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

Authors: A. T. Ademola; S. Moyo; B. S. Ogundare; M. O. Ogundiran; O. A. Adesina
Journal: International Journal of Analysis
 (2016)

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## Abstract

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.


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## Body

## 1. Introduction

Differential equations of second-order have generated a great deal of applications in various fields of science and technology such as biology, chemistry, physics, mechanics, control technology, communication network, automatic regulation, economy, and ecology to mention few. In addition, the study of problems that involve the behaviour of solutions of ordinary differential equations (ODE), delay or functional differential equations (DDE), and stochastic differential equations (SDE) has been dealt with by many outstanding authors; see, for instance, Arnold [1], Burton [2, 3], Hale [4], Oksendal [5], Shaikihet [6], and Yoshizawa [7, 8], which contain the background to the study and the expository papers of Abou-El-Ela et al. [9, 10], Ademola et al. [11, 12], Alaba and Ogundare [13], Burton and Hatvani [14], Cahlon and Schmidt [15], Caraballo et al. [16], Domoshnitsky [17], Gikhman and Skorokhod [18, 19], Grigoryan [20], Ivanov et al. [21], Jedrzejewski and Brochard [22], Jin and Zengrong [23], Kolarova [24], Kolmanovskii and Shaikhet [25, 26], Kroopnick [27], Liu and Raffoul [28], Mao [29], Ogundare et al. [30–32], Raffoul [33], Rezaeyan and Farnoosh [34], Tunç [35–43], Wang and Zhu [44], Xianfeng and Wei [45], Yeniçerioğlu [46, 47], Yoshizawa [48], Zhu et al. [49], and the references cited therein.The authors in [18, 19] investigated the second-order linear scalar equations of the form(1)Yt′′+at+btηiY˙t=0,t≥t0,where η˙i is a general disturbance process (the derivative of a martingale). In [11, 12] the authors discussed stability, boundedness, and periodic solutions to the following second-order ordinary and delay differential equations: (2)ϕxtx′t′+gt,xt,x′tx′t+φthxt=pt,xt,x′t,(3)x′′t+ϕtfxt,xt-τt,x′t,x′t-τt+gxt-τt=pt,xt,x′t,respectively, where f, g, p, h, ϕ, and φ are continuous functions in their respective arguments. In their contributions, the authors in [9, 10] investigated asymptotic stability and boundedness of solutions of the following second-order stochastic delay differential equations:(4)x′′t+ax′t+bxt-h+σxtω′t=0,(5)x′′t+ax′t+fxt-h+σxt-τω′t=0,(6)x′′t+gx′t+bxt-h+σxtω′t=pt,xt,x′t,x′t-h,respectively, where a, b, and σ are positive constants; h, τ are delay constants; f, g, and p are continuous functions in their respective arguments and w(t)∈Rm is an m-dimensional standard Brownian motion defined on the probability space (also called Wiener process). Recently, in 2016 the authors in [43] discussed global existence and boundedness of solutions of a certain nonlinear integrodifferential equation of second-order with multiple deviating arguments(7)pxtx′t′+atft,xt,x′tx′t+btgt,x′t+∑i=1ncithixt-τi=∫0tct,sx′sds,where τi(i=1,2,…,n) are positive constants, a, b, and c are defined on R+, and f,g,h, and p are continuous functions defined in their respective arguments.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:(8)x′′t+gxt,x′tx′t+fxt+σxtω′t=pt,xt,x′t,where σ is a positive constant, the functions g, f, and p are continuous in their respective arguments on R2,R, and R+×R2, respectively, with R≔(-∞,∞), R+≔[0,∞), and ω (a standard Wiener process, representing the noise) is defined on R. Furthermore, it is assumed that the continuity of the functions g, f, and p is sufficient for the existence of solutions and the local Lipschitz condition for (8) to have a unique continuous solution denoted by (x(t),y(t)). The primes denote differentiation with respect to the independent variable t∈R+. If x′(t)=y(t), then (8) is equivalent to the system:(9)x′t=yt,y′t=pt,xt,yt-fx-gxt,ytyt-σxtω′t,where the derivative of the function f (i.e., f′) exists and is continuous for all x. 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 σ=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.


