Mean-Square Stability of Milstein Methods for Stochastic Pantograph Equations

Abstract

This paper deals with nonlinear stochastic pantograph equations. For solving the equations, a class of extended Milstein methods are suggested. A mean-square stability criterion for this type of equations is presented. It is proved that under the suitable conditions the Milstein methods preserve the mean-square stability. Numerical examples further illustrate the obtained theoretical results.

1. Introduction

Stochastic delay differential equations (SDDEs) are often used to model some problems with aftereffect in many scientific fields such as physics, biology, mechanics, finance, and control theory. Generally speaking, it is hard to obtain the analytical solutions of SDDEs. Hence, recently, many researchers began to study their numerical solutions, and hence, some significant results have been achieved.

The stability analysis plays an important role in construction of excellent numerical algorithms for SDDEs. Hence, it has received wide attention of researchers. The early related results can be found in Mao [1, 2], Baker and Buckwar [3], Buckwar [4, 5], Küchler and Platen [6], and the references therein. More recently, for the linear SDDEs, Cao et al. [7], Liu et al. [8], and Wang and Zhang [9] studied mean-square stability (MS-stability) of Euler-Maruyama, semi-implicit Euler-Maruyama, and Milstein methods, respectively. Taking use of the Halanay inequality, Baker and Buckwar [10] extended the MS-stability analysis of Euler-Maruyama methods to nonlinear SDDEs. Moreover, Wang and Zhang [11] also dealt with nonlinear MS-stability of Milstein methods.

We note that the above numerical stability investigations were mainly devoted to the case of constant delay. Although the deterministic delay differential equations with variable delays have been widely studied (see, e.g., [12, 13] and the references therein), the case of variable delay of SDDEs was rarely concerned. Fan and Liu [14] first studied linear stochastic pantograph equations and gave MS-stability criteria of semi-implicit Euler methods. Also, by taking use of the analytical and discrete Razumikhin theorems, they dealt with 𝛼-moment stability of linear stochastic pantograph equations and their semi-implicit Euler method (cf. [15]). Recently, Xiao et al. [16, 17] gave sufficient MS-stability conditions of backward Euler method and semi-implicit Euler method with variable stepsize for linear stochastic pantograph differential equations. In the present paper, we will investigate the MS-stability of nonlinear stochastic pantograph equations and their Milstein methods. Some criteria for MS-stability of the analytical and numerical solutions will be derived. Numerical experiments will be used to illustrate the obtained theoretical results.

2. MS-Stability of the Analytical Solutions

Let (Ω,𝒜,𝑃) be a complete probability space with a filtration (𝒜_𝑡)_𝑡≥0, which is right-continuous and satisfies that each 𝒜_𝑡(𝑡≥0) contains all 𝑃-null sets in 𝒜, and 𝑤 is a one-dimensional Brownian motion defined on the probability space. Moreover, we introduce the following notations: |⋅|:|𝐴|=trace𝐴^𝑇𝐴();𝐿thetracenormofmatrix𝐴_𝑝Ω,𝑅^𝑑:thefamilyof𝑅^𝑑|𝑥|-valuerandomvariable𝑥with𝐸^𝑝ℒ<∞;_𝑝[𝑎,𝑏],𝑅^𝑑:thefamilyof𝑅^𝑑-value𝒜_𝑡{𝑥(𝑡-adaptedprocesses)}_{𝑎≤𝑡≤𝑏}with^𝑏_𝑎|𝑥(𝑡)|^𝑝ℳ𝑑𝑡<∞a.s.;_𝑝[𝑎,𝑏],𝑅^𝑑:thefamilyofprocesses{𝑥(𝑡)}_𝑡≥0∈ℒ^𝑝[𝑎,𝑏],𝑅^𝑑with𝐸^𝑏_𝑎|𝑥(𝑡)|^𝑝ℒ𝑑𝑡<∞;_𝑝𝑅₊,𝑅^𝑑:thefamilyofprocesses{𝑥(𝑡)}_𝑡≥0with{𝑥(𝑡)}_{0≤𝑡≤𝑇}∈ℒ^𝑝[0,𝑇],𝑅^𝑑ℳ∀𝑇>0;_𝑝𝑅₊,𝑅^𝑑:thefamilyofprocesses{𝑥(𝑡)}_𝑡≥0with{𝑥(𝑡)}_{0≤𝑡≤𝑇}∈ℳ^𝑝[0,𝑇],𝑅^𝑑∀𝑇>0. (1)

