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A novel general stability analysis scheme based on a non-Lyapunov framework is explored. Several easy-to-check sufficient conditions for exponential
*p*-stability are formulated in terms of
*M*-matrices. Stability analysis of applied second-order Itô equations with delay is provided as well. The linearization technique, in combination with the tests obtained in this paper, can be used for local stability analysis of a wide class of nonlinear stochastic differential equations.

Stochastic differential equations (SDE) of the second and higher order with or without time-varying delays, naturally appear in multiple applications, where deterministic models are perturbed by the white noise or its generalizations. A classical example is the Langevin equation (see e.g. [

Stochastic high-order models of processes related to abrasive waterjet milling or fluid energy milling (batch grinding) are well-known as well (see e.g. [

Several definitions of stochastic Lyapunov stability are used in the literature, e.g. stability in probability, stability in the mean and almost sure stability, stability of the p-th mean (p-stability), and even more. For applications to real systems, stability properties that are close to deterministic stability (almost sure sample stability) are the most desired, while conditions for p-stability are technically easier to obtain.

In this paper we study the global p-stability of the linear n-th order Itô delay equation

d x ( n − 1 ) ( t ) = [ − ∑ j = 0 n − 1 a j 0 x ( j ) ( t ) + ∑ j = 1 m 0 c j 0 x ( t − τ j 0 ) ] d t + ∑ l = 1 m [ − ∑ j = 0 n − 1 a j l ( t ) x ( j ) ( t ) + ∑ j = 1 m l c j l ( t ) x ( t − τ j l ) ] d B l ( t ) ( t ≥ 0 ) , (1)

and its non-autonomous generalization

d x ( n − 1 ) ( t ) = [ − ∑ j = 0 n − 1 a j 0 ( t ) x ( j ) ( t ) + ∑ j = 1 m 0 c j 0 ( t ) x ( h j 0 ( t ) ) ] d t + ∑ l = 1 m [ − ∑ j = 0 n − 1 a j l ( t ) x ( j ) ( t ) + ∑ j = 1 m l c j l ( t ) x ( h j l ( t ) ) ] d B l ( t ) ( t ≥ 0 ) , (2)

where B l ( t ) are independent scalar Brownian motions defined on a probability space.

Stability of stochastic first-order differential equations with delays, as well as systems of equations, has been extensively studied (see [

In the recent paper [

In conclusion, we stress that even if this paper studies stochastic linear equations, the various linearization criteria for a nonlinear stochastic differential equation (see e.g. [

Let ( Ω , F , ( F ) t ≥ 0 , P ) be a stochastic basis, where Ω is set of elementary probability events, F is a σ-algebra of all events on Ω, ( F ) t ≥ 0 is a right continuous family of s-subalgebras of F, P is a probability measure on F; all the above s-algebras are assumed to be complete w.r.t. P, i.e. containing all subsets of zero measure; the symbol E stands below for the expectation related to the probability measure P. The expectation (the integral w.r.t. the measure P) is denoted by E.

We will use the following notations:

- | . | is an arbitrary yet fixed norm in R^{n}, ‖ . ‖ being the associated matrix norm.

- μ is the Lebesgue measure on [ 0, + ∞ ) .

- ‖ . ‖ X is the norm in a normed space X.

- p is an arbitrary real number satisfying 1 ≤ p < ∞ .

- ( B 1 , ⋯ , B m ) is the standard m-dimensional Brownian motion (i.e. the scalar Brownian motions B l are all independent).

Recall that the classic Marcinkiewicz-Zygmund inequality

( E | ∑ X i | 2 p ) 1 / 2 p ≤ ρ p ( E | ∑ X i 2 | p ) 1 / 2 p , (3)

where X i are independent random variables with the zero mean, can be extended to the integral form

( E | ∫ 0 t f ( s ) d B l ( s ) | 2 p ) 1 / 2 p ≤ ρ p ( E ( ∫ 0 t | f ( s ) | 2 d s ) p ) 1 / 2 p (4)

for any predictable stochastic process f ( s ) ( 0 ≤ s ≤ t ), any t > 0 and any component B l ( s ) ( 1 ≤ l ≤ m ) of the Brownian motion B. The inequality (4) is often used in this paper. In 1988 D.L. Burkholder proved (see for example [

