The Qualitative Theory Of Stochastic Pantograph Equations

Since the 1940s, stochastic differential equations is a very active, eye-catching young subject, this field is extremely broad. It has a wide range of applications in mathematics and other areas and plays an effective role in connecting many branches of mathematics. Since the day of its birth, many international famous mathematicians devoted themselves to the research in this field and obtained brilliant results.When we take the environmental noise as well as the delayed impact for system into account, the stochastic delay differential equations are more realistic mathematical models for describing the state and change of objective things. Accordingto the types of delay, we can divide stochastic delay differential equations into stochastic differential equations with bounded delay and stochastic differentialequations with unbounded delay. For stochastic delay differential equationswith bounded delay, in particular, stochastic differential equations with a constant delay, there are in-depth studies at present. Moreover, stochastic pantographequations are special stochastic differential equations with unbounded delay, it also can be seen as deterministic pantograph equations after added randomperturbations. Deterministic pantograph equations arise in quite different fields of pure and applied mathematics such as number theory, dynamical systems,probability, quantum mechanics and electrodynamics. In recent years the research of stochastic pantograph equations have gradually become a new hot spot and made some notable progress. In the paper, we will introduce the latest research results of the existence and uniqueness, polynomial asymptotic behavior and stability for the solutions of stochastic pantograph equations, based on the main line of the qualitative theory of the stochastic equations.It is well known that in the deterministic situation there is a very special pantograph equationwhere q∈(0,1), in which it is conventional to take x'(t) to denote the right-hand derivative of x.If we take into account action of the external noise for this system and its internal parameters a and b, a linear stochastic pantograph equation will be obtained.where q∈(0,1), a given Brownian motion B(t).Since qt < t when t≥0 , equations (1) and (2) are differential equations with time lag. The quantity x(t -ζ(t)) in the delayed argument of x(t -ζ(t)) will be called the (variable) lag. We note that the argument qt satisfies qt→∞as t→∞so the lag is unbounded, that is t - qt→∞as t→∞. Equations (2) are stochastic differential equations with unbounded delay of Ito type. We shall refer to (1) as the underlying version of (2), and to (2) as the stochastic analogue of(1).The existence and uniqueness of solutions for stochastic differential equationsare the foundations of other theoretical research and practical work, and always is important subjects for stochastic differential equations in theoretical studies. Therefore, first we will introduce the existence and uniqueness of solutionsfor stochastic pantograph equations. In order to facilitate the explanation we will briefly state the conclusions of existence-uniqueness for stochastic differ- ential equations and stochastic functional differential equations.In this paper we assume (Ω, F, P) be a complete probability space with a filtration {F_t}_t≥0 satisfying the usual conditions. Let B(t)_t≥0 be a given one dimensional Brownian motion defined on complete probability space. We give the thm for existence and uniqueness of the strong solutions of the equations (2), and introduce the properties of a strong solution with lemma.Theorem 1. If E|X₀|²<∞, then there exists a path-wise unique strong solution to equation (2).Lemma 1. The solution of equation (2) has the propertyE((?)