Consider the following nonlinear stochastic pantograph equations: (𝑡)(𝑡)(𝑡)(𝑡)𝑥(0)𝑑𝑥=𝑓𝑡,𝑥,𝑥𝑝𝑡𝑑𝑡+𝑔𝑡,𝑥,𝑥𝑝𝑡𝑑𝑤,𝑡>0,=𝜉, (2) where 𝑥(𝑡) is a 𝑅^𝑑-value random process, 𝑝∈(0,1) denotes a given constant, 𝑓∶𝑅₊×𝑅^𝑑×𝑅^𝑑→𝑅^𝑑 and 𝑔∶𝑅₊×𝑅^𝑑×𝑅^𝑑→𝑅^𝑑 are two given Borel-measurable functions, 𝜉 is an 𝒜₀-measurable 𝑅^𝑑-value random variable, and 𝜉∈𝐿²(Ω,𝑅^𝑑). Throughout this paper, we always assume that (2) has a unique solution 𝑥(𝑡)∈ℳ²(𝑅₊,𝑅^𝑑).

Definition 1. The solution of (2) is said to be MS-stable if lim_𝑡→+∞𝐸|𝑥(𝑡)|²=0. (3)

Theorem 2. Assume that there exist constants 𝛼>0, 𝛽≥0, and 𝛾≥0 such that 𝑥^𝑇𝑓(𝑡,𝑥,𝑢)≤−𝛼|𝑥|²+𝛽|𝑢|²,∀𝑥,𝑢∈𝑅^𝑑,𝑔(𝑡,𝑥,𝑢)²≤𝛾|𝑥|²+|𝑢|²,∀𝑥,𝑢∈𝑅^𝑑. (4)(5) Then, the solution of (2) is MS-stable whenever 𝛾−2𝛼+𝛾+2𝛽𝑝<0. (6)

Proof. By the Itô formula (cf. [1]), we have 𝑑|𝑥(𝑡)|²=2𝑥^𝑇(𝑡)𝑓𝑡,𝑥(𝑡),𝑥𝑝𝑡+𝑔𝑡,𝑥(𝑡),𝑥𝑝𝑡²𝑑𝑡+2𝑥^𝑇(𝑡)𝑔𝑡,𝑥(𝑡),𝑥𝑝𝑡𝑑𝑤(𝑡). (7) Integrating from 0 to 𝑡 on both sides of the equality (7) and then taking expectation yield that 𝐸|𝑥(𝑡)|²=𝐸𝜉²+𝐸^𝑡₀2𝑥^𝑇(𝑠)𝑓(𝑠)+𝑔(𝑠)𝑠,𝑥,𝑥𝑝𝑠𝑠,𝑥,𝑥𝑝𝑠²𝑑𝑠+𝐸^𝑡₀2𝑥^𝑇(𝑠)𝑔𝑠,𝑥(𝑠),𝑥𝑝𝑠𝑑𝑤(𝑠). (8) Since 𝑥(𝑡)∈ℳ²(𝑅₊,𝑅^𝑑), we further have 𝐸|𝑥(𝑡)|²=𝐸𝜉²+𝐸^𝑡₀2𝑥^𝑇(𝑠)𝑓(𝑠)+𝑔(𝑠)𝑠,𝑥,𝑥𝑝𝑠𝑠,𝑥,𝑥𝑝𝑠²𝑑𝑠. (9) Applying the conditions (4) and (5) to (9), it follows that 𝐸|𝑥(𝑡)|²≤𝐸𝜉²+𝛾−2𝛼𝐸^𝑡₀|𝑥(𝑠)|²+𝐸𝑑𝑠𝛾+2𝛽^𝑡₀𝑥𝑝𝑠²𝜉𝑑𝑠≤𝐸²+𝛾−2𝛼+𝛾+2𝛽𝑝𝐸^𝑡₀|𝑥(𝑠)|²𝑑𝑠, (10) which gives −𝛾−2𝛼+𝛾+2𝛽𝑝𝐸^𝑡₀|𝑥(𝑠)|²𝑑𝑠≤𝐸𝜉²,∀𝑡>0. (11) This, together with (6), implies lim_𝑡→∞𝐸|𝑥(𝑡)|²=0. Therefore, the theorem is proven.