Equation (2) is assumed to be equipped with the initial conditions

x ( t ) = φ ( t ) ( t < 0 ) , (5)

and

x ( j ) ( 0 ) = b j + 1 , j = 0 , ⋯ , n − 1 , (6)

where

1) a j l , l = 0 , ⋯ , m , j = 0 , ⋯ , n − 1 , c j l , l = 0 , ⋯ , m , j = 1 , ⋯ , m l are Lebesgue measurable functions defined on [ 0, ∞ ) ; in addition, we assume that 0 < a ^ j 0 ≤ a j 0 ( t ) ≤ A j 0 μ―almost everywhere for some positive constants a ^ j 0 , A j 0 , j = 0 , ⋯ , n − 1 , | a j l ( t ) | ≤ A j l μ―almost everywhere for some positive constants A j l , l = 1 , ⋯ , m , j = 0 , ⋯ , n − 1 , and | c j l ( t ) | ≤ c j l μ―almost everywhere for some positive constants c j l , l = 0 , ⋯ , m , j = 0 , ⋯ , m l .

2) h j l , l = 0 , ⋯ , m , j = 1 , ⋯ , m l are Lebesgue measurable functions defined on [ 0, ∞ ) and satisfying the estimates 0 ≤ t − h j l ( t ) ≤ τ j l μ―almost everywhere for some positive constants τ j l , l = j , ⋯ , m , j = 1 , ⋯ , m l .

3) φ is an F 0 ―measurable, scalar stochastic process defined on [ − σ ,0 ) , where σ = max { τ j l , l = 0 , ⋯ , m , j = 1 , ⋯ , m l } .

4) b i is an F 0 ―measurable random variable for i = 1 , ⋯ , n .

We define a solution of the initial value problem (2), (5), (6) to be a predictable stochastic process x ( t ) , t ≥ − σ , which is ( n − 1 ) ―times differentiable on ( 0, ∞ ) and which satisfies the initial conditions (5), (6) and the integral equation

x ( n − 1 ) ( t ) = b n + ∫ o t [ − ∑ j = 0 n − 1 a j 0 ( s ) x ( j ) ( s ) + ∑ j = 1 m 0 c j 0 ( s ) x ( h j 0 ( s ) ) ] d s + ∑ l = 1 m ∫ o t [ − ∑ j = 0 n − 1 a j l ( s ) x ( j ) ( s ) + ∑ j = 1 m l c j l ( s ) x ( h j l ( s ) ) ] d B l ( s ) ( t ≥ 0 ) , (7)

where the integrals are understood in the Lebesgue and the Itô sense, respectively, and x ( h j l ( s ) ) = φ ( h j l ( s ) ) if h j l ( s ) < 0 .

The initial value problem (2), (5), (6) has a unique (up to the natural P-equivalency) solution x ( t , b , φ ) (see e.g. [

We will write b : = ( b 1 , ⋯ , b n ) ∈ k n , where k n denotes the linear space of all n―dimensional, F 0 ―measurable random values. In addition, we define the following normed space:

k p n = { α : α ∈ k n , ‖ α ‖ k p n ≡ ( E | α | p ) 1 / p < ∞ } . (8)

Definition 1 [

( E | x ( t , b , φ ) | p ) 1 / p ≤ K e x p { − λ t } ( ‖ b ‖ k p n + ess sup t < 0 ( E | φ ( t ) | p ) 1 / p ) ( t ≥ 0 ) . (9)