## 2. Preliminary Results

Let(Ω,F,{Ft}t>0,P) be a complete probability space with a filtration {Ft}t>0 satisfying the usual conditions (i.e., it is right continuous and {F0} contains all P-null sets). Let B(t)=(B1(t),…,Bm(t))T be an m-dimensional Brownian motion defined on the probability space. Let · denotes the Euclidean norm in Rn. If A is a vector or matrix, its transpose is denoted by AT. If A is a matrix, its trace norm is denoted by (10)A=traceATA.For more exposition in this regard, see Mao [29] and Arnold [1]. Now let us consider a nonautonomous n-dimensional stochastic differential equation(11)dXt=Ft,Xtdt+Gt,XtdBton t>0 with initial value X(0)=X0∈Rn. Here F:R+×Rn→Rn and G:R+×Rn→Rn×m are measurable functions. Suppose that both F and G are sufficiently smooth for (11) to have a unique continuous solution on t≥0 which is denoted by X(t,X0), if X(0)=0. Assume further that (12)Ft,0=Gt,0=0for all t≥0. Then, the stochastic differential equation (11) admits zero solution X(t,0)≡0.Definition 1 (see [1]).
The zero solution of the stochastic differential equation (11) is said to bestochastically stable orstable in probability, if for every pair of ϵ∈(0,1) and r>0, there exists a δ0=δ0(ϵ,r)>0 such that (13)Pr⁡Xt;X0<r∀t≥0≥1-ϵwheneverX0<δ0.Otherwise, it is said to bestochastically unstable.Definition 2 (see [1]).
The zero solution of the stochastic differential equation (11) is said to bestochastically asymptotically stable if it is stochastically stable and in addition if for every ϵ∈(0,1) and r>0, there exists a δ=δ(ϵ)>0 such that (14)Pr⁡limt→∞⁡Xt;X0=0≥1-ϵwheneverX0<δ.Definition 3.
A solutionX(t0,X0) of the stochastic differential equation (11) is said to bestochastically bounded orbounded in probability, if it satisfies(15)EX0Xt,X0≤Ct0,X0,∀t≥t0,where EX0 denotes the expectation operator with respect to the probability law associated with X0,C:R+×Rn and R+ is a constant depending on t0 and X0.Definition 4.
The solutionsX(t0,X0) of the stochastic differential equation (11) are said to beuniformly stochastically bounded if C in inequality (15) is independent of t0.Forh>0, let Uh={X∈Rn:X<h}⊂Rn and let C1,2(Uh×R+,R+) denote the family of all nonnegative functions V(t,X(t)) (Lyapunov function) defined on R+×Uh which are twice continuously differentiable in X and once in t. By Itô’s formula we have (16)dVt,Xt=LVt,Xtdt+Vxt,XtGt,XtdBt,where(17)LVt,Xt=∂Vt,Xt∂t+∂Vt,Xt∂xiFt,Xt+12traceGTt,XtVxxt,XtGt,Xt.Furthermore, (18)Vxxt,Xt=∂2Vt,Xt∂xi∂xjn×n,i,j=1,…,n.In this study we will use the diffusion operator LV(t,X(t)) defined in (17) to replace V′(t,X(t))=(d/dt)V(t,X(t)). We now present the basic results that will be used in the proofs of the main results.Lemma 5 (see [1]).
