Stability of numerical method for semi-linear stochastic pantograph differential equations
- Yu Zhang^{1} and
- Longsuo Li^{1}Email author
https://doi.org/10.1186/s13660-016-0971-x
© Zhang and Li 2016
Received: 7 August 2015
Accepted: 14 January 2016
Published: 28 January 2016
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 semi-linear stochastic pantograph differential equations. Some suitable conditions for the mean-square stability of an analytical solution are obtained. Then we proved the general mean-square stability of the exponential Euler method for a numerical solution of semi-linear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is mean-square stable for arbitrary step-size \(h>0\). Numerical examples further illustrate the obtained theoretical results.
Keywords
semi-linear stochastic pantograph differential equations exponential Euler method mean-square stability general mean-square stability1 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 mean-square 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 mean-square stability of the Milstein method for stochastic pantograph differential equations under suitable conditions. Zhou [15] showed that the Euler-Maruyama method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions, and the backward Euler-Maruyama 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 step-size \(h>0\). Namely, the exponential Euler method for semi-linear stochastic pantograph differential equations is general mean-square stable.
2 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 P-null 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 n-dimensional 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 \(\min\{a,b\}\).
3 Mean-square stability of analytical solution
In this part, we illustrate the mean-square stability of the analytical solution for semi-linear 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]
- (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$$\begin{aligned}& \bigl|f(t,x_{1},y_{1})-f(t,x_{2},y_{2})\bigr|^{2} \vee\bigl|g(t,x_{1},y_{1})-g(t,x_{2},y_{2})\bigr|^{2} \\& \quad\leq K\bigl(|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2} \bigr); \end{aligned}$$(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 Lthere 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 \(E\int_{0}^{t}|x(t)|^{2}<\infty\).$$ \bigl|f(t,x,y)\bigr|^{2}\vee\bigl|g(t,x,y)\bigr|^{2}\leq L \bigl(1+|x|^{2}+|y|^{2}\bigr), $$(7)
Definition 3.1
Definition 3.2
[21]
Theorem 3.1
Proof
4 General mean-square stability of numerical solution of the exponential Euler method
We introduce the exponential Euler method for semi-linear stochastic pantograph differential equations in this section.
Definition 4.1
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 mean-square stable.
Proof
Remark 4.1
Remark 4.2
5 Numerical example
6 Conclusions
In this paper, we investigate the stability of analytical solutions and numerical solutions for a class of semi-linear stochastic pantograph differential equations. We not only obtain the mean-square stability of the analytical solution under some sufficient conditions but we also prove the general mean-square stability of numerical solution. That is, if the semi-linear stochastic pantograph differential equation is stable, then the exponential Euler method applied to the system is mean-square stable for any step-size \(h>0\).
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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