The analysis of the exponential p-stability of Equation (2) will be performed via an equivalent first order system of Itô equations. The technique of reduction of a high-order linear differential equation to a system by the substitution x ( k ) = x k + 1 is quite common, and for system (2) it yields

x ′ j ( t ) = x j + 1 ( t ) ( t ≥ 0 ) , j = 1 , ⋯ , n − 1 , d x n ( t ) = [ − ∑ j = 0 n − 1 a j 0 ( t ) x j + 1 ( t ) + ∑ j = 1 m 0 c j 0 ( t ) x 1 ( h j 0 ( t ) ) ] d t + ∑ l = 1 m [ − ∑ j = 0 n − 1 a j l ( t ) x j + 1 ( t ) + ∑ j = 1 m l c j l ( t ) x 1 ( h j l ( t ) ) ] d B l ( t ) ( t ≥ 0 ) , (10)

and

x 1 ( t ) = φ ( t ) ( t < 0 ) , (11)

x j ( 0 ) = b j , j = 1 , ⋯ , n , (12)

where the first component x 1 of the solution ( x 1 , ⋯ , x n ) of initial value problem (10), (11), (12) coincides with the solution x of the initial value problem (2), (5), (6), so that the exponential p-stability of Equation (2) follows from the exponential p-stability of system (10); and the latter can be, at least in the theory, studied by the Lyapunov-Razumikhin method of the stability analysis of stochastic delay equations. This method is based on finding a suitable Lyapunov function satisfying special conditions (see e.g. [

Below we use the generalized reduction technique based on a set of positive parameters, which can be chosen arbitrarily. Adapting this set to the coefficients of the given stochastic equation considerably increases, and this will be shown in the paper, flexibility of the reduction method. In addition, we will combine this technique with the method of stability analysis based on positive invertible matrices [

Let q j ( j = 1 , ⋯ , n − 1 ) be some positive numbers. Consider the following generalization of system (10):

x ′ j ( t ) = − q j x j ( t ) + x j + 1 ( t ) ( t ≥ 0 ) , j = 1 , ⋯ , n − 1 , d x n ( t ) = [ − ( a ( n − 1 ) 0 ( t ) − ∑ j = 1 n − 1 q j ) x n ( t ) + ∑ j = 1 n − 1 g j 0 ( t ) x j ( t ) + ∑ j = 1 m 0 c j 0 ( t ) x 1 ( h j 0 ( t ) ) ] d t + ∑ l = 1 m [ − a ( n − 1 ) l ( t ) x n ( t ) + ∑ j = 1 n − 1 g j l ( t ) x j ( t ) + ∑ j = 1 m l c j l ( t ) x 1 ( h j l ( t ) ) ] d B l ( t ) ( t ≥ 0 ) , (13)

where g j 0 ( t ) = ∑ i = 1 j q i S j − i , n − i , i − ∑ i = j − 1 n − 1 a i 0 ( t ) S j − 1 , i , 1 ( j = 1 , ⋯ , n − 1 ), g j l ( t ) = − ∑ i = j − 1 n − 1 a i l ( t ) S j − 1 , i , 1 ( l = 1 , ⋯ , m , j = 1 , ⋯ , n − 1 ) and

S r , k , i = ( − 1 ) k − r ∑ k 0 + ⋯ + k r = k − r ∏ j = 0 r q i + j k j , S k , k , i = 1 (14)

for i + r < i + k ≤ n , and the other entries are obtained from Equation (2). System (13) is supposed to be equipped with the initial conditions (11), (12).

Let us make some comments on this reduction technique. According to the paper [

Lemma 1 Let g ( s ) be a scalar function which is square integrable on [ 0, ∞ ) , f ( s ) be a predictable stochastic process satisfying sup s ≥ 0 ( E | f ( s ) | 2 p ) 1 / 2 p < ∞ . Then

sup s ≥ 0 ( E | ∫ 0 t g ( s ) f ( s ) d s | 2 p ) 1 / 2 p ≤ sup t ≥ 0 ( ∫ 0 t | g ( s ) | d s ) sup t ≥ 0 ( E | f ( t ) | 2 p ) 1 / 2 p (15)

and

sup t ≥ 0 ( E | ∫ 0 t ( g ( s ) ) 2 ( f ( s ) ) 2 d s | p ) 1 / 2 p ≤ sup t ≥ 0 ( ∫ 0 t ( g ( s ) ) 2 d s ) 1 / 2 sup t ≥ 0 ( E | f ( t ) | 2 p ) 1 / 2 p . (16)

