.1)} = [v(x_{k+1},\eta^{(j)})-v(x_k,\eta^{(j)})]\frac{(1-\alpha_{ij}h)}{h} + \]
\[ +\sum_{r\ne j}^{\infty}\alpha_{jr}[v(x_{k+1},\eta^{(r)})-v(x_k,\eta^{(j)})] = \]
\[ = \left[ \sum_{i=1}^{n} \frac{\partial v(x_k,\eta^{(j)})}{\partial x^{(i)}} \left[ \sum_{m=1}^{n} a_{im}(\eta^{(j)})x_k^{(m)}+Q^{(i)}(x_k,\eta^{(j)}) \right]h \right. \]
\[ \left. + \sum_{i,s=1}^{n} \frac{\partial^2 v(x_k,\eta^{(j)})}{\partial x^{(i)}\partial x^{(s)}} \left[ \sum_{m=1}^{n}a_{im}(\eta^{(j)})x_k^{(m)}+Q^{(i)}(x_k,\eta^{(j)}) \right]\times \right. \]
\[ \left. \times \left[ \sum_{p=1}^{n}a_{sp}(\eta^{(j)})x_k^{(p)}+Q^{(s)}(x_k,\eta^{(j)}) \right]\frac{h^2}{2} \right]\frac{(1-\alpha_{ij}h)}{h} + \]
\[ + \sum_{r\ne j}^{\infty}\alpha_{jr} \left[ v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})-v(x_k,\eta^{(j)}) \right] + \]
\[ + \sum_{r\ne j}^{\infty}\alpha_{jr} \sum_{i=1}^{n} \frac{\partial v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})}{\partial x^{(i)}} Q^{(i)}(x_k,\eta^{(r)})h + \]
\[ + \sum_{r\ne j}^{\infty}\alpha_{jr} \sum_{i,s=1}^{n} \frac{\partial^2 v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})}{\partial x^{(i)}\partial x^{(s)}} Q^{(i)}(x_k,\eta^{(r)})\times \]
\[ \times Q^{(s)}(x_k,\eta^{(r)})\frac{h^2}{2} \leqslant \left[\frac{\Delta M[v]}{h}\right]_{(1.4)} + \sum_{i=1}^{n} \left| \frac{\partial v(x_k,\eta^{(j)})}{\partial x^{(i)}} \right| \times \]

\[ \begin{aligned} &\times \left| Q^{(i)}(x_k,\eta^{(j)}) \right| (1-\alpha_{jj}h) +\sum_{i,s=1}^{n}\left| \frac{\partial^2 v(x_k,\eta^{(j)})}{\partial x^{(i)}\partial x^{(s)}} \right| \times \\ &\times \left[ \sum_{m=1}^{n} |a_{im}(\eta^{(j)})| \cdot |x_k^{(m)}| \cdot |Q^{(s)}(x_k,\eta^{(j)})|+ \right.\\ &\qquad\left. +\sum_{p=1}^{n} |a_{sp}(\eta^{(j)})| \cdot |x_k^{(p)}| \cdot |Q^{(i)}(x_k,\eta^{(j)})|+ \right.\\ &\qquad\left. + |Q^{(i)}(x_k,\eta^{(j)})| \cdot |Q^{(s)}(x_k,\eta^{(j)})| \right]\frac{(1-\alpha_{jj}h)h}{2}+\\ &\quad+\sum_{r\ne j}^{\infty}\alpha_{jr}h \sum_{i=1}^{n}\left| \frac{\partial v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})}{\partial x^{(i)}} \right| \cdot \left| Q^{(i)}(x_k,\eta^{(r)}) \right|+\\ &\quad+\frac{h}{2}\sum_{r\ne j}^{\infty}\alpha_{jr}h \sum_{i,s=1}^{n}\left| \frac{\partial^2 v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})}{\partial x^{(i)}\partial x^{(s)}} \right| \times\\ &\qquad\qquad\times |Q^{(i)}(x_k,\eta^{(r)})|\cdot |Q^{(s)}(x_k,\eta^{(r)})|. \end{aligned} \tag{1.7} \]