3. MS-Stability of the Numerical Solutions

For the stability analysis, we introduce the following notational conventions: 𝑔^′₁()=()𝑡,𝑥,𝑢𝜕𝑔𝑡,𝑥,𝑢𝜕𝑥,𝑔^′₂()=()𝑡,𝑥,𝑢𝜕𝑔𝑡,𝑥,𝑢,𝐼𝜕𝑢₁=^𝑡𝑛+1_𝑡𝑛^𝑠_𝑡𝑛𝑑𝑤(𝑟)𝑑𝑤(𝑠)=Δ𝑤_𝑛²−ℎ2,𝐼₂=^𝑡𝑛+1_𝑡𝑛^𝑠_𝑡𝑛𝑑𝑤𝑝𝑟𝑑𝑤(𝑠), (12) where Δ𝑤_𝑛∶=∫^𝑡𝑛+1_𝑡𝑛𝑑𝑤(𝑠)=𝑤(𝑡_𝑛+1)−𝑤(𝑡_𝑛), denoting independent 𝑁(0,ℎ)-distributed Gaussian random variables. Moreover, on space 𝑅^𝑑, we define an inner product ⟨⋅,⋅⟩ and the corresponding induced norm |⋅| as follows: ⟨𝑈,𝑉⟩=^𝑑_𝑖=1𝑢_𝑖𝑣_𝑖,|𝑈|=^𝑑_𝑖=1𝑢²_𝑖, (13) where 𝑈=(𝑢₁,𝑢₂,…,𝑢_𝑑)^𝑇, 𝑉=(𝑣₁,𝑣₂,…,𝑣_𝑑)^𝑇∈𝑅^𝑑.

Applying the Milstein method to (2) derives the following numerical scheme: 𝑥_𝑛+1=𝑥_𝑛+ℎ𝑓𝑡_𝑛,𝑥_𝑛,𝑥_𝑛+𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛Δ𝑤_𝑛+𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₁+𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₂𝑥,𝑛≥0,₀=𝜉, (14) where ℎ>0 is the computational stepsize, Δ𝑤_𝑛=𝑤(𝑡_𝑛+1)−𝑤(𝑡_𝑛), and 𝑥_𝑛, 𝑥_𝑛, and 𝑥_𝑛 are approximations to 𝑥(𝑡_𝑛), 𝑥(𝑝𝑡_𝑛), and 𝑥(𝑝²𝑡_𝑛), respectively. When set 𝑝𝑡_𝑛=𝑛−𝜈_𝑛ℎ+𝛿_𝑛ℎ,𝑝²𝑡_𝑛=𝑛−𝜈_𝑛ℎ+𝛿_𝑛ℎ, (15) where 𝜈_𝑛,𝜈_𝑛∈ℕ and 𝛿_𝑛,𝛿_𝑛∈[0,1), the approximations of 𝑥(𝑝𝑡_𝑛) and 𝑥(𝑝²𝑡_𝑛) can be defined as follows: 𝑥_𝑛=𝛿_𝑛𝑥_{𝑛−𝜈𝑛+1}+1−𝛿_𝑛𝑥_{𝑛−𝜈𝑛},𝑥_𝑛=𝛿_𝑛𝑥_{𝑛−𝜈𝑛+1}+1−𝛿_𝑛𝑥_{𝑛−𝜈𝑛},𝑛≥0. (16) In this way, an extended Milstein method, composed by (14) and (16), is obtained.

Definition 3. An extended Milstein method (14)–(16) is said to be MS-stable if there exists an ℎ₀>0 such that lim_𝑛→+∞𝐸𝑥_𝑛²=0,ℎ∈0,ℎ₀. (17)

Lemma 4. The Itô-type double integrals 𝐼₁, 𝐼₂ have the following properties: 𝐸𝐼₁=𝐸𝐼₂=0,𝐸𝐼₁²=ℎ²2,𝐸𝐼₂²=𝑝ℎ²2. (18)