Assume that there existV∈C1,2(R+×Uh,R+) and ϕ∈K such that (i)
V(t,0)=0;(ii)
V(t,X(t))>ϕ(X(t));(iii)
LV(t,X(t))≤0 for all (t,X)∈R+×Uh.Then the zero solution of stochastic differential equation (11) isstochastically stable.Lemma 6 (see [1]).
Suppose that there existV∈C1,2(R+×Uh,R+) and ϕ0,ϕ1,ϕ2∈K such that (i)
V(t,0)=0;(ii)
ϕ0(X(t))≤V(t,X(t))≤ϕ1(X(t)), ϕ0(r)→∞ as r→∞;(iii)
LV(t,X(t))≤-ϕ2(X(t)) for all (t,X)∈R+×Uh.Then the zero solution of stochastic differential equation (11) isuniformly stochastically asymptotically stable in the large.Assumption 7 (see [28, 33]).
LetV∈C1,2(R+×Rn;R+), and suppose that for any solutions X(t0,X0) of stochastic differential equation (11) and for any fixed 0≤t0≤T<∞, we have(19)EX0∫t0TVxi2t,XtGik2t,Xtdt<∞,1≤i≤n,1≤k≤m.Assumption 8 (see [28, 33]).
A special case of the general condition (19) is the following condition. Assume that there exits a function σ(t) such that(20)Vxit,XtGikt,Xt<σt,X∈Rn1≤i≤n,1≤k≤m,and for any fixed 0≤t0≤T<∞,(21)∫t0Tσ2tdt<∞.Lemma 9 (see [28, 33]).
Assume there exists a Lyapunov functionV(t,X(t))∈C1,2(R+×Rn;R+), satisfying Assumption 7, such that, for all (t,X)∈R+×Rn, (i)
X(t)p≤V(t,X(t))≤X(t)q,(ii)
LV(t,X(t))≤-α(t)X(t)r+β(t),(iii)
V(t,X(t))-Vr/q(t,X(t))≤γ,where α,β∈C(R+;R+), p, q, and r are positive constants, p≥1, and γ is a nonnegative constant. Then all solutions of the stochastic differential equation (11) satisfy(22)EX0Xt,X0≤Vt0,X0e-∫t0tαsds+∫t0tγαu+βue-∫utαsdsdu1/p,for all t≥t0.Lemma 10 (see [28, 33]).
Assume there exists a Lyapunov functionV(t,X(t))∈C1,2(R+×Rn;R+), satisfying Assumption 7, such that, for all (t,X)∈R+×Rn, (i)
X(t)p≤V(t,X(t)),(ii)
LV(t,X(t))≤-α(t)Vq(t,X(t))+β(t),(iii)
V(t,X(t))-Vq(t,X(t))≤γ,where α,β∈C(R+;R+), p, q are positive constants, p≥1, and γ is a nonnegative constant. Then all solutions of the stochastic differential equation (11) satisfy (22) for all t≥t0.Corollary 11 (see [28, 33]).
(i) Assume that hypotheses (i) to (iii) of Lemma9 hold. In addition,(23)∫t0tγαu+βue-∫utαsdsdu≤M,∀t≥t0≥0,for some positive constant M; then all solutions of stochastic differential equation (11) areuniformly stochastically bounded.
(ii) Assume that hypotheses (i) to (iii) of Lemma10 hold. If condition (23) is satisfied, then all solutions of the stochastic differential equation (11) arestochastically bounded.