From the theory of quadratic forms we have

\[ \left(\sum_{z=1}^{n}|x^{(z)}|\right)^2 \leq n\sum_{z=1}^{n}(x^{(z)})^2 = n\|x\|^2. \tag{1.8} \]

Now, using inequalities (1.3), (1.6), (1.8) and taking into account that

\[ \sum_{r\ne j}^{\infty}\alpha_{jr}h<1,\qquad 1-\alpha_{jj}h<1, \]

from (1.7) we obtain the estimate

\[ \left[\frac{\Delta M[v]}{h}\right]_{(1.1)} \leq [-c_3+N_1\gamma^2+N_2\gamma]\|x\|^2, \]

where \(N_1, N_2\) are constants.

If the inequality

\[ N_1\gamma^2+N_2\gamma<c_3, \tag{1.9} \]

is satisfied, then the quantity \(\left[\dfrac{\Delta M[v]}{h}\right]_{(1.1)}\) is a negative-definite function. Solving inequality (1.9), one can choose \(\gamma^0\) so that for \(\gamma\leq \gamma^0\) this inequality is satisfied. Then the function \(v(x,\eta)\) will satisfy all the conditions of the theorems on strong asymptotic stability in the large in probability [17] and on exponential stability in the mean [16]. This also proves here our Theorem 1.1.

The following assertion is also valid. Suppose that the solution \(x=0\) of system (1.4) is asymptotically stable in the mean. One can specify a positive number \(\gamma\) such that the solution of the full system (1.1) will be strongly asymptotically stable in the large in probability, provided only that conditions (1.3) are satisfied.

The validity of this assertion follows from Lemma 3.1 of [16] and from the theorem 1.1 proved above.

§ 2. We now consider a stochastic system of difference equations of perturbed motion of the form

\[ x_{k+1}-x_k=\bigl[A(\eta_{k+1})x_k+Q(x_k,(k+1)h,\eta_{k+1})\bigr]h, \tag{2.1} \]

where \(A(\eta)\) is the same matrix as in system (1.4), and the functions \(Q^{(i)}\), in contrast to (1.1), are assumed to depend explicitly on \(k\).

In addition, we shall assume that for the functions \(Q^{(i)}\) the conditions [4, 10, 13]

\[ |Q^{(i)}(x,(k+1)h,\eta)|\leq \varphi_{k+1}\|x\| \tag{2.2} \]

are fulfilled. Here \(\varphi\) is a certain uniformly bounded function of \(k\), satisfying the estimates: for all \(\Theta>0\) the inequalities

\[ \frac{1}{K_0}\sum_{k=\Theta}^{\Theta+K_0}\varphi_k<\gamma_1,\qquad \frac{1}{K_0}\sum_{k=\Theta}^{\Theta+K_0}\varphi_k^2<\gamma_2 \quad(\gamma_1,\gamma_2=\operatorname{const}>0), \tag{2.3} \]

hold, where \(K_0\) is some constant natural number.

Conditions (2.2), (2.3) impose restrictions on the mean value of the absolute magnitude of the perturbations \(Q^{(i)}\) on an interval of length \(K_0h\), but do not require these perturbations to be small at each instant \(k\), as was the case for the restriction (1.3) in § 1.

We shall prove the following theorem.

Theorem 2.1. Suppose that for every solution of the system of the first approximation (1.4) with sufficiently small step \(h>0\), for arbitrary initial data from the domain (1.2), the inequality (1.5) holds.

Then one can specify positive numbers \(\gamma_1\) and \(\gamma_2\) such that the solution \(x=0\) of system (2.1) will be strongly asymptotically stable in the large in probability, and also exponentially stable in the mean, for any choice of functions \(Q^{(i)}\) satisfying in the domain (1.2) conditions (2.2) and (2.3).