Proof. The equalities 𝐸[𝐼₁]=𝐸[𝐼₂]=0 can be derived directly from the properties of martingales. Moreover, by the equality 𝐼₁=[(Δ𝑤)²−ℎ]/2, we have 𝐸𝐼₁²=14𝐸Δ𝑤_𝑛²−ℎ²=ℎ²2. (19) Also, it follows from the properties of Itô integral that 𝐸𝐼₂²=𝐸^𝑡𝑛+1_𝑡𝑛^𝑠_𝑡𝑛𝑑𝑤(𝑝𝑟)𝑑𝑤(𝑠)²=^𝑡𝑛+1_𝑡𝑛𝐸^𝑠_𝑡𝑛𝑑𝑤𝑝𝑟²𝑑𝑠=^𝑡𝑛+1_𝑡𝑛^𝑠_𝑡𝑛𝑑𝑝𝑟𝑑𝑠=𝑝ℎ²2. (20) This completes the proof.

Let 𝑞=1−𝑝. Then, we have the following lemma.

Lemma 5. Assume that there exist positive integers 𝑟, 𝜈_𝑖 and 𝛿_𝑖∈[0,1) such that 1𝑟≤𝑞<𝑟+1,𝑖𝑞=𝜈_𝑖−𝛿_𝑖,𝑖=0,1,2,…. (21) Then, the sequence {𝜈_𝑖} is monoincreasing and has at most 𝑟+1 equal components.

Proof. It follows from 𝑖𝑞=𝜈_𝑖−𝛿_𝑖 that 𝜈_𝑖+1+𝛿_𝑖=𝜈_𝑖+𝛿_𝑖+1+𝑞,𝑖=0,1,2,…. (22) Let ⌊⋅⌋ denote the integer part of a real number. Then, by 𝑞,𝛿_𝑖∈[0,1) and 𝜈_𝑖∈ℕ, we have for all 𝑖 that 𝜈_𝑖+1+𝛿_𝑖=𝜈_𝑖+1,𝜈_𝑖+𝛿_𝑖+1+𝑞≥𝜈_𝑖. (23) Hence, it holds that 𝜈_𝑖+1≥𝜈_𝑖,𝑖=0,1,2,…, (24) This shows that the sequence {𝜈_𝑖} is monoincreasing.
For proving the second part of this lemma, we use reduction to absurdity. If the sequence {𝜈_𝑖} has 𝑟+2 components which satisfy that 𝜈_𝑖0=𝜈_𝑖1=⋯=𝜈_𝑖𝑟+1,where0≤𝑖₀<𝑖₁<⋯<𝑖_𝑟+1, (25) then, by 𝜈_𝑖𝑟+1−𝜈_𝑖0=0, 𝛿_𝑖𝑟+1∈[0,1), and 𝑞>1/(𝑟+1), we have 𝛿_𝑖0=𝛿_𝑖𝑟+1+𝑖_𝑟+1−𝑖₀𝑖𝑞≥^𝑟+1−𝑖₀𝑟+1≥1. (26) This is contrary to 𝛿_𝑖0∈[0,1). Hence, Lemma 5 is proven.

With the above lemmas, the main result can be stated as follows.

Theorem 6. Assume that the conditions (4) and (5) hold and that there exist constants 𝜅, 𝑀, and 𝑁≥0 such that 𝑓(𝑡,𝑥,𝑢)²≤𝜅|𝑥|²+|𝑢|²,𝑥,𝑢∈𝑅^𝑑,𝑔^′₁(𝑡,𝑥,𝑢)≤𝑀,𝑔^′₂(𝑡,𝑥,𝑢)≤𝑁,𝑥,𝑢∈𝑅^𝑑. (27)(28) Then, the extended Milstein method (14)–(16) is MS-stable whenever 𝑐₁+2𝑐₂(𝑟+1)+2𝑐₃(𝑟+1)<0, (29) where 𝑐₁=−2𝛼−2𝛾−𝑀²𝛾,𝑐₂=2𝛽+2𝛾+𝑀²𝛾+𝑁²𝛾𝑝,𝑐₃=2𝑁²𝛾𝑝. (30)

