On the Stochastic Stability and Boundedness of Solutions for Stochastic Delay Differential Equation of the Second Order

We present two qualitative results concerning the solutions of the following equation: ?̈?(t) + g(?̇?(t)) + bx(t − h) + σx(t)?̇?(t) = p(t, x(t), ?̇?(t), x(t − h)); the first result covers the stochastic asymptotic stability of the zero solution for the above equation in case p ≡ 0, while the second one discusses the uniform stochastic boundedness of all solutions in case p ̸ ≡ 0. Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.


Introduction
During the last twenty years, the theory of stochastic differential equations has successfully attracted considerable attentions of scholars; for example, see [1][2][3][4][5][6][7][8][9][10][11][12][13]. Since then, the number of contributions to statistics, numerics, and control theory of stochastic differential equations has been rapidly increasing, since stochastic modelling plays an important role in formulation and analysis in modelling physical, technical, biological and economical dynamical systems in which significant uncertainty is present.
Stochastic delay differential equation, also known as stochastic functional differential equation, is a natural generalization of stochastic ordinary differential equation by allowing the coefficients to depend on the past values. The Razumikhin argument, generalized Itô formula, and Euler-Maruyama formula play an important role in studying stochastic differential equations. Unfortunately, it is generally not possible to give explicit expressions for the solutions to stochastic differential equations. Therefore, most of the papers are interested in being able to characterize at least qualitatively the behaviour of the solutions. Thus, Lyapunov theory is a powerful tool for qualitative analysis of stochastic differential equations, since the advantage of this method can judge the behaviour of systems without any prior knowledge of the explicit solutions, while the greatest disadvantage of the Lyapunov approach is that no universal method has been given, which enables us to find a Lyapunov function or determine that no such function exists.
It is worth mentioning that there are few results on the stability and boundedness of solutions for first-order stochastic delay differential equations; for example, see [8,[14][15][16][17][18].
In 2004, Kolmanovskii and Shaikhet [8] investigated the conditions of asymptotic mean-square stability for first-order stochastic delay differential equation of neutral type: where and are two positive constants and ( ) is a standard Wiener process. Later, in 2006, Rodkina and Basin [17] obtained global asymptotic stability conditions for nonlinear stochastic systems with state delay as follows: The Lyapunov Krasovskii and degenerate functionals techniques are used. In addition, nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given. On the other hand, the corresponding problem for the stability and boundedness of solutions of second-order stochastic delay differential equations was studied far less often. So our main aim in this paper is to establish new results on the stability and boundedness for solutions of secondorder stochastic delay differential equation of the typë where and are two positive constants, and ℎ is a positive constant delay; and are two continuous functions with (0) = 0; ( ) = ( 1 ( ), 2 ( ), . . . , ( )) ∈ R is an -dimensional standard Brownian motion defined on the probability space (also called Wiener process), a stochastic process representing the noise [19].
In this section, for the stability result, we impose Standing Hypothesis (H2): Hence, the stochastic delay differential equation (4) admits the zero solution ( ; 0) ≡ 0, for any given initial value Definition 1. The zero solution of the stochastic differential equation is said to be stochastically stable or stable in probability, if for every pair of ∈ (0, 1) and > 0, there exists a = ( , ) > 0 such that Otherwise, it is said to be stochastically unstable.

Definition 2.
The zero solution of the stochastic differential equation is said to be stochastically asymptotically stable, if it is stochastically stable, and moreover for every ∈ (0, 1), there exists a 0 = 0 ( ) > 0 such that Let K denote the family of all continuous nondecreasing functions : where It should be emphasized that the operator L is defined on R + × R × R , although the function is defined only on R + × R .
To prove our main stability result, we shall introduce the following theorems.
Now we present the main stability result of (3) with ≡ 0.
Then, the zero solution of (3) is stochastically asymptotically stable, provided that Proof of Theorem 5. In fact, (3) with ≡ 0 can be transformed into an equivalent system of the following form: Define the Lyapunov functional 1 ( , ) as where = ( + ), ≤ 0, and is a positive constant, which will be determined later.
From (14) and (13) and by using Itô formula, we find since | ( )| ≤ , > 0 and ( )/ ≥ > 0 by (i), then Thus, by using the inequality | V| ≤ (1/2)( 2 + V 2 ), we obtain Let us choose = ( + + 1)/2 > 0. Then, it is easy to see that Now, in view of (18), one can conclude for some positive constant that provided that Since ∫ 0 −ℎ ∫ + 2 ( ) is nonnegative, then we find from ( )/ ≥ > 0, therefore, we get Then, there exists a positive constant such that Also, since | ( )| ≤ , (0) = 0, and by using the meanvalue theorem, we get ( ) ≤ . So we can rewrite (14) in the following form: It follows that, by using the inequality 2 V ≤ 2 + V 2 , we have Therefore, we obtain Then, there exists a positive constant ] satisfying Thus, from the results (19), (23), and (27), we note that all the conditions of Theorem 4 are satisfied, and then the zero solution of (3) is stochastically asymptotically stable. This completes the proof of Theorem 5.

Further Preliminaries and Boundedness Result
Consider the -dimensional stochastic delay differential equation (SDDE): where : R + × R × R → R and : R + × R × R → R × . In order to solve the equation, we need to know the initial data and we assume that they are given by initial value of { ( ) : − ≤ ≤ 0}, 0 ∈ C([− , 0]; R ).
In [23], Liu and Raffoul use Lyapunov second method to determine sufficient conditions for stochastic boundedness of system (28). The theorems in [23] will make significant contribution to the theory of stochastic differential equations, when dealing with equations that might contain unbounded terms.

Definition 6.
A solution ( ; 0 , 0 ) of (28) is said to be stochastically bounded or bounded in probability, if it satisfies where 0 denotes the expectation operator with respect to the probability law associated with 0 , : R + × R + → R + is a constant depending on 0 and 0 . We say that the solutions of (28) are uniformly stochastically bounded, if is independent of 0 . Assumption 7. We assume that for any solution ( ) of (28) and for any fixed 0 ≤ 0 ≤ < ∞, the following condition holds: and for any fixed 0 ≤ 0 ≤ < ∞, Here, we will present the main theorems used in our main boundedness result. Theorem 9. Assume there exists a function ( , ) in C 1,2 (R + × R ; R + ) satisfying Assumption 7, such that for all where , ∈ (R + ; R + ), 1 , 2 , and are positive constants, 1 ≥ 1, and is a nonnegative constant. Then, all solutions of (28) satisfy for all ≥ 0 .
for some positive constant M; then, all the solutions of (28) are uniformly stochastically bounded. (2) Assume that the hypotheses of Theorem 10 hold. If condition (35) is satisfied, then all the solutions of (28) are stochastically bounded.
The following theorem is the second main result for (3) with ̸ ≡ 0.

Illustrative Examples
In this section, we display two examples to illustrate the application of the results we obtained in the previous sections.