Proof. By the condition of the theorem, the solution \(x=0\) of system (1.4) is exponentially stable in the mean. Then the equations (1.4) in the domain (1.2) admit a quadratic-form Lyapunov function

\[ v(x,\eta)=\sum_{i,s=1}^{n}d_{is}(\eta)x^{(i)}x^{(s)}, \]

satisfying the estimates (1.6).

To investigate system (2.1), as a Lyapunov function we construct the function \(W(x,kh,\eta)=v(x,\eta)e^{\nu_k}\) and indicate conditions which the quantity \(\nu\) must satisfy so that, along the trajectories of system (2.1), the function \(W\) be positive definite, admit an infinitely small upper and an infinitely large lower bound, and have a negative-definite first averaged difference

\[ \frac{\Delta M[W]}{h}. \]

Let us compute the first difference \(\dfrac{\Delta M[W]}{h}\) along the trajectories of system (2.1):

\[ \left[\frac{\Delta M[W]}{h}\right]_{(2.1)} = \left[ \frac{M[W(x_{k+1},(k+1)h,\eta_{k+1})/x_k,\eta_k]-W(x_k,kh,\eta_k)}{h} \right]_{(2.1)} = \]

\[ = \frac{M[v(x_{k+1},\eta_{k+1})/x_k,\eta_k]\bigl(e^{\nu_{k+1}}-e^{\nu_k}\bigr)}{h} + \]

\[ + \frac{e^{\nu_k}\bigl[M\{v(x_{k+1},\eta_{k+1})/x_k,\eta_k\}-v(x_k,\eta_k)\bigr]}{h} = \]

\[ = e^{\nu_k}\left[\frac{\Delta M[v]}{h}\right]_{(2.1)} + \frac{e^{\nu_k}\bigl[e^{\nu_{k+1}-\nu_k}-1\bigr]M\{v(x_{k+1},\eta_{k+1})/x_k,\eta_k\}}{h}. \tag{2.4} \]

On the basis of formula [16] for the first difference \(\left[\dfrac{\Delta M[v]}{h}\right]_{(2.1)}\), we have

\[ \begin{aligned} \left[\frac{\Delta M[v]}{h}\right]_{(2.1)} ={}& \Biggl[\sum_{i=1}^{n}\frac{\partial v(x_k,\eta^{(j)})}{\partial x^{(i)}}\,h \left[ \sum_{m=1}^{n} a_{im}(\eta^{(j)})x_k^{(m)} + Q^{(i)}(x_k,(k+1)h,\eta^{(j)}) \right] \\ &+ \sum_{i,s=1}^{n} \frac{\partial^2 v(x_k,\eta^{(j)})}{\partial x^{(i)}\partial x^{(s)}} \left[ \sum_{m=1}^{n} a_{im}(\eta^{(j)})x_k^{(m)} + Q^{(i)}(x_k,(k+1)h,\eta^{(j)}) \right] \\ &\qquad\qquad\times \left[ \sum_{p=1}^{n} a_{sp}(\eta^{(j)})x_k^{(p)} + Q^{(s)}(x_k,(k+1)h,\eta^{(j)}) \right]\frac{h^2}{2} \Biggr]\frac{(1-a_{jj}h)}{h} \\ &+ \sum_{r\ne j}^{\infty} a_{jr} \bigl[v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})-v(x_k,\eta^{(j)})\bigr] \\ &+ \sum_{r\ne j}^{\infty} a_{jr}\sum_{i=1}^{n} \frac{\partial v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})}{\partial x^{(i)}} \,hQ^{(i)}(x_k,(k+1)h,\eta^{(r)}) \\ &+ \sum_{r\ne j}^{\infty} a_{jr}\sum_{i,s=1}^{n} \frac{\partial^2 v(x_k+A(\eta^{(r)})x_kh,\eta^{(r)})}{\partial x^{(i)}\partial x^{(s)}} Q^{(i)}(x_k,(k+1)h,\eta^{(r)}) \\ &\qquad\qquad\times Q^{(s)}(x_k,(k+1)h,\eta^{(r)})\frac{h^2}{2}. \end{aligned} \]