Proof. Once we prove the inequality (15), the inequality (16) can be justified similarly.

sup t ≥ 0 ( E | ∫ 0 t g ( s ) f ( s ) d s | 2 p ) 1 / 2 p ≤ sup t ≥ 0 ( E ( ∫ 0 t | g ( s ) | | f ( s ) | d s ) 2 p ) 1 / 2 p ≤ sup t ≥ 0 ( E ( ∫ 0 t | g ( s ) | ( 2 p − 1 ) / 2 p | g ( s ) | 1 / 2 p | f ( s ) | d s ) 2 p ) 1 / 2 p ≤ sup t ≥ 0 ( E ( ( ∫ 0 t | g ( s ) | d s ) 2 p − 1 ∫ 0 t | g ( s ) | | f ( s ) | 2 p d s ) 1 / 2 p ) ≤ sup t ≥ 0 ( ( ∫ 0 t | g ( s ) | d s ) 2 p − 1 ∫ 0 t | g ( s ) | E | f ( s ) | 2 p d s ) 1 / 2 p ≤ sup t ≥ 0 ( ∫ 0 t | g ( s ) | d s ) s u p t ≥ 0 ( E | f ( t ) | 2 p ) 1 / 2 p . (17)

An n × n -matrix Γ = ( γ i j ) i , j = 1 n is called nonnegative if γ i j ≥ 0 , i , j = 1 , ⋯ , n , and positive if γ i j > 0 , i , j = 1 , ⋯ , n .

Definition 2 A matrix Γ = ( γ i j ) i , j = 1 n is called an M-matrix if γ i j ≤ 0 for i , j = 1 , ⋯ , n , i ≠ j and one of the following conditions is satisfied:

- G has a positive inverse matrix G^{−1};

- the principal minors of the matrix G are positive.

Now we define an n × n -matrix G which plays a crucial role in the theorem below. Let

- γ i i = 1 , γ i ( i + 1 ) = − 1 q i ( i = 1 , ⋯ , n − 1 ) ,

- γ i j = 0 ( i = 1 , ⋯ n − 1 , j = 1 , ⋯ , n , i ≠ j , j − i ≠ 1 ) ,

- γ n 1 = − G 10 + ∑ j = 1 m 0 c j 0 q n − ρ p ∑ l = 1 m [ G 1 l + ∑ j = 1 m l c j l ] 2 q n ,

- γ n j = − G j 0 q n − ρ p ∑ l = 1 m G j l 2 q n ( j = 2 , ⋯ , n − 1 ) ,

- γ n n = 1 − ρ p ( ∑ l = 1 m A ( n − 1 ) l ) 2 q n .

Here q n = a ^ ( n − 1 ) 0 − ∑ j = 1 n − 1 q j and G j l = sup t ≥ 0 | g j l ( t ) | for all j = 1 , ⋯ , n − 1 , l = 0 , ⋯ , m . These numbers can be expressed via the constants a ^ j 0 , A j l and q j from assumption 1 in Section 2. Thus, the matrix G becomes

Γ = ( 1 − 1 q 1 0 ⋯ 0 0 0 1 − 1 q 2 ⋱ 0 0 0 0 1 − 1 q 3 ⋱ 0 ⋮ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 0 ⋯ 1 − 1 q n − 1 γ n 1 γ n 2 γ n 3 ⋯ γ n , n − 1 γ n n ) . (18)

Theorem 1 Assume that 1 ≤ p < ∞ and there exist positive numbers q j , ( j = 1 , ⋯ , n − 1 ) such that q n > 0 and

γ n n + ∑ j = 1 n − 1 γ n j ∏ r = j n − 1 1 q r > 0. (19)

Then system (13), and hence Equation (2), is exponentially 2p-stable.

Proof. First of all, we observe that the determinant of the matrix G is equal to the left-hand side of the equality (19), while the other principal minors are all equal to 1. Hence G is an M-matrix.

