Abstract
As a particular expression of stochastic delay differential equations, stochastic pantograph differential equations have been widely used in nonlinear dynamics, quantum mechanics, and electrodynamics. In this paper, we mainly study the stability of analytical solutions and numerical solutions of semilinear stochastic pantograph differential equations. Some suitable conditions for the meansquare stability of an analytical solution are obtained. Then we proved the general meansquare stability of the exponential Euler method for a numerical solution of semilinear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is meansquare stable for arbitrary stepsize \(h>0\). Numerical examples further illustrate the obtained theoretical results.
Introduction
Stochastic delay differential equations played an important role in application areas, such as physics, biology, economics, and finance [1–4]. Stochastic pantograph differential equations are particular cases of stochastic unbounded delay differential equations, Ockendon and Tayler [5] found how the electric current is collected by the pantograph of an electric locomotive, therefore one speaks of stochastic pantograph differential equations.
In recent years, as one of the most important characteristics of stochastic systems, the stability analysis caused much more attention [6–10]. Generally speaking, due to the characteristics of stochastic differential equations themselves, it is difficult for us to get analytical solution of equations, therefore, researching the proper numerical methods for a numerical solution has certain theoretical value and practical significance. However, the research for the numerical solution of stochastic pantograph differential equations is still rare. Fan [11] investigated meansquare asymptotic stability of the θ method for linear stochastic pantograph differential equations. Hua [12, 13] developed an almost surely asymptotic stability analytical solution and numerical solution for neutral stochastic pantograph differential equations. Xiao [14] proved meansquare stability of the Milstein method for stochastic pantograph differential equations under suitable conditions. Zhou [15] showed that the EulerMaruyama method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions, and the backward EulerMaruyama method can reproduce almost surely exponential stability for highly nonlinear stochastic pantograph differential equations. The numerical research for stochastic pantograph differential equations has just begun, and the stability analysis of the numerical solution of the equations needs further perfection and development.
Unfortunately, some conditions of stability are somewhat restrictive as applied to practical applications. This paper mainly proves that if an analytical solution is stable, then so is the exponential Euler method applied to the system for any stepsize \(h>0\). Namely, the exponential Euler method for semilinear stochastic pantograph differential equations is general meansquare stable.
Exponential Euler scheme for stochastic pantograph differential equations
Throughout this paper, unless otherwise specified, let \((\Omega,\mathcal {F},P)\) be a complete probability space with a filtration \((\mathcal {F}_{t})_{t\geq0}\), which is increasing and right continuous, and \(\mathcal{F}_{0}\) contains all Pnull sets. \(W(t)\) is Wiener process defined on the probability space, which may be \(\mathcal{F}_{t}\)adapted and independent of \(\mathcal{F}_{0}\). Let \(\mid\cdot\mid\) be the Euclidean norm. The inner product of x, y in \(\mathbb{R}^{n}\) is denoted by \(\langle x,y\rangle\) or \(x^{T}y\), \(\mathbb{R}^{n}\) is ndimensional Euclidean space. If A is a vector or matrix, its transpose is denoted by \(A^{T}\), if A is a matrix, the trace norm of the matrix A is \(A=\sqrt{\operatorname{trace}(A^{T}A)}\). We use \(a\vee b\) and \(a\wedge b\) to denote \(\max\{a,b\}\) and \(\min\{a,b\}\).
We first introduce the exponential Euler method [16, 17] for a semilinear ordinary differential equation,
Making use of method of variation of constant, the expression of the solution is
Applying the exponential RungeKutta method to equation (1),
where
\(L_{j}(\tau)\) is the Lagrange interpolating polynomial, \(c_{1}, c_{2},\ldots, c_{p}\) are nodes, \(u_{n}\), \(u_{n,i}\) are approximate values of \(u(t_{n})\), and \(u(t_{n}+c_{i}h)\), letting \(B_{n,i}=g(t_{n}+c_{i}h,u_{n,i})\), then the numerical scheme can be written as
When \(i=1\), the numerical scheme of the firstorder exponential RungeKutta method is
That is,
Hence, \(u_{n+1}=e^{Ah}u_{n}+e^{Ah}g(t_{n},u_{n})h\) is called the numerical scheme of the exponential Euler method.