## 3. Main Results

Let(x(t),y(t)) be any solution of the stochastic differential equation (9); the main tool employed in the proofs of our results is the continuously differentiable function V=V(t,x(t),y(t)) defined as(24)2V=b2x2+by2+2xfx+ax+y2,where a and b are positive constants and the function f is as defined in Section 1.Theorem 12.
Suppose thata, b, σ, and M0 are positive constants such that (i)
a≤g(x,y) for all x and y,(ii)
bx≤f(x)≤Bx for all x≠0 and σ2<2ab(b+1)-1,(iii)
pt,x,y≤M0 for all t≥0,x and y.Then solution (x(t),y(t)) of the stochastic differential equation (9) is uniformly stochastically bounded.Remark 13.
We note the following:(i)
Whenever the functionsg(x,x′)=a,f(x)=bx and ω′=p(t,x,x′)=0, then the stochastic differential equation (8) becomes a second-order linear ordinary differential equation(25)x′′+ax′+bx=0,and conditions (i) to (iii) of Theorem12 reduce to Routh Hurwitz criteria a>0 and b>0 for the asymptotic stability of the second-order linear differential equation (25).(ii)
The termσx(t)ω′(t) in the stochastic differential equation (8) is an extension of the ordinary case discussed recently by authors in [11, 18, 23, 31, 32, 35–37, 40].We shall now state and prove a result that will be used in the proofs of our results.Lemma 14.
Under the hypotheses of Theorem12, there exist positive constants D0=D0(a,b) and D1=D1(a,b,B) such that(26)D0x2t+y2t≤Vt,xt,yt≤D1x2t+y2t,for all t≥0, x, and y. In addition, there exist positive constants D2=D2(a,b,σ) and D3=D3(a,b) such that(27)LVt,xt,yt≤-D2x2t+y2t+D3xt+ytpt,xt,yt,for all t≥0,x, and y.Proof.
Let(x(t),y(t)) be any solution of the stochastic differential equation (9); since X=(x,y)∈R2, it follows from (24) that(28)Vt,0,0=0,for all t≥0. Moreover, from (24) and the fact that f(x)≥ax for all x≠0, there exists a positive constant δ0 such that(29)Vt,X≥δ0x2+y2,for all t≥0, x, and y, where(30)δ0≔min⁡b2+2b+min⁡a,1,b+min⁡a,1.It is clear from inequality (29) that (31)Vt,X=0⟺x2+y2=0,Vt,X>0⟺x2+y2≠0,(32)Vt,X⟶+∞asx2+y2⟶∞.Furthermore, since f(x)≤Bx for all x≠0, it follows from (24) that there exists a positive constant δ1 such that(33)Vt,X≤δ1x2+y2,for all t≥0, x, and y, where (34)δ1≔max⁡b2+2B+max⁡a,1,b+max⁡a,1.From inequalities (29) and (33), we have(35)δ0x2+y2≤Vt,X≤δ1x2+y2,for all t≥0, x, and y. It is not difficult to see that estimates (35) satisfy inequalities (26) of Lemma 14 with δ0 and δ1 equivalent to D0 and D1, respectively.
Moreover, applying Itô’s formula in (24) using system (9), we find that(36)LVt,X=12afxx-12σ2b+1x2-12b+1gx,y-ay2-Wi+ax+b+1ypt,x,y,i=1,2,where (37)W1≔14afxx-12σ2b+1x2+4agx,y-a2+b2xy+b+1gx,y-ay2,W2≔14afxx-12σ2b+1x2+4afxx-f′xxy+b+1gx,y-ay2.It is clear from the inequalities(38)4agx,y-a2+b22<afxx-12σ2b+1b+1gx,y-a,4afxx-f′x<afxx-12σ2b+1b+1gx,y-athat(39)W1=W2≥afxx-12σ2b+1x-b+1gx,y-ay2≥0,for all x and y. Using inequality (39) and hypotheses (i) and (ii) of Theorem 12 in (36), there exist positive constants δ2 and δ3 such that(40)LVt,X≤-δ2x2+y2+δ3x+ypt,x,y,for all t≥0, x, and y, where (41)δ2≔12min⁡ab-12σ2b+1,ab,δ3≔max⁡a,b+1.Inequality (40) satisfies inequality (27) with δ2 and δ3 equivalent to D2 and D3, respectively. This completes the proof of Lemma 14.Proof of Theorem12.
