# Mean-Square Stability of Milstein Methods for Stochastic Pantograph Equations

Authors: Feiyan Xiao; Tingting Qin; Chengjian Zhang
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 (𝒜t)t≥0, which is right-continuous and satisfies that each 𝒜t(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:
(1)|·|:|A|=trace(ATA)(thetracenormofmatrixA);Lp(Ω,Rd):thefamilyofRd-valuerandomvariablexwithE|x|p<∞;ℒp([a,b],Rd):thefamilyofRd-value𝒜t-adaptedprocesses{x(t)}a≤t≤bwith∫ab‍|x(t)|pdt<∞a.s.;ℳp([a,b],Rd):thefamilyofprocesses{x(t)}t≥0∈ℒp([a,b],Rd)withE∫ab‍|x(t)|pdt<∞;ℒp(R+,Rd):thefamilyofprocesses{x(t)}t≥0with{x(t)}0≤t≤T∈ℒp([0,T],Rd)fggggffgffgffffff∀T>0;ℳp(R+,Rd):thefamilyofprocesses{x(t)}t≥0with{x(t)}0≤t≤T∈ℳp([0,T],Rd)fggggfgfgffffffff∀T>0.Consider the following nonlinear stochastic pantograph equations:(2)dx(t)=f(t,x(t),x(pt))dt+g(t,x(t),x(pt))dw(t),t>0,x(0)=ξ,
where x(t) is a Rd-value random process, p∈(0,1) denotes a given constant, f:R+×Rd×Rd→Rd and g:R+×Rd×Rd→Rd are two given Borel-measurable functions, ξ is an 𝒜0-measurable Rd-value random variable, and ξ∈L2(Ω,Rd). Throughout this paper, we always assume that (2) has a unique solution x(t)∈ℳ2(R+,Rd).Definition 1.
The solution of (2) is said to be MS-stable if
(3)limt→+∞E|x(t)|2=0.Theorem 2.
Assume that there exist constantsα>0, β≥0, and γ≥0 such that
(4)xTf(t,x,u)≤-α|x|2+β|u|2,∀x,u∈Rd,(5)|g(t,x,u)|2≤γ(|x|2+|u|2),∀x,u∈Rd.
Then, the solution of (2) is MS-stable whenever
(6)γ-2α+γ+2βp<0.Proof.
By the Itô formula (cf. [1]), we have
(7)d|x(t)|2=[2xT(t)f(t,x(t),x(pt))+|g(t,x(t),x(pt))|2]dt+2xT(t)g(t,x(t),x(pt))dw(t).
Integrating from 0 to t on both sides of the equality (7) and then taking expectation yield that
(8)E|x(t)|2=E|ξ|2+E∫0t‍[2xT(s)f(s,x(s),x(ps))ffffffff+|g(s,x(s),x(ps))|2]ds+E∫0t‍2xT(s)g(s,x(s),x(ps))dw(s).
Since x(t)∈ℳ2(R+,Rd), we further have
(9)E|x(t)|2=E|ξ|2+E∫0t[2xT(s)f(s,x(s),x(ps))ffffffff+|g(s,x(s),x(ps))|2]ds.
Applying the conditions (4) and (5) to (9), it follows that
(10)E|x(t)|2≤E|ξ|2+(γ-2α)E∫0t‍|x(s)|2ds+(γ+2β)E∫0t‍|x(ps)|2ds≤E|ξ|2+(γ-2α+γ+2βp)E∫0t‍|x(s)|2ds,
which gives
(11)-(γ-2α+γ+2βp)E∫0t‍|x(s)|2ds≤E|ξ|2,∀t>0.
This, together with (6), implies limt→∞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:(12)g1′(t,x,u)=∂g(t,x,u)∂x,g2′(t,x,u)=∂g(t,x,u)∂u,I1=∫tntn+1‍∫tns‍dw(r)dw(s)=(Δwn