Proof. By (14), we have 𝑥_𝑛+1²≤𝑥_𝑛²+2𝑥_𝑛,ℎ𝑓𝑡_𝑛,𝑥_𝑛,𝑥_𝑛+𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛Δ𝑤_𝑛+𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₁+𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₂+ℎ𝑓𝑡_𝑛,𝑥_𝑛,𝑥_𝑛+𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛Δ𝑤_𝑛+𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₁+𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₂²≤𝑥_𝑛²+2ℎ𝑥^𝑇_𝑛𝑓𝑡_𝑛,𝑥_𝑛,𝑥_𝑛+2𝑥^𝑇_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛Δ𝑤_𝑛+2𝑥^𝑇_𝑛𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₁+2𝑥^𝑇_𝑛𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₂+4ℎ²𝑓𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²+4𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²Δ𝑤_𝑛²+4𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝐼₁²+4𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝐼₂². (31) Using conditions (4) and (27) generates 𝐸𝑥^𝑇_𝑛𝑓𝑡_𝑛,𝑥_𝑛,𝑥_𝑛≤−𝛼𝐸𝑥_𝑛²+𝛽𝐸𝑥_𝑛²,𝐸𝑓𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²≤𝜅𝐸𝑥_𝑛²+𝐸𝑥_𝑛², (32)(33) respectively. Moreover, the 𝒜_𝑡𝑛-measurability implies that 𝐸𝑥^𝑇_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛Δ𝑤_𝑛=𝐸𝑥^𝑇_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐸Δ𝑤_𝑛∣𝒜_𝑡𝑛𝐸𝑔𝑡=0,_𝑛,𝑥_𝑛,𝑥_𝑛²Δ𝑤_𝑛²=𝐸𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝐸Δ𝑤_𝑛²∣𝒜_𝑡𝑛≤𝛾ℎ𝐸𝑥_𝑛²+𝐸𝑥_𝑛²,𝐸𝑥^𝑇_𝑛𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₁=𝐸𝑥^𝑇_𝑛𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐸𝐼₁∣𝒜_𝑡𝑛𝐸𝑥=0,_𝑇𝑛𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐼₂=𝐸𝑥^𝑇_𝑛𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛𝐸𝐼₂∣𝒜_𝑡𝑛=0, (34) and a combination of Lemma 4, (5), and (28) gives 𝐸𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝐼₁²=𝐸𝑔^′₁𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝐸𝐼₁²∣𝒜_𝑡𝑛≤12𝑀²𝛾ℎ𝐸𝑥_𝑛²+𝐸𝑥_𝑛²,𝐸𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝐼₂²=𝐸𝑔^′₂𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝑔𝑡_𝑛,𝑥_𝑛,𝑥_𝑛²𝐸𝐼₂²∣𝒜_𝑡𝑛≤12𝑁²𝛾𝑝ℎ𝐸𝑥_𝑛²+𝐸𝑥_𝑛². (35)(36) Taking expectation on both sides of (31) and then substituting (32)–(36) into the obtained inequality yield 𝐸𝑥_𝑛+1²≤𝐸𝑥_𝑛²+𝑐₁+4𝜅ℎℎ𝐸𝑥_𝑛²+𝑐₂+4𝜅ℎℎ𝐸𝑥_𝑛²+𝑐₃ℎ𝐸𝑥_𝑛². (37) Combining (16) and (37) derives 𝐸𝑥_𝑛+1²≤𝐸𝑥_𝑛²+𝑐₁+4𝜅ℎℎ𝐸𝑥_𝑛²+𝑐₂+4𝜅ℎℎ𝛿_𝑛𝐸𝑥_{𝑛−𝜈𝑛+1}²+𝑐₂+4𝜅ℎ×ℎ1−𝛿_𝑛𝐸𝑥_{𝑛−𝜈𝑛}²+𝑐₃ℎ𝛿_𝑛𝐸𝑥_{𝑛−𝜈𝑛+1}²+𝑐₃ℎ1−𝛿_𝑛𝐸𝑥_{𝑛−𝜈𝑛}². (38) An induction to (38) yields 𝐸𝑥_𝑛+1²≤𝐸𝜉²+𝑐₁+4𝜅ℎℎ^𝑛_𝑖=1𝐸𝑥_𝑖²+𝑐₂+4𝜅ℎℎ^𝑛_𝑖=1𝛿_𝑖𝐸𝑥_{𝑖−𝜈𝑖+1}²+𝑐₂+4𝜅ℎℎ×^𝑛_𝑖=11−𝛿_𝑖𝐸𝑥_{𝑖−𝜈𝑖}²+𝑐₃ℎ^𝑛_𝑖=1𝛿_𝑖𝐸𝑥_{𝑖−𝜈𝑖+1}²+𝑐₃ℎ^𝑛_𝑖=11−𝛿_𝑖𝐸𝑥_{𝑖−𝜈𝑖}². (39) Applying Lemma 5 to (39), it follows that 𝐸𝑥_𝑛+1²≤𝐸𝜉²+𝑐₁+4𝜅ℎℎ^𝑛_𝑖=1𝐸𝑥_𝑖²+𝑐₂+4𝜅ℎ(𝑟+1)ℎ^𝑛_𝑖=1𝐸𝑥_𝑖²+𝑐₂+4𝜅ℎ(𝑟+1)ℎ^𝑛_𝑖=0𝐸𝑥_𝑖²+𝑐₃(𝑟+1)ℎ^𝑛_𝑖=1𝐸𝑥_𝑖²+𝑐₃ℎ(𝑟+1)^𝑛_𝑖=0𝐸𝑥_𝑖²≤1+𝑐₂+4𝜅(𝑟+1)+𝑐₃(𝑟+1)𝐸𝜉²+ℎ𝑐₁+2𝑐₂(𝑟+1)+2𝑐₃()()ℎ𝑟+1+4𝜅ℎ+8𝜅𝑟+1^𝑛_𝑖=1𝐸𝑥_𝑖². (40) This shows that the positive series ∑^𝑛_𝑖=1𝐸|𝑥_𝑖|² is bounded when (29) holds and ℎ∈(0,ℎ₀), where ℎ₀=min1,−𝑐₁−2𝑐₂(𝑟+1)−2𝑐₃(𝑟+1)[(.4𝜅1+2𝑟+1)] (41) Therefore, it holds that lim_𝑛→∞𝐸|𝑥_𝑛|²=0. This completes the proof.