Let(x(t),y(t)) be any solution of system (9). From inequality (40) and assumption (iii) of Theorem 12, we have (42)LVt,X≤-12δ2x2+y2-12δ2M0x-δ2-1δ32+y-δ2-1δ32+M0δ2-1δ32,for t≥0, x, and y. Since δ2,δ3, and M0 are positives and (43)x-δ2-1δ32+y-δ2-1δ32≥0,for all x and y, there exist positive constants δ4 and δ5 such that(44)LVt,X≤-δ4x2+y2+δ5,for all t≥0, x,y, where δ4≔(1/2)δ2 and δ5≔M0δ2-1δ32. Hence, condition (ii) of Lemma 9 is satisfied with α(t)≔δ4, r≔2 and β(t)≔δ5. Also from inequality (35), hypotheses (i) and (iii) of Lemma 9 hold with p=q=2 so that γ=0.
Furthermore, from inequality (23) we have(45)∫t0tγαu+βue-δ4∫utαsdsdu=∫t0tδ5e-δ4∫utdsdu=δ4-1δ51-e-δ4t-t0≤δ4-1δ5,for all t≥t0≥0. Inequality (45) satisfies estimate (23) with M≔δ4-1δ5=2M0δ2-2δ32>0. Moreover, from (9) and (24) there exists a positive constant δ6 such that(46)Vxit,XGikt,X≤12σ2a+b+1x2+b+1y2≤δ6x2+y2≔λt,where (47)δ6≔12σmax⁡2a+b+1,b+1.Also,(48)∫t0Tδ62x2t+y2t2dt<∞,for any fixed 0≤t0≤T<∞. Thus, from inequalities (46) and (48) estimates (20) and (21) hold, respectively. Finally, from inequalities (33) and (45), we have(49)EX0Xt,X0≤δ1X02+2M0δ2-2δ321/2,for all t≥t0≥0, where X0≔(x02+y02) and C≔δ1. Thus, the solutions (x(t),y(t)) of the stochastic differential equation (9) are uniformly stochastically bounded.Theorem 15.
If assumptions of Theorem12 hold, then the solution (x(t),y(t)) of the stochastic differential equation (9) is stochastically bounded.Proof.
Suppose that(x(t),y(t)) is any solution of the stochastic differential equation (9). From inequalities (33) and (44) there exists a positive constant δ7 such that(50)LVt,X≤-δ7Vt,X+δ5for all t≥0,x, and y, where δ7≔δ1-1δ4. Hence, from inequalities (29) and (50) hypotheses of Lemma 10 hold. Moreover, from inequalities (45), (46), (48), and (49) assumption (ii) of Corollary 11 holds. Thus, by Corollary 11, all solutions of the stochastic differential equation (9) are stochastically bounded. This completes the proof of Theorem 15.Next, we shall discuss the stability of the trivial solution of the stochastic differential equation (8). Suppose that p(t,x,x′)=0, (8) specializes to(51)x′′t+gxt,x′tx′t+fxt+σxtω′t=0.Equation (51) has the following equivalent system:(52)x′t=yt,y′t=-fx-gxt,ytyt-σxtω′t,where the functions f,g, and ω are defined in Section 1.Theorem 16.
If assumptions (i) and (ii) of Theorem12 hold, then the trivial solution of the stochastic differential equation (52) is stochastically stable.Proof.
Let(x(t),y(t)) 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 V(t,X) is positive definite. Furthermore, using Itô’s formula along the solution path of (52), we obtain(53)LVt,X≤-δ2x2t+y2t≤0,for all t≥0, x, and y, where δ2 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 Theorem12 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(x(t),y(t)) be any solution of the stochastic differential equation (52). In view of (28) and estimate (29), the function V(t,X) is positive definite. Furthermore, estimate (32) and inequality (33) show that the function V(t,X) 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 functionp(t,x,x′) is replaced by p(t)∈C(R+,R+), we have the following special case:(54)x′′t+gxt,x′tx′t+fxt+σxtω′t=pt,of (8). Equation (54) has the following equivalent system:(55)x′t=yt,y′t=pt-fx-gxt,ytyt-σxtω′t,with the following result.Corollary 18.
If assumptions (i) and (ii) of Theorem12 hold and hypothesis (iii) is replaced by the boundedness of the function p(t), then the solutions (x(t),y(t)) of the stochastic differential equation (55) are not only stochastically bounded but also uniformly stochastically bounded.Proof.
The proof of Corollary18 is similar to the proof of Theorems 12 and 15. This completes the proof of Corollary 18.


