Stability and Boundedness of Solutions to a Certain Second-Order Nonautonomous Stochastic Differential Equation

This paper focuses on stability and boundedness of certain nonlinear nonautonomous second-order stochastic differential equations. Lyapunov’s second method is employed by constructing a suitable complete Lyapunov function and is used to obtain criteria, on the nonlinear functions, that guarantee stability and boundedness of solutions. Our results are new; in fact, according to our observations from the relevant literature, this is the first attempt on stability and boundedness of solutions of second-order nonlinear nonautonomous stochastic differential equations. Finally, examples together with their numerical simulations are given to authenticate and affirm the correctness of the obtained results.

Although second-order stochastic delay differential equations have started receiving attention of authors, according to our observation from relevant literature, there is no previous literature available on the stability and boundedness of solutions of second-order nonlinear nonautonomous stochastic differential equation.The aim of this paper is to bridge this gap.Consider the following second-order nonlinear nonautonomous stochastic differential equation: where  is a positive constant, the functions , , and  are continuous in their respective arguments on R 2 , R, and and  (a standard Wiener process, representing the noise) is defined on R. Furthermore, it is assumed that the continuity of the functions , , and  is sufficient for the existence of solutions and the local Lipschitz condition for (8) to have a unique continuous solution denoted by ((), ()).The primes denote differentiation with respect to the independent variable  ∈ R + .If   () = (), then ( 8) is equivalent to the system:   () =  () ,   () =  (,  () ,  ()) −  () −  ( () ,  ())  () −  ()   () , (9) where the derivative of the function  (i.e.,   ) exists and is continuous for all .Despite the applicability of these classes of equations, there is no previous result on nonautonomous second-order nonlinear stochastic differential equation (8).The motivation for this investigation comes from the works in [9-12, 18, 19].If  = 0 in (8), then we have a general second-order nonlinear ordinary differential equation which has been discussed extensively in relevant literature.The remaining parts of this paper are organized as follows.In Section 2, we give the preliminary results on stochastic differential equations.Main results and their proofs are presented in Section 3 while examples and simulation of solutions are given in Section 4 to validate our results.

Preliminary Results
Let (Ω, F, {F  } >0 , P) be a complete probability space with a filtration {F  } >0 satisfying the usual conditions (i.e., it is right continuous and {F 0 } contains all P-null sets).Let () = ( 1 (), . . .,   ())  be an -dimensional Brownian motion defined on the probability space.Let |⋅| denotes the Euclidean norm in R  .If  is a vector or matrix, its transpose is denoted by   .If  is a matrix, its trace norm is denoted by For more exposition in this regard, see Mao [29] and Arnold [1].Now let us consider a nonautonomous -dimensional stochastic differential equation on  > 0 with initial value (0) =  0 ∈ R  .Here  : R + × R  → R  and  : R + ×R  → R × are measurable functions.Suppose that both  and  are sufficiently smooth for (11) to have a unique continuous solution on  ≥ 0 which is denoted by (,  0 ), if X(0) = 0. Assume further that for all  ≥ 0.Then, the stochastic differential equation (11) admits zero solution (, 0) ≡ 0.
Otherwise, it is said to be stochastically unstable.
International Journal of Analysis 3 Definition 2 (see [1]).The zero solution of the stochastic differential equation ( 11) is said to be stochastically asymptotically stable if it is stochastically stable and in addition if for every  ∈ (0, 1) and  > 0, there exists a  = () > 0 such that Definition 3. A solution ( 0 ,  0 ) of the stochastic differential equation ( 11) 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  and R + is a constant depending on  0 and  0 .
Definition 4. The solutions ( 0 ,  0 ) of the stochastic differential equation ( 11) are said to be uniformly stochastically bounded if  in inequality ( 15) is independent of  0 .
For ℎ > 0, let where Furthermore, In this study we will use the diffusion operator (, ()) defined in (17) to replace   (, ()) = (/)(, ()).We now present the basic results that will be used in the proofs of the main results.
Lemma 6 (see [1]).Suppose that there exist Then the zero solution of stochastic differential equation ( 11) is uniformly stochastically asymptotically stable in the large.
(ii) Assume that hypotheses (i) to (iii) of Lemma 10 hold.If condition (23) is satisfied, then all solutions of the stochastic differential equation ( 11) are stochastically bounded.