|X(t)|²)≤C₁(T),withC₁(T) := (1/2+ 3E|X₀|²)exp(18K(T + 4)t), K:=max{|a|,|b|, |σ₁|,|σ₂|,|σ₃|}.Moreover, for any 0≤s < t≤T with t - s < 1,E|X(t)-X(s)|²â‰¤C₂(T)(t-s),whereC₂(T):=12K(1 + 2C₁(T)).Lemma 2. As a consequence of the preceding theorem we obtain the estimate for all t∈[0, T]E|aX(t) + bX(qt)|≤(?),with L := 2max{|a|, |b|}.We consider the following nonlinear stochastic pantograph equation where q∈(0,1). Let B(t) = (B₁(t), B₂(t),…, B_m(t))^T, t≥0, be a m-dimensional Brownian motion complete probability space. Let T > 0, X₀ be a F₀-measurable R^d-valued random variable such that E(|X₀|²) <∞. Let f: R^d×R^d×[0, T]→R^dand g: R^d×R^d×[0, T]→R^d×m be both Borel measurable. We give the following the thm existence-uniqueness theorem and property of solutions.Theorem 2. Assume that there exists a positive constant K such that (i)(Lipschitz condition) For all X₁,Y₁, X₂, Y₂∈R^d and t∈[0, T]|f(X₁,Y₁,t)-f(X₂,Y₂,t)|²âˆ¨|g(X₁,Y₁,t)-g(X₂,Y₂,t)|²â‰¤K(|X₁-X₂|²+|Y₁-Y₂|²).(ii) (Linear growth condition) For all (X, Y, t)∈R^d×R^d×[0, T] |f(X,Y, t)|²âˆ¨|g(X, Y, t)|²â‰¤K(1 +|X|² + |Y|²).Then there exists a unique solution X(t) to equation(3) and E( (?)|X(t)|²)≤∞.Theorem 3. If the Lipschitz condition (i) of theorem 2 is replaced with the following the local Lipschitz condition: for every integer n≥1 there exists a positive constant K_n such that for all t∈[0,T] and X₁, Y₁,X₂, Y₂∈R^d with |X₁|∨|Y₁|∨|X₂|∨|Y₂|≤n|f(X₁,Y₁,t) - f(X₂,Y₂,t)|²âˆ¨|g(X₁,Y₁,t) - g(X₂,Y₂,t)|²â‰¤K_n(|X₁-X₂|²+|Y₁-Y₂|²),then there exists a unique solution X(t) to equation(3) and E( (?) |X(t)|²)≤∞.Lemma 3. Assume that the linear growth condition (ii) holds and X(t) is a solution of equation(3), thenE((?)|X(t)|²)≤(1 + 3E|X₀|²)e^6KT(T+4). A wealth of literature now exists on the non-exponential (general) rates of decay to equilibrium of solutions of differential and functional differential equations,both for deterministic and stochastic equations. Three types of equations which exhibit such general (non-exponential) rates of decay have attracted much attention. These are(i) Non-autonomous perturbations or forcing terms added to linear or near-linear problems (such as quasi-linear, or semi-linear equations).(ii) Nonlinear equations (which have no linear, or linearisable terms near equilibrium), giving rise to weak exponential asymptotic stability.(iii) Certain types of linear equations with unbounded delay.In the deterministic theory, all three mechanisms have been studied extensively.For stochastic differential equations, and functional differential equations, several authors have obtained results in categories (i), (ii), but comparatively few results have been established in category (iii). In category (iii), less is known about the non-exponential asymptotic behaviour of linear stochastic differentialequations with unbounded delay. An important subclass of non-exponential asymptotic behaviour is the so-called polynomial asymptotic behaviour.In the paper, the following consideration is a linear stochastic pantograph equation with multiplicative noise.where a, b,σ₁ andσ₂ are real constants and q∈(0,1). Let (B(t))_t≥0 is a standard one dimensional Brownian motion in complete probability space. The initial value X₀ satisfies .E|X₀|² <∞and is independent of B(t). We expound the asymptotic growth and decay properties of solutions of equation(4). We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in p-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in p-th mean and in the almost sure sense. For a different set of parameters we establish exponential growth rates of solutions in p-th mean and an almost sure sense. