The Global Solutions and Moment Boundedness of Stochastic Multipantograph Equations

We consider the existence of global solutions and their moment boundedness for stochastic multipantograph equations. By the idea of Lyapunov function, we impose some polynomial growth conditions on the coefficients of the equation which enables us to study the boundedness more applicably. Methods and techniques developed here have the potential to be applied in other unbounded delay stochastic differential equations.


Introduction
Delay differential equations (DDEs) play an important role in applied mathematics owing to providing a powerful model of many phenomena, such as some physical applications with noninstant transmission phenomena, neural networks, or other memory processes, and specially biological motivations (e.g., [1][2][3]) like species' growth or incubating time on disease models among many others.
An interesting case of DDEs which is the subject of a lot of papers is the pantograph equation: ẋ () =  (,  () ,  ()) ,  ≥ 0 where 0 <  ≤ 1,  0 ∈ R  .The name ℎ originated from the work of Ockendon and Tayler [4] on the collection of current by the pantograph head of an electric locomotive.The pantograph equations appeared in modeling of various problems such as number theory, astrophysics, nonlinear dynamical systems, biology, economy, quantum mechanics, and electrodynamics.For some applications of this type of equations, we refer to [4][5][6][7][8].
Since any realistic systems are inevitably subject to environmental noise, the stochastic pantograph equation  () =  (,  () ,  ())  +  (,  () ,  ())  () ,  ≥ 0,  (0) =  0 (2) therefore receives more and more attention.Fan et al. [9] have given the sufficient conditions of existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for (2).Appleby and Buckwar [10] have studied the asymptotic growth and decay properties of solutions of the linear stochastic pantograph equation with multiplicative noise.For more literatures we refer the interested reader to [11][12][13].
However, to the best of our knowledge, there are no corresponding numerical and analytical results on stochastic multipantograph equations which also have numerous applications as (2) in engineering and science.It has the form  () =  (,  () ,  ( 1 ) , . . .,  (  ))  +  (,  () ,  ( 1 ) , . . .,  (  ))  () , where 0 <  1 <  2 < ⋅ ⋅ ⋅ <   < 1,  : R + × R (+1) → R  , and  : R + × R (+1) → R × .In this paper, we mainly study the asymptotic properties of the analytic solution of (4).Owing to the fact that the delay is unbounded, many methods which are useful for the bounded delay systems are inefficient or impossible for these systems.For example, some classical techniques such as Lyapunov direct methods in [21][22][23] cannot be transferred directly to the study of boundedness properties for unbounded delay equation (4).By introducing a decay function to control the unbounded delay term, we develop the traditional techniques like Lyapunov direct methods to be applied in the pantograph equations' cases.
It is well known for stochastic differential equations that the linear growth condition plays an important role in suppressing the potential explosion of solutions and guarantees the existence of the global solutions (cf.[22][23][24][25]).This paper, without the linear growth condition, shows that (4) almost surely makes a global solution (,  0 ) and this solution is bounded in the sense lim sup where , ,   , and   are positive constants independent of the initial data  0 .The content of the paper is as follows.In Section 2, we give some necessary notations and useful lemmas.Section 3 is devoted to presenting a general theorem for the existence and boundedness of the global solution.In Section 4, we apply Theorem 4 to obtain two useful criteria which can be easily verifiable in applications.Two examples are provided to show how our results will be applied in Section 5. Further remarks are made to conclude the paper in the final section.

Some Preliminaries
Throughout this paper, unless otherwise specified, we use the following notations.Let (Ω, F, P) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions; that is, it is right continuous and increasing while F 0 contains all P-null sets.Let () be an -dimensional Brownian motion defined on the probability space.Let | ⋅ | be 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 where  = ( where () = ( 1 () ,  2 () , . . .,   ()) ,   () =  (  ) ,  = 1, . . ., .
For coefficients  and , we will impose the following standing assumptions.
To proceed, we need a lemma which will play a crucial role in overcoming the difficulties for the existence of unbounded delays.For the sake of simplicity, we denote where   ,   (1 ≤  ≤ ) are all positive constants and  ≥ 0,   = 1 −   .If  = 0, then we have Lemma 3. Assume that 0 ≤  ≤ .If () is a solution to (4) with initial data  0 ∈ R  , then Proof.Let  denote the left side of (17).We compute The proof is complete.