Main Results
Let ((), ()) be any solution of the stochastic differential equation ( 9); the main tool employed in the proofs of our results is the continuously differentiable function  = (, (), ()) defined as where  and  are positive constants and the function  is as defined in Section 1.
Remark 13.We note the following: (i) Whenever the functions (,   ) = , () =  and   = (, ,   ) = 0, then the stochastic differential equation ( 8) becomes a second-order linear ordinary differential equation and conditions (i) to (iii) of Theorem 12 reduce to Routh Hurwitz criteria  > 0 and  > 0 for the asymptotic stability of the second-order linear differential equation (25).
(51) Equation ( 51) has the following equivalent system: where the functions , , and  are defined in Section 1.

Theorem 16. If assumptions (i) and (ii) of Theorem 12 hold, then the trivial solution of the stochastic differential equation (52) is stochastically stable.
Proof.Let ((), ()) be any solution of the stochastic differential equation (52).From equation (28) and estimate ( 29) assumptions (i) and (ii) of Lemma 5 hold so that the function (, ) is positive definite.Furthermore, using Itô's formula along the solution path of (52), we obtain for all  ≥ 0, , and , where  2 is defined in (40).Inequality (53) satisfies hypothesis (iii) of Lemma 5; hence, by Lemma 5 the trivial solution of the stochastic differential equation ( 52) is stochastically stable.This completes the proof of Theorem 16.
Theorem 17.If assumptions (i) and (ii) of Theorem 12 hold, then the trivial solution of the stochastic differential equation ( 52) is not only uniformly stochastically asymptotically stable, but also uniformly stochastically asymptotically stable in the large.
Proof.Let ((), ()) be any solution of the stochastic differential equation ( 52).In view of (28) and estimate (29), the function (, ) is positive definite.Furthermore, estimate (32) and inequality (33) show that the function (, ) is radially unbounded and decrescent, respectively.It follows from (28), estimate (32), inequality (35), and the first inequality in (53) that all assumptions of Lemma 6 hold.Thus, by Lemma 6 the trivial solution of the stochastic differential equation ( 52) is uniformly stochastically asymptotically stable in the large.If estimate ( 32) is omitted then the trivial solution of the stochastic differential equation ( 52) is uniformly stochastically asymptotically stable.This completes the proof of Theorem 17.
Next, if the function (, ,   ) is replaced by () ∈ (R + , R + ), we have the following special case: of (8).Equation (54) has the following equivalent system: with the following result.

Corollary 18. If assumptions (i) and (ii) of Theorem 12 hold and hypothesis (iii) is replaced by the boundedness of the function 𝑝(𝑡)
, then the solutions ((), ()) of the stochastic differential equation ( 55) are not only stochastically bounded but also uniformly stochastically bounded.
Proof.The proof of Corollary 18 is similar to the proof of Theorems 12 and 15.This completes the proof of Corollary 18.

Examples
In this section we shall present two examples to illustrate the applications of the results we obtained in the previous section.

Simulation of Solutions.
In what follows, we shall now simulate the solutions of (56) (resp., system (57)) and (78) (resp., system (79)).Our approach depends on the Euler-Maruyama method which enables us to get approximate numerical solution for the considered systems.It will be seen from our figures that the simulated solutions are bounded which justifies our given results.For instance, when  = 0.1, the numerical solutions of (56) in three-dimensional space are shown in For the case of (78), Figure 8 shows the closeness of the solution (()) and the perturbed solution (  ()) for a very large  which implies asymptotic stability in the large for the considered SDE.

Figure 4 .
If we vary the value of the noise in the numerical solution ((), ()) of system (57), as  = 0.1 and  = 1.0, we have Figures5(a) and 5(b), respectively.It can be seen that, when the noise is increased, the stochasticity becomes more pronounced.The behaviour of the numerical solution ((), ()) of system (57) when  = 0.5 and  = 2.0 is shown in Figures6(a) and 6(b), respectively.The behaviour of the numerical solution ((), ()) of system (57) for  = 0 and  = 5.0 is shown in Figures7(a) and 7(b), respectively.