Now, system (13) with the conditions (11) can be rewritten as follows:

x ¯ ′ j ( t ) = − q j x ¯ j ( t ) + x ¯ j + 1 ( t ) ( t ≥ 0 ) , j = 1 , ⋯ , n − 1 , d x ¯ n ( t ) = [ − ( a ( n − 1 ) 0 ( t ) − ∑ j = 1 n − 1 q j ) x ¯ n ( t ) + ∑ j = 1 n − 1 g j 0 ( t ) x ¯ j ( t ) + ∑ j = 1 m 0 c j 0 ( t ) ( x ¯ 1 ( h j 0 ( t ) ) + φ ¯ ( h j 0 ( t ) ) ) ] d t + ∑ l = 1 m [ − a ( n − 1 ) l ( t ) x ¯ n ( t ) + ∑ j = 1 n − 1 g j l ( t ) x ¯ j ( t ) + ∑ j = 1 m l c j l ( t ) ( x ¯ 1 ( h j l ( t ) ) + φ ¯ ( h j l ( t ) ) ) ] d B l ( t ) ( t ≥ 0 ) , (20)

where x ¯ i ( t ) is an unknown scalar predictable stochastic process on ( − ∞ , ∞ ) such that x ¯ i ( t ) = 0 for t < 0 , and φ ¯ ( t ) a known scalar predictable stochastic process on ( − ∞ , ∞ ) such that φ ¯ ( t ) = φ ( t ) for t ∈ [ − σ ,0 ) and φ ¯ ( t ) = 0 outside the interval [ − σ ,0 ) .

Let x ¯ ( t ) = ( x ¯ 1 ( t ) , ⋯ , x ¯ n ( t ) ) be the solution of (20) satisfying the initial conditions (12). A straightforward calculation shows that x ¯ ( t ) coincides with the solution of the initial value problem (11), (12), (13) for t ≥ 0 (but not necessarily for t < 0 , of course).

We choose a positive number λ < min { q 1 , ⋯ , q n } for all i = 1 , ⋯ , n and make the following substitution into system (20): x ¯ ( t ) = e x p { − λ t } y ( t ) , where y ( t ) = ( y 1 ( t ) , ⋯ , y n ( t ) ) is an unknown predictable stochastic process defined on ( − ∞ , ∞ ) . By the definition, y i ( t ) = 0 for t < 0 and 1 ≤ i ≤ n , thus

y ′ j ( t ) = ( λ − q j ) y j ( t ) + y j + 1 ( t ) ( t ≥ 0 ) , j = 1 , ⋯ , n − 1 , d y n ( t ) = [ ( λ − ( a ( n − 1 ) 0 ( t ) − ∑ j = 1 n − 1 q j ) ) y n ( t ) + ∑ j = 1 n − 1 g j 0 ( t ) y j ( t ) + ∑ j = 1 m 0 c j 0 ( t ) exp { λ t } ( exp { − λ h j 0 ( t ) } y 1 ( h j 0 ( t ) ) + φ ¯ ( h j 0 ( t ) ) ) ] d t + ∑ l = 1 m [ − a ( n − 1 ) l ( t ) y n ( t ) + ∑ j = 1 n − 1 g j l ( t ) y j ( t ) + ∑ j = 1 m l c j l ( t ) exp { λ t } ( exp { − λ h j l ( t ) } y 1 ( h j l ( t ) ) + φ ¯ ( h j l ( t ) ) ) ] d B l ( t ) ( t ≥ 0 ) . (21)

Let η ( t ) = ( a ( n − 1 ) 0 ( t ) − ∑ j = 1 n − 1 q j ) − λ . Then by using (12), we rewrite system (21)

y j ( t ) = exp { − ( q j − λ ) t } b i + ∫ 0 t exp { − ( q j − λ ) ( t − s ) } y j + 1 ( s ) d s ( t ≥ 0 ) , j = 1 , ⋯ , n − 1 ,