## 4. Examples

In this section we shall present two examples to illustrate the applications of the results we obtained in the previous section.Example 1.
Consider the second-order nonlinear nonautonomous stochastic differential equation(56)x′′+3+cos⁡xx′x′+x+sin⁡x+0.1xω′t=1+2t+xx′-1.Equation (56) is equivalent to system(57)x′=y,y′=1+2t+xy-1-x+sin⁡x-3+cos⁡xyy-0.1xω′t.Now from systems (9) and (57) we have the following relations: (i)
The function(58)gx,y≔3+cos⁡xy.Noting that(59)cos⁡xy≥0for allx and y, it follows that (60)gx,y=3+cos⁡xy≥a=3,for allx and y. The behaviour of the function g(x,y) is shown below in Figure 1.(ii)
The function(61)fx≔x+sin⁡x.Since(62)-0.2≤Fx=sin⁡xx≤1for allx≠0, then we have (63)1=b≤fxx=1+sin⁡xx≤B=2,for allx≠0 and since σ≔0.1 it follows that σ2<2ab(b+1)-1 implies that 0<2.99. The function f(x)/x and its bounds are shown in Figure 2.(iii)
The function(64)pt,x,y≔11+2t+xy.Clearly,(65)pt,x,y=11+2t+xy≤1=M0,for allt≥0, x, and y.Now from items (i), (ii) above and (24), the continuously differentiable function V(t,X) used for system (57) is(66)2Vt,X=3x2+y2+3x+y2.Different views of the function V(t,X) are shown in Figure 3. From (66), it is not difficult to show that(67)x2+y2≤Vt,X≤3x2+y2,for all t≥0,x, and y. From (35) and (67) we have δ0=1, δ1=3, p=2, and q=2, and thus, inequalities (67) satisfy condition (i) of Lemma 9. Also, from the first inequality in (67), we have(68)Vt,X⟶+∞asx2+y2⟶∞.Estimate (68) verifies (32) (i.e., the function V(t,X) defined by (66) is radially unbounded). Next, applying Itô’s formula in (66) using system (57), we find that(69)LVt,X=12xy+3y2-x3x+2y1+sin⁡xx-y3x+2y3+cosxy+1100x2-x103x+2y+3x+2y1+2t+xy-1.Using the estimates in items (i) to (iii) of Example 1 and the inequality 2x1x2≤x12+x22 in (69), we obtain(70)LVt,X≤-2.9x2+y2+3x+y,for all t≥0,x, and y. Inequality (70) satisfies inequality (40) where δ2=2.9 and δ3=3. Since (71)x-1.052+y-1.052≥0,for all x and y, it follows from inequality (70) that(72)LVt,X≤-1.45x2+y2+3.2,for all t≥0, x, and y. Inequality (72) satisfies assumption (ii) of Lemma 9 and estimate (44) with α(t)=δ4=1.45 and β(t)=δ5=3.2. Since r=p=q=2, it follows that γ=0, so that assumption (iii) of Lemma 9 holds. In addition,(73)∫t0tγαu+βue-∫ut⁡αsdsdu≤1.6,for all t≥t0≥0. Estimate (73) satisfies (23) and (45), with M=2.6. Furthermore,(74)Vxit,XGikt,X=-1103x2+2xy,and (75)Vxit,XGikt,X≤25x2+y2,for all t≥0, x, and y. Inequality (75) satisfies inequalities (20) and (21) with (76)λt=25x2+y2.Hence, by Corollary 11 (i), all solutions of stochastic differential equation (57) are uniformly stochastically bounded.Figure 1
Behaviour of the functiong(x,y).Figure 2
Bounds on the functionf(x)/x.Figure 3
The behaviour of the functionV(t,X).Example 2.
Ifp(t,x,x′)=p(t,x,y)=0 in (56) and system (57), we have the following stochastic differential equation:(77)x′′+3+cos⁡xx′x′+x+sin⁡x+0.1xω′t=0.Equation (77) is equivalent to system(78)x′=y,y′=-x+sin⁡x-3+cos⁡xyy-0.1xω′t.Now from systems (52) and (78) items (i) and (ii) of Example 1 hold. Also, equations (66), (67) and estimate (68) hold: that is,(79)2Vt,X=3x2+y2+3x+y2,Vt,0=0,∀t≥0;x2+y2≤Vt,X≤3x2+y2∀t≥0,x,y,Vt,X⟶+∞asx2+y2⟶∞.Furthermore, application of Itô’s formula in (66) and using system (78) yield(80)LVt,X≤-2.9x2+y2,for all t≥0, x,y and thus(81)LVt,X≤0,for all t≥0, x, and y. Moreover, from (79) and (80) all assumptions of Theorem 17 and Lemma 6 are satisfied. Thus, by Lemma 6 the trivial solution of system (78) is not only uniformly stochastically asymptotically stable but also uniformly stochastically asymptotically stable in the large. Finally, from (79) and (81) the function V(t,X) is positive definite and (82)LVt,X≤0,∀t,X∈R+×R2.Hence, assumptions of Theorem 17 and Lemma 5 hold; by Theorem 17 and Lemma 5 the trivial solution of system (78) is stochastically stable.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 σ=0.1, the numerical solutions of (56) in three-dimensional space are shown in Figure 4. If we vary the value of the noise in the numerical solution (x(t),y(t)) of system (57), as σ=0.1 and σ=1.0, we have Figures 5(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 (x(t),y(t)) of system (57) when σ=0.5 and σ=2.0 is shown in Figures 6(a) and 6(b), respectively. The behaviour of the numerical solution (x(t),y(t)) of system (57) for σ=0 and σ=5.0 is shown in Figures 7(a) and 7(b), respectively. For the case of (78), Figure 8 shows the closeness of the solution (x(t)) and the perturbed solution (xϵ(t)) for a very large t which implies asymptotic stability in the large for the considered SDE.Figure 4
Graph of solutions of (56) in 3D.Figure 5 (a) (b)Figure 6 (a) (b)Figure 7 (a) (b)Figure 8
Graph of solutions of (78).



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*Source: 2012315-2016-12-25.xml*