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.In this paper, we show that, in common with the deterministic pantograph equation studied in [29], solutions of the stochastic pantograph equation (4) can be bounded by polynomials in both a p-th mean and almost sure sense, and, for values of the parameters a, b,σ₁ andσ₂, we establish polynomial asymptotic stabilityin these senses. Furthermore, it appears, in common with the deterministic pantograph equation, that the polynomial asymptotic behaviour is determined only by the values of the parameters associated with the non-delay terms. We also observe, when the noise intensitiesσ₁,σ₂ are small, that the polynomial asymptotic behaviour of the stochastic problem can be inferred from that of the corresponding deterministic equation. Our analysis involves obtaining estimates on the second mean of the solution of (4) using comparison principle arguments (as in [14, 31]), and then using these estimates to obtain upper bounds on the solution in an almost sure sense, using an idea of Mao [32].We state the definitions for the asymptotic behaviour and results for the deterministicpantograph equation given in Liu and Mao [17] and Mao [33]. Then we concentrate on giving sufficient conditions under which the asymptotics behaviourof the process satisfying (1) are polynomial bounded or stable, in both a p-th mean (p= 1, 2) and almost sure sense.The following theorem will give the sufficient conditions for parameters a, b,σ₁ andσ₂, when the solutions for (4) are polynomial bounded in the first and second mean.Theorem 4. Let (X(t))_t≥0 be the unique process satisfying (4).(i) Letσ₂ = 0, and E(|X₀|²) <∞. If a < 0, there exists a real constantα and a positive constant C such thatE(|X(t)|)≤CE(|X₀|)t^α, for t≥0,whereαis given byα:=1/logq log(-a/|b|) (5)(ii) Letσ₂≠0, and E(|X₀|⁴) <∞. If 2a +σ₁² < 0, there exists a real constantα, and a positive constant C such thatE(|X(t)|²)≤CE(|X₀|²)t^α, for t≥0,whereαis given byα:=2/logq log(1/(σ₂)²((?)-|b +σ₁σ₂|)). (6)We give polynomial stable conditions in the following theorem.Theorem 5. Let (X(t))_t≥0 be the process uniquely defined by (4).(i) Letσ₂= 0, and E(|X₀|²) <∞. If a + |b| < 0, there exists C > 0 andα< 0 such thatE(|X(t)|)≤CE(|X₀|)t^α, t≥0,with a given by (5).(ii) Letσ₂≠0, and E(|X₀|⁴) <∞.If 2a +σ₁² +σ₂² + 2|b +σ₁σ₂| < 0, thereexists C > 0,α< 0 such thatE(|X(t)|²)≤CE(|X₀|²)t^α, t≥0, with a given by (6).In theorem 6, we concentrate on the a.s. polynomial boundedness.Theorem 6. Let (X(t))_t≥0 be the process uniquely defined by (4). (1) Letσ₂ = 0, E(|X₀|²) <∞. If a < 0, then(?)log|X(t)|/logt≤α+1,a.s., where a is defined by (5).(ii) Letσ₂≠0, E(|X₀|⁴) <∞. If 2a +σ₁²< 0, then(?)log|X(t)|/logt≤1/2(α+1),a.s.,whereαis defined by (6).Theorem 7 concerns the a.s. polynomial stability of solutions for (4).Theorem 7. Let (X(t))_t≥0 be the process uniquely defined by (4).(i) Letσ₂ = 0, E(|X₀|²) <∞. If a + |b|/q < 0, thenα, defined by (5), satisfiesα< -1, and we have(?)log|X(t)|/logt≤α+1,a.s.,so X(t)→0, as t→∞, a.s.(ii) Letσ₂≠0, E(|X₀|⁴) <∞. If2a +σ₁²+σ₂²/q+2/(?)