Then, consider the following semilinear stochastic pantograph differential equations:
where \(t>0\), \(0< p<1\), ξ is the initial function, \(W(t)\) is a Wiener process, \(f:\mathbb{R}^{+}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) and \(g:\mathbb{R}^{+}\times\mathbb{R}^{n}\times\mathbb {R}^{n}\rightarrow\mathbb{R}^{n}\) are two given Borelmeasurable functions, and f and g are called drift coefficient and diffusion coefficient, respectively. \(A\in\mathbb{R}^{n\times n}\) is the generator of a strongly continuous analytical semigroup \(S=(S(t))_{t\geq0}\) [18]. By the definition of the stochastic differential equations, equation (2) can be rewritten as the following stochastic integral equation:
We can derive numerical schemes by [19]. From this, we have
if we choose the interval to approximate the integrals in the drift and diffusion terms, we obtain
where the initial value \(\xi=x_{0}\), \(x_{n}\) is an approximation to analytical solution \(x(t_{n})\), which is \(\mathcal {F}_{t_{n}}\)measurable, \(h>0\) is the given stepsize, and \(h=t_{n+1}t_{n}\), \(\triangle W_{n}=W(t_{n+1})W(t_{n})\) are independent \(N(0,h)\) distributed stochastic variables. So equation (5) is called the exponential Euler scheme for semilinear stochastic pantograph differential equations.
Meansquare stability of analytical solution
In this part, we illustrate the meansquare stability of the analytical solution for semilinear stochastic pantograph differential equations under some suitable conditions. First of all, in order to consider the existence and uniqueness of the solution for equation (2), we impose the following assumption.
Assumption 3.1
[20]
We assume that f, g are sufficiently smooth and satisfy the Lipschitz condition and the linear growth condition, that is,

(1)
(Lipschitz condition) for all \(x_{1}, x_{2}, y_{1}, y_{2}\in\mathbb {R}^{n}\), there exist a positive constant K, and \(t\in[0,T]\), such that
$$\begin{aligned}& \biglf(t,x_{1},y_{1})f(t,x_{2},y_{2})\bigr^{2} \vee\biglg(t,x_{1},y_{1})g(t,x_{2},y_{2})\bigr^{2} \\& \quad\leq K\bigl(x_{1}x_{2}^{2}+y_{1}y_{2}^{2} \bigr); \end{aligned}$$(6) 
(2)
(linear growth condition) for all \((t,x,y)\in[0,T]\times\mathbb {R}^{n}\times\mathbb{R}^{n}\), and assuming there exists a positive constant L
$$ \biglf(t,x,y)\bigr^{2}\vee\biglg(t,x,y)\bigr^{2}\leq L \bigl(1+x^{2}+y^{2}\bigr), $$(7)there exists a unique solution \(x(t)\) to equation (2) and the solution belongs to \(\mathcal{M}^{2}([0,T];\mathbb{R})\), namely \(x(t)\) satisfies \(E\int_{0}^{t}x(t)^{2}<\infty\).
Definition 3.1
The solution of equation (2) is said to be meansquare stable if
Definition 3.2
[21]
\(\mu[A]\) is a logarithmic norm of the matrix A, the definition is as follows:
Particularly, if \(\\cdot\\) denotes the inner norm, \(\mu [A]\) can be written
Theorem 3.1
Assume that the condition (6) holds, assume \(\mu[A]\) and K satisfy
Then the analytical solution of equation (2) is meansquare stable.
Proof
By the Itô formula [22], we have
According to condition (6) and the inequality \(2ab\leq a^{2}+b^{2}\), we have
Combining with Definition 3.2, we can obtain
Integrating from 0 to t on both sides of the above inequality, it turns into
Taking the expectation,
Together with condition \(1+2\mu[A]+2K+\frac{2K}{p}<0\), we have
The analytical solution is meansquare stable. Therefore, the theorem is proven. □
General meansquare stability of numerical solution of the exponential Euler method
We introduce the exponential Euler method for semilinear stochastic pantograph differential equations in this section.
Definition 4.1
For any stepsize \(h>0\), if the exponential Euler method to equation (2) generates a numerical approximation that satisfies
then the numerical method applied to equation (2) is said to be general meansquare stable.
Theorem 4.1
Suppose that the conditions (6) and (9) hold, for arbitrary \(h>0\), then the numerical solution of the exponential Euler method is general meansquare stable.
Proof
According to equation (5) and taking squares on both sides, we can get
Taking the expectation and substituting condition (6), we obtain
We still note that \(E(\triangle W_{n})=0\), \(E[(\triangle W_{n})^{2}]=h\), and \(x_{n}\), \(x_{[pn]}\) are \(\mathcal{F}_{t_{n}}\) measurable, then
Similarly
Equation (11) turns into
where
Then
and if
the exponential Euler method is meansquare stable. Then we verify that (12) holds under the conditions (9) and the following inequality. We all know that
if
Simplifying equation (13), we obtain
Let
Due to \(0< p<1\) and condition (9),
We can see \(\mu[A]<0\), it is easy to know that
when \(h>0\). We have the monotonicity of the function, namely, \(m(h)< m(0)\) and \(m(0)=1+2\mu[A]+4K<0\). Hence, equation (13) holds and this implies that
and (12) holds.