Now, substituting the value of \(\left[\dfrac{\Delta M[v]}{h}\right]_{(2.1)}\) into expression (2.4) and estimating the resulting expression by virtue of (1.6), (2.2), we obtain the following estimate:

\[ \left[\frac{\Delta M[W]}{h}\right]_{(2.1)} \leq W\left[ \frac{(e^{\nu_{k+1}-\nu_k}-1)M\{v(x_{k+1},\eta_{k+1})/x_k,\eta_k\}}{h\,v(x_k,\eta_k)} - \frac{c_3}{c_2} + N_1\varphi_{k+1}^{2} + N_2\varphi_{k+1} \right], \tag{2.5} \]

where \(N_1\) and \(N_2\) are constants. It can be verified that for sufficiently small \(h\) the quantity

\[ \rho=\frac{M\{v(x_{k+1},\eta_{k+1})/x_k,\eta_k\}}{v(x_k,\eta_k)} \]

is close to unity.

Now, using this circumstance and relying on (2.5), it is verified, similarly to how this was done in [4], that one can specify a sequence of numbers \(|\nu_k|\leq N_3\) such that the inequality will be satisfied at all times—

then

\[ \frac{\Delta M[W]}{h}\leq -N_4W, \]

where \(N_4>0\)—const., and, consequently, for these \(\nu_k\) the function \(W\) will satisfy all the conditions of Theorem 3.2 [17]. This also proves our Theorem 2.1.

Let us note, moreover, that for any solution of system (2.1) the inequality

\[ M\left[\left\|x_k(x_{k_0},\eta_{k_0})\right\|^2/x_{k_0},\eta_{k_0}\right]\leq B_1\left\|x_{k_0}\right\|^2 e^{-\beta_1(k-k_0)h} \quad \text{for all } \quad k\geq k_0 \]

will also hold.

§ 3. Consider a system described by a stochastic vector difference equation of the form

\[ x_{k+1}-x_k=\left[A(\eta_{k+1})x_k+bR_{k+1}\right]h, \tag{3.1} \]

where \(b\) is an \(n\)-dimensional constant vector \(\{b^{(1)},\ldots,b^{(n)}\}\); \(R=\{R^{(1)},\ldots,R^{(n)}\}\) is a random vector function describing constantly acting perturbations.

We shall assume that the function \(R_{k+1}\) is a sequence of random impulses

\[ R_{k+1}=l_{k+1} \tag{3.2} \]

(here \(l_{k+1}\) are mutually independent random variables having one and the same distribution function \(F(l)\)). It is also assumed that the variables \(\eta\) and \(l_{k+1}\) are independent, and that the variables \(l_{k+1}\) have zero mean value and finite variances \(M[l_{k+1}^2]=\tau\). Under these conditions the functions \(R^{(i)}\) are a random set of impulses.

We introduce the concept of stability in probability under constantly acting perturbations.

Definition 3.1. The solution \(x=0\) of system (1.4) is called stable in probability under random constantly acting perturbations if, for any positive numbers \(\varepsilon,\mu_3\), there exist numbers \(\gamma_3>0,\gamma_4>0\) such that every solution of the system of equations (3.1), for each \(k\geq k_0\), satisfies the inequality

\[ P\left[\left\|x_k(x_{k_0},\eta_{k_0})\right\|<\varepsilon\right]>1-\mu_3, \tag{3.3} \]

provided only that the initial values satisfy the conditions \(\|x_{k_0}\|<\gamma_3,\ \eta_{k_0}\in Y\) and \(\tau\leq \gamma_4\).

Now, relying on the theorem on the existence of Lyapunov functions [16], we shall show that for stochastic difference equations the following theorem holds, corresponding to theorems from [3—4, 8—9, 10—12].

Theorem 3.1. If the solution \(x=0\) of the linear system (1.4) with sufficiently small step \(h\) is asymptotically stable in the mean, then stability in probability under random constantly acting perturbations holds.