4. Numerical Illustration

In this section, we give a numerical example to illustrate the obtained theoretical results. Consider the following stochastic pantograph equation: (𝑡)1𝑑𝑥=−4𝑥(𝑡)1+cos²𝑥𝑡2+1𝑑𝑡5𝑥(𝑡)𝑥𝑡2(𝑡)𝑥(0)𝑑𝑤,𝑡>0,=2. (42) It is easy to verify that the conditions of Theorems 2 and 6 can be satisfied with parameters 1𝛼=41,𝛽=0,𝛾=150,𝜅=,116𝑟=1,𝑀=𝑁=5,ℎ₀=428>162502⁴. (43) Hence, both the solution of (42) and its solving method (14) are all MS-stable.

Applying the extended Milstein method (14)–(16), with stepsizes ℎ=1/2⁴,1/2⁵,1/2⁶,1/2⁷∈(0,ℎ₀], respectively, to (42) on interval [0,20], we can obtain four groups of numerical solutions (see Figure 1), where we take the average of 1000 block samples. Figure 1 shows that the numerical solutions are all stable. However, if we take a larger stepsize, then the numerical stability cannot be assured. This is shown in Figure 2, where stepsizes ℎ=1/2²,1/2³∉(0,ℎ₀] are used, which leads to two groups of unstable solutions.

(a)

(b)

(c)

(d)

Figure 1 

Numerical solutions with stepsizes ℎ=1/24, 1/25, 1/26, 1/27.

(a)

(b)

Figure 2 

Numerical solutions with stepsizes ℎ=1/22, 1/23.

5. Conclusions

In this paper, a class of extended Milstein methods for solving nonlinear stochastic pantograph equations are suggested. A mean-square stability criterion for this type of equations is presented. It is proved that, under the suitable conditions, if the stepsize satisfies the sufficient condition ℎ≤ℎ₀, where ℎ₀ is given by (41), then the Milstein methods preserve the mean-square stability. How does one obtain an exact critical stepsize ℎ₀ such that the method is stable for ℎ∈(0,ℎ₀] and unstable for ℎ∈(ℎ₀,+∞)? This is a difficult problem which keeps open at present. We will work on it in the future research.

Acknowledgments

This work is supported by NSFC (nos. 11171125, 91130003, 11201162, and 11301099), NSFH (no. 2011CDB289), Humanities and Social Science Research Projects in Ministry of Education (13YJA910003), the Doctor Scientific Research Foundation of Guangxi Normal University, and the Fundamental Research Funds for the Central Universities, HUST (2011QN168).