So
Because of \(B_{1}+B_{2}<1\), it is not difficult to see that \(Ex_{n}^{2}\leq Ex_{[pn]}^{2}\), therefore
as k tends to infinity, \((B_{1}+B_{2})^{k}<1\). Then \(\lim_{n\rightarrow\infty }Ex_{n}^{2}=0\), the exponential Euler method is general meansquare stable. This completes the proof. □
Remark 4.1
When \(p=1\), equation (2) turns into
For convenience, we consider the scalar semilinear stochastic pantograph differential equation
where \(a<0\), if conditions (6) and \(2a+2\sqrt{K}+K<0\) hold, for any stepsize \(h>0\), the exponential Euler method is stable. This result was demonstrated by Shi and Xiao [23].
Remark 4.2
Consider the following scalar stochastic pantograph differential equation:
Take \(a=5\), \(b=1\), \(c=2\). It is easy to see the coefficients satisfy the condition \(a< \frac{1}{2}(b+c)^{2}\). Using the exponential Euler method for (17), we get
Squaring both sides of (18), taking the expectation, and using the inequality \(2ab\leq a^{2}+b^{2}\), we have
Namely
Use the inequality \(e^{10h}>1+10h\). So
The coefficients \(\frac{1+9h}{1+10h}<1\). According to Theorem 4.1, the numerical solution produced by the exponential Euler method is meansquare stable for any stepsize \(h>0\).
Numerical example
We will use numerical example to prove the effectiveness of the exponential Euler method. Consider the following stochastic pantograph differential equation:
If the coefficients of equation (19) satisfy
then the solution of (19) is meansquare stable.
Case 1. We choose the coefficients of the test equation (19) as \(a_{1}=1.5\), \(a_{2}=3\), \(b_{1}=1\), \(b_{2}=0.5\), and \(p=0.5\). Obviously, the coefficients do not satisfy the condition (20). Numerical solutions produced by the exponential Euler method with \(h_{1}=0.05\), \(h_{2}=0.5\) are shown Figure 1. It is easy to see that numerical solutions are not meansquare stable.
Case 2. Taking the coefficients as \(a_{1}=6.5\), \(a_{2}=1\), \(b_{1}=1\), \(b_{2}=0.5\), and \(p=0.5\). The coefficients satisfy the condition (20). Namely, the analytical solution is stable. We used Matlab to randomly generate 50,000 discrete trajectories, getting the meansquare value of 50,000 trajectories at the same time, that is,
where \(y_{j}^{i}\) is numerical solution of i trajectories at the time \(t_{j}\). Apply the exponential Euler method with stepsize \(h_{3}=0.05\), \(h_{4}=0.5\), \(h_{5}=1.5\), and \(h_{6}=2.5\) as shown Figure 2. We observe that numerical solutions produced by the exponential Euler method with arbitrary stepsizes \(h>0\) are all stable.
Case 3. Considering (17) and taking \(p=0.5\). We can know that numerical solutions produced by the Euler Maruyama method are not stable under \(h=0.2\), \(h=0.5\) (see [24]). While, under the same stepsize, the exponential Euler numerical solutions are stable as shown Figure 3. It is proved that the exponential Euler method is more advantageous than the Euler Maruyama method in certain cases.
Conclusions
In this paper, we investigate the stability of analytical solutions and numerical solutions for a class of semilinear stochastic pantograph differential equations. We not only obtain the meansquare stability of the analytical solution under some sufficient conditions but we also prove the general meansquare stability of numerical solution. That is, if the semilinear stochastic pantograph differential equation is stable, then the exponential Euler method applied to the system is meansquare stable for any stepsize \(h>0\).
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YZ carried out stochastic differential equations studies, analyzed, and drafted the manuscript. LSL participated in its design and coordination and helped to analyze the manuscript. All authors read and approved the final manuscript.
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Zhang, Y., Li, L. Stability of numerical method for semilinear stochastic pantograph differential equations. J Inequal Appl 2016, 30 (2016). https://doi.org/10.1186/s136600160971x
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Keywords
 semilinear stochastic pantograph differential equations
 exponential Euler method
 meansquare stability
 general meansquare stability