Proof. According to Lemmas 3.1 and 3.2 of [16], the Lyapunov function may be chosen in the form of a positive-definite quadratic form

\[ v(x,\eta)=\sum_{i,s=1}^{n} d_{is}(\eta)x^{(i)}x^{(s)}, \]

which satisfies the condition

\[ \frac{\Delta M[v]}{h} = - \|x\|^2 . \tag{3.4} \]

Taking (3.4) into account, we compute the first difference of the function \(v(x,\eta)\) along the trajectories of system (3.1)

\[ \left[\frac{\Delta M[v]}{h}\right]_{(3.1)} = -\|x\|^2+\frac{1}{h}\{M[v[(x_{k+1})_{(3.1)},\eta_{k+1}]- \]

\[ - v[(x_{k+1})_{(1.4)},\eta_{k+1}]/(x_k)_{(1.4)}=(x_k)_{(3.1)},\eta_k\}, \tag{3.5} \]

where \((x_k)_{(3.1)}\), \((x_k)_{(1.4)}\) denote the trajectories of systems (3.1) and (1.4), respectively.

Let us compute the second term on the right-hand side of (3.5). Denote

\[ \Delta x^{(i)}=(x_{k+1}^{(i)})_{(3.1)}-(x_{k+1}^{(i)})_{(1.4)}=b^{(i)}l_{k+1}. \tag{3.6} \]

To compute \(M[\Delta v]\), we use the expansion of the quadratic form \(\Delta v\). Then

\[ M[\Delta v]= M\left[ \sum_{i=1}^{n}\frac{\partial v}{\partial x^{(i)}}\Delta x^{(i)} + \sum_{i,s=1}^{n} \frac{\partial^2 v}{\partial x^{(i)}\partial x^{(s)}} \Delta x^{(i)}\Delta x^{(s)} \right]. \tag{3.7} \]

The first term on the right-hand side of (3.7) is equal to zero in consequence of (3.6), since \(M[l_{k+1}]=0\). Therefore,

\[ M[\Delta v] = M\left[ \sum_{i,s=1}^{n} d_{is}(\eta)\Delta x^{(i)}\Delta x^{(s)} \right] = \]

\[ = M\left\{ \sum_{i,s=1}^{n} d_{is}(\eta)b^{(i)}b^{(s)} \right\}M[l_{k+1}^{2}], \tag{3.8} \]

since the random variables \(l_{k+1}\) and \(\eta\) are assumed independent.

Now, substituting expression (3.8) into (3.5), we obtain

\[ \left[\frac{\Delta M[v]}{h}\right]_{(3.1)} = -\|x\|^2+\frac{1}{h}\lambda\tau \quad (\lambda=\mathrm{const}). \tag{3.9} \]

We note that the second term on the right-hand side of (3.9) is positive, since \(v(x,\eta)\) is a positive-definite quadratic form.

Now, taking (1.6) into account, one can write the inequality

\[ \left[\frac{\Delta M[v]}{h}\right]_{(3.1)} \leq -c_4v+\frac{\lambda\tau}{h}, \tag{3.10} \]

where \(c_4\) is a positive number.

Construct the function

\[ v_k=M[V(x_k,\eta_k)/x_{k_0},\eta_{k_0}]. \]

From (3.10), by the formula of iterated mathematical expectations, we have

\[ v_{k+1}-v_k\leq -hc_4v_k+\lambda\tau. \tag{3.11} \]

It follows from (3.11) that, for sufficiently small \(\tau\), the quantity \(v_k\), if it is large, decreases as \(k\) increases until it becomes less than some number \(\varepsilon_1 > 0\), which can be chosen in advance to be arbitrarily small. Further, for small \(\tau\) and a sufficiently small initial value \(v_{k_0}\), the quantity \(v_k\) for \(k \gg k_0\) will all the time satisfy the inequality \(v_k < \varepsilon_1\). Hence, on the basis of (1.6) and Chebyshev’s inequality, we are convinced of the validity of inequality (3.3), which proves the theorem.

I consider it my duty to express my gratitude to N. N. Krasovskii for valuable advice and guidance.

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Received by the editors
July 28, 1965

Ural State University
named after A. M. Gorky