y n ( t ) = exp { − ∫ 0 t μ ( s ) d s } b n + ∫ 0 t exp { − ∫ s t μ ( ζ ) d ζ } [ ∑ j = 1 n − 1 g j 0 ( s ) y j ( s ) + ∑ j = 1 m 0 c j 0 ( s ) exp { λ s } ( exp { − λ h j 0 ( s ) } y 1 ( h j 0 ( s ) ) + φ ¯ ( h j 0 ( s ) ) ) ] d s + ∫ 0 t exp { − ∫ s t μ ( ζ ) d ζ } ∑ l = 1 m [ − a ( n − 1 ) l ( t ) y n ( s ) + ∑ j = 1 n − 1 g j l ( s ) y j ( s ) + ∑ j = 1 m l c j l ( s ) exp { λ s } ( exp { − λ h j l ( s ) } y 1 ( h j l ( s ) ) + φ ¯ ( h j l ( s ) ) ) ] d B l ( s ) ( t ≥ 0 ) . (22)

Denote y ^ i = s u p t ≥ 0 ( E | y i ( t ) | 2 p ) 1 / 2 p , and φ ^ = ess s u p t ≥ 0 ( E | φ ( t ) | 2 p ) 1 / 2 p

From the first ( n − 1 ) equations in (22) we obtain

y ^ i ≤ ‖ b i ‖ k 2 p 1 + 1 q i − λ y ^ i + 1 , i = 1 , ⋯ , n − 1. (23)

The estimate (4) and the last equation in (22) yield

y ^ n ≤ ‖ b n ‖ k 2 p 1 + [ ∑ j = 1 n − 1 G j 0 y ^ j + ∑ j = 1 m 0 c j 0 e x p { λ τ j 0 } ( y ^ 1 + φ ^ ) ] s u p t ≥ 0 ∫ 0 t e x p { − ∫ s t μ ( ζ ) d ζ } d s + ρ p ∑ l = 1 m [ A ( n − 1 ) l y ^ n + ∑ j = 1 n − 1 G j l ( s ) y ^ j ( s ) + ∑ j = 1 m l c j l e x p { λ τ j l } ( y ^ 1 + φ ^ ) ] × s u p t ≥ 0 ( ∫ 0 t e x p { − 2 ∫ s t μ ( ζ ) d ζ } d s ) 1 / 2 . (24)

Since

sup t ≥ 0 ∫ 0 t exp { − ∫ s t μ ( ζ ) d ζ } d s = sup t ≥ 0 ∫ 0 t [ exp { − ∫ s t μ ( ζ ) d ζ } μ ( s ) ] / μ ( s ) d s ≤ 1 a ^ ( n − 1 ) 0 − ∑ j = 1 n − 1 q j − λ (25)

and

sup t ≥ 0 ( ∫ 0 t e x p { − 2 ∫ s t μ ( ζ ) d ζ } d s ) 1 / 2 = sup t ≥ 0 ( ∫ 0 t [ e x p { − 2 ∫ s t μ ( ζ ) d ζ } 2 μ ( s ) ] / ( 2 μ ( s ) ) d s ) 1 / 2 ≤ 1 2 ( a ^ ( n − 1 ) 0 − ∑ j = 1 n − 1 q j − λ ) , (26)

the inequality (24) yields

y ^ n ≤ ‖ b n ‖ k 2 p 1 + ∑ j = 1 n − 1 G j 0 y ^ j + ∑ j = 1 m 0 c j 0 exp { λ τ j 0 } y ^ 1 a ^ ( n − 1 ) 0 − ∑ j = 1 n − 1 q j − λ + ρ p ∑ l = 1 m [ A ( n − 1 ) l y ^ n + ∑ j = 1 n − 1 G j l ( s ) y ^ j ( s ) + ∑ j = 1 m l c j l exp { λ τ j l } y ^ 1 ] 2 ( a ^ ( n − 1 ) 0 − ∑ j = 1 n − 1 q j − λ ) + M ( λ ) φ ^ , (27)

where

M ( λ ) : = ∑ j = 1 m 0 c j 0 exp { λ τ j 0 } a ^ ( n − 1 ) 0 − ∑ j = 1 n − 1 q j − λ + ρ p ∑ l = 1 m ∑ j = 1 m l c j l exp { λ τ j l } 2 ( a ^ ( n − 1 ) 0 − ∑ j = 1 n − 1 q j − λ ) . (28)