|b+σ₁σ₂|<0,thenα,defined by (6), satisfiesα< - 1, and(?)log|X(t)|/logt≤1/2(α+1),a.s.,so X(t)→0, as t→∞, a.s.In [10], it is shown that stochastic delay differential equations, or stochastic functional differential equations with bounded delay, are a.s. bounded by increasingexponential functions, provided that the coefficients of the equation satisfy global linear bounds. More precisely, Mao shows that the top Lyapunov exponentis bounded almost surely by a finite constant. For the stochastic pantograph equation, we similarly show that all solutions have top a.s. and p-th mean (p= 1, 2) Lyapunov exponents which are bounded by finite constants. We consider only parameter regions in which the polynomial boundedness of the solution of (4) has not been established, as the a.s. exponential upper bound (respectively, the p-th mean exponential upper bound) is a direct consequence of the a.s. polynomial boundedness (respectively, p-th mean polynomial boundedness) of the solution.Theorem 8. Let (X(t))_t≥0 be the process satisfying (4).(i) Letσ₂ = 0, E(|X₀|²) <∞. If a>0, thenE(|X(t)|)≤CE(|X₀|e^at,and(?)1/t log |X(t)|≤a a.s.(ii) Letσ₂≠0, E(|X₀|²) <∞. If 2a +σ₁² > 0, then E(|X(t)|²)≤CE(|X₀|²)e^{(2a+σ₁2)t},and(?)1/t log |X(t)|≤a +1/2σ₁² a.s.Finally, we will briefly review stochastic differential equations and stochastic differential delay equations in perspective course of research and development, give analysis of the current research situation, and introduce asymptotic stability and pth moment stability of stochastic pantograph equations.Consider the d-dimensional stochastic pantograph differential equationwith q∈(0,1). Let B(t) = (B₁(t), B₂(t),…, B_m(t))^T, t≥0, be a m-dimensional Brownian motion defined on complete probability space.f:Rⁿ×Rⁿ×R⁺→Rⁿ, g: Rⁿ×Rⁿ×R⁺→R^n×m. Let X(t) be the process satisfying (7), continuous sample vand F_t-measurable.We will give this equation the asymptotically convergence theorem of LaSalle-type. Theorem 9. Let X(t) be the process satisfying (7). Assume that there exists V∈C^2,1(Rⁿ×R⁺; R⁺), r∈L¹(R⁺; R⁺),ω∈C(Rⁿ;R⁺) and a constantδ> 1 such thatLV(x, y, t)≤r(t) -δω(x) + qω(y), (x, y, t)∈Rⁿ×Rⁿ×R⁺, then every X(t), defined by (7), satisfies E(|X₀|²) <∞, and we have(?)V(X(t),t) <∞a.s.and∫₀^∞ω(X(t))dt<∞a.s.Using the theorem 9, we will obtain the asymptotically stable conditions for solutions.Theorem 10. Let X(t) be the process satisfying (7). Assume that thereexists V∈C^2,1(Rⁿ×R⁺; R⁺), r∈L¹(R⁺;R⁺),ω∈C(Rⁿ;R⁺) and a constantδ> 1 such thatLV(x, y, t)≤r(t) -δω(x) + qω(y), (x, y, t)∈Rⁿ×Rⁿ×R⁺,Futhermore assume that there exists continuous and strictly monotone rising u₁, u₂, u₃ : R⁺→R⁺, satisfies u₁(0) = u₂(0) = u₃(0),we haveu₁(|x|)≤V(x, t)≤u₂(|x|) and u₃(|x|)≤ω(x), for x∈Rⁿ and t≥0. So for X₀, satisfies E(|X₀|²) <∞, we have(?)|X(t)| = 0 a.s.We have the following the asymptotically stable conditions of solutions for the linear stochastic pantograph equation (4).Theorem 11. Let X(t) be the process satisfying (4). There existsδ> 1 for (4) such that2a+|b| +σ₁² + |σ₁σ₂| < -δand |b| +|σ₁σ₂|+σ₂²< q, then(?)X(t) = 0 a.s.Let B(t)_t≥0 be a one dimensional Brownian motion defined on complete probabilityspace, and E|X₀|^p <∞,p > 0. We will obtain the following conclusions for the equation (7) using Razumikhin skills.Theorem 12. Assume that there exists positive constants p,λ,C₁,C₂(C₂ > C₁) and continuous V(x, t): R×R⁺→R⁺ with partial derivative V_x,V_t, V_xx such thatC₁|x|^p≤V(x,t)≤C₂|x|^p, for all t∈R⁺,x∈R,then we have(i) The zero solutions of (7) are pth moment stable, ifELV(X(t), t)≤0, for EV(X(qt), qt)≤EV(X(t), t)holds.(ii)The zero solutions of (7) are large-scale pth moment stable, if there existsη>1, such thatELV(X(t), t)≤-λEV(X(t), t), for EV(X(qt), qt) <ηEV(X(t), t), where LV(x, t) = V_xf + V_t + 1/2V_xxg².