A General Theorem
In this section, by Lyapunov function techniques, we establish a general theorem for the existence and boundedness of the global solution to (4).
Theorem 4. Assume that there exist positive constants , , , letting the function where Φ  is defined by (15) and L(, , ) is defined by (7).
Proof.For initial data  0 ∈ R  , the proof will be divided into three steps.
Step 1 (existence of the global solution).By Assumption 1, there exists a unique maximal local solution () = (,  0 ) (− 0 ≤  < ) to ( 4), where  is the explosion time.Let  0 be a sufficiently large positive number such that | 0 | ≤  0 .For each integer  ≥  0 , define the stopping time: Clearly,   is increasing and   →  ∞ ≤  as  → ∞.If we can show  ∞ = ∞, a.s., then  = ∞ a.s., which implies the desired result.This is also equivalent to proving that, for any  > 0, P(  ≤ ) → 0 as  → ∞.Letting   =  ∧   , by the Itô formula (19), and noting that Φ  is decreasing in , we have where we have used Lemmas 2 and 3.In this paper, const and  always represent some positive constants whose values are not important.
Journal of Control Science and Engineering Therefore, as required.
Step 2 (moment boundedness).Applying the Itô formula and connecting (19) with Lemmas 2 and 3, we compute which implies that lim sup as desired.
Step 3 (moment boundedness average in time).By (19), applying the Itô formula to () yields which implies the desired (6).The proof is complete.After completing the proof of the general theorem, we continue to examine it in both ways.On one hand, ( 5) and ( 6) are the two main results whose understanding can be enriched as the following corollary shows.Corollary 6.Let () be a positive stochastic process with properties ( 5) and (6).If 0 <  <  < ∞ and 0 <  <  < ∞, then where   and   are positive constants, which may be dependent on   ,   , and , .
Proof.By the Lyapunov inequality, for any th integrable random variable , we have which gives the first result.By the Lyapunov inequality, the Hölder inequality, and (6), lim sup This completes the proof.
On the other hand, condition ( 19) is inconvenient to be checked because it is unrelated to functions  and  explicitly.To make Theorem 4 more applicable, one natural alternative is to look for other simplified conditions on  and .Applying where  ∈ R × is defined in (11).By (19), if we can test where  and  represent positive constants,  > , (||  ) denotes some ℎ() ∈ (R  ) which satisfies || − ℎ() → 0 as || → ∞.By Lemma 2 we can easily decide that sup ∈R  [−||  +(||  )] < ∞, which implies that Theorem 4 will hold.
In the next section, we give some alternative conditions to guarantee Theorem 4, which shows coefficients  and  how to determine existence of global solution to (4) and boundedness of this solution.

Main Results
To match (30), we will impose the following two groups of conditions on the functions  and , which shows that the growth of both  and  is polynomial or controlled by polynomial speed.
For any  ∈ R  ,  ∈ R × , and  ≥ 0, where all parameters are positive.Since  −   is decreasing in , the above conditions will still hold when  is replaced by any   ∈ (0, ].
If  ≤ 2, for any initial data  0 ∈ R  × , we give the following lemma for existence of global solution to (4).
Applying this lemma, we may obtain the following theorem.
Theorem 7 shows that the drift coefficient  makes a dominant role when  ≥ 2; in particular,  only needs to satisfy condition (A 2 ) when  > 2, while in Theorem 9 the diffusion coefficient  is dominant.That is, we depend on the environmental noise to suppress the explosion of solutions and guarantee the boundedness.However, we obtained that the order of moment is lower than Theorem 7.

Further Remarks
This paper is devoted to the asymptotic properties of the stochastic multipantograph equations.We investigate the existence and uniqueness of the global solution and its moment boundedness.Besides obtaining a general theorem, we obtain two sufficient criteria which can be much more easily verifiable than the general theorem.Two examples demonstrate our results.
Since (3) arises in the analysis of the dynamics of an overhead current collection system for an electric locomotive and applied to engineering and applied mathematics fields, the asymptotic behavior of stochastic multipantograph equation has meaningful interpretations (cf.[4,26]).The main idea and the method developed in this paper have the potential to investigate some other unbounded delay stochastic systems, such as neural networks, infinite-delay Kolmogorov-type systems, and Volterra equations in mathematical biology.

Remark 5 .
In existence of the global solution, it is not necessary to specify () = ||  .If the function () satisfies lim inf ||→∞ () = ∞, existence of the global solution still holds.