Denote y ^ = ( y ^ 1 , ⋯ , y ^ n ) and define the n × n --matrix Γ ( λ ) = ( γ i j ( λ ) ) i , j = 1 n by putting

- γ i i ( λ ) = 1 , γ i ( i + 1 ) ( λ ) = 1 λ − q i ( i = 1 , ⋯ , n − 1 ) ,

- γ i j ( λ ) = 0 ( i = 1 , ⋯ , n − 1 , j = 1 , ⋯ n , i ≠ j , j − i ≠ 1 ) ,

- γ n 1 ( λ ) = − G 10 exp { λ τ j 0 } + ∑ j = 1 m 0 c j 0 q n − λ − ρ p ∑ l = 1 m [ G 1 l + ∑ j = 1 m l c j l exp { λ τ j l } ] 2 q n − λ ,

- γ n j ( λ ) = − G j 0 q n − λ − ρ p ∑ l = 1 m G j l 2 q n − λ ( j = 2 , ⋯ , n − 1 ) ,

- γ n n ( λ ) = 1 − ρ p ∑ l = 1 m A ( n − 1 ) l 2 q n − λ .

Then from (27) we obtain the componentwise vector inequality

Γ ( λ ) y ^ ≤ ( ‖ b n ‖ k 2 p n + M ( λ ) φ ^ ) v , (29)

where v = ( 1, ⋯ ,1 ) T is an n-dimentional column vector.

Since Γ ( 0 ) = Γ is an M-matrix, Γ ( λ ) is also an M-matrix for small l. Therefore there exists a number λ = λ 0 such that the matrix Γ ( λ 0 ) is positive invertible. The inequality (29) justifies

| y ^ | ≤ K ( ‖ b n ‖ k 2 p n + φ ^ ) , (30)

where K = ‖ Γ ( λ 0 ) − 1 v ‖ max { 1 , | M ( λ 0 ) | } .

Recall

x ¯ ( t ) = exp { − λ t } y ( t ) , s u p t ≥ 0 ( E | y ( t ) | 2 p ) 1 / 2 p ≤ | y ^ | , φ ^ = ess s u p t ≥ 0 ( E | φ ( t ) | 2 p ) 1 / 2 p . (31)

Based on the inequality (30), we conclude that the solution x ¯ ( t ) of the initial value problem (13), (11), (12) satisfies

( E | x ¯ ( t ) | 2 p ) 1 / 2 p ≤ K e x p { − λ t } ( ‖ b ‖ k 2 p n + ess s u p t ≥ 0 ( E | φ ( t ) | 2 p ) 1 / 2 p ) ( t ≥ 0 ) , (32)

where λ = λ 0 , K = ‖ Γ ( λ 0 ) − 1 v ‖ m a x { 1, | M ( λ 0 ) | } . Therefore system (13) is exponentially 2p-stable. Theorem 1 is proven.

In this section we consider a second order equation (a particular case of Equation (10) if n = 2)

d x ′ ( t ) = [ − a 10 ( t ) x ′ ( t ) + ∑ j = 1 m 0 c j 0 ( t ) x ( h j 0 ( t ) ) ] d t + ∑ l = 1 m [ − a 0 l ( t ) x ′ ( t ) + ∑ j = 1 m l c j l ( t ) x ( h j l ( t ) ) ] d B l ( t ) ( t ≥ 0 ) , (33)

that we transform into

x ′ 1 ( t ) = − q 1 x 1 ( t ) + x 2 ( t ) ( t ≥ 0 ) , d x 2 ( t ) = [ − ( a 10 ( t ) − q 1 ) x 2 ( t ) + g 10 ( t ) x 1 ( t ) + ∑ j = 1 m 0 c j 0 ( t ) x 1 ( h j 0 ( t ) ) ] d t + ∑ l = 1 m [ − a 1 l ( t ) x 2 ( t ) + g 1 l ( t ) x 1 ( t ) + ∑ j = 1 m l c j l ( t ) x 1 ( h j l ( t ) ) ] d B l ( t ) ( t ≥ 0 ) , (34)

where q_{1} is some positive number and g 10 ( t ) = q 1 ( q 1 − a 10 ( t ) ) , g 1 l ( t ) = − q 1 a 1 l ( t ) − a 0 l ( t ) for l = 1 , ⋯ , m .

The matrix G is now defined as

Γ = ( 1 − 1 q 1 γ 21 γ 22 ) , (35)

where

- γ 11 = 1 , γ 12 = − 1 q 1 ,

- γ 21 = − G 10 + ∑ j = 1 m 0 c j 0 q n − ρ p ∑ l = 1 m [ G 1 l ( s ) + ∑ j = 1 m l c j l ] 2 q 2 ,

- γ 22 = 1 − ρ p ∑ l = 1 m A 1 l 2 q 2

and a 10 ( t ) ≥ a ¯ 10 > 0 , q 2 = a ¯ 10 − q 1 , G 1 l = s u p t ≥ 0 | g 1 l ( t ) | for all l = 0 , ⋯ , m .

Corollary 1 Assume that 1 ≤ p < ∞ and there exists a positive number q 1 < a ¯ 10 such that q 1 γ 22 − γ 21 > 0 . Then system (33) is exponentially 2p-stable.

Proof. The statement follows from Theorem 1 and the observation that the determinant of the matrix G is equal to γ 22 − q 1 − 1 γ 21 > 0 .

To demonstrate the efficiency of Corollary 1, let us consider the following particular case of Equation (33):

d x ′ ( t ) = [ − a x ′ ( t ) − b x ( t ) + c x ( t − τ ) ] d t + [ − δ x ′ ( t ) − e x ( t ) ] d B ( t ) ( t ≥ 0 ) , (36)

where B is the standard scalar Brownian motion, a , b , c , δ , e , τ are positive real numbers. Then a straightforward application of Corollory 1 yields.

Corollary 2 Assume that 1 ≤ p < ∞ and there exists a positive number q 1 < a such that

1 − ρ p ( δ + e ) 2 ( a − q 1 ) − 1 q 1 [ q 1 ( a − q 1 ) + b + c a − q 1 + ρ p ( δ + e ) 2 ( a − q 1 ) ] > 0. (37)

Then Equation (36) is exponentially 2p-stable.

We studied Lyapunov stability of high-order linear stochastic Itô equations with delay using a non-Lyapunov approach, which combines the method described in the review paper [

Solution of the following problems will complement the results of the present paper:

-Find explicit stability conditions for the 3rd and higher order stochastic delay equations in terms of their coefficients.

-Find sufficient conditions for the stability of the linear hybrid SDE with delay

d x ( n − 1 ) ( t ) = [ − ∑ j = 0 n − 1 a j 0 ( r ( t ) ) x ( j ) ( t ) + ∑ j = 1 m 0 c j 0 ( r ( t ) ) x ( h j 0 ( t ) ) ] d t + ∑ l = 1 m [ − ∑ j = 0 n − 1 a j l ( r ( t ) ) x ( j ) ( t ) + ∑ j = 1 m l c j l ( r ( t ) ) x ( h j l ( t ) ) ] d B l ( t ) ( t ≥ 0 ) , (38)

where r ( t ) is a Markov chain with its state space S, which is independent of the Brownian motions B l ( t ) and which represents random switchings between different delay equations.

-Generalize the suggested framework to the case of high-order SDEs driven by an arbitrary semimartingale, rather than by the Brownian motion.

-For the van der Pol-type SDE delayed equation under perturbations of white noise

d x ′ ( t ) + a x 2 n ( t ) d x ( t ) + b d x ( t − τ ) = c x n + 1 ( t ) d B ( t ) ( n ∈ N ) , (39)

examine stability by using the linearization criteria introduced in [

The work of the first and the third author was partially supported by the grant DDG-2015-00046 of NSERC. The work of the second and the third author was partially supported by the grant \#239070 of the Norwegian Research Council.

Idels, L., Kadiev, R. and Ponosov, A. (2018) Stability of High-Order Linear Itô Equations with Delays. Applied Mathematics, 9, 250-263. https://doi.org/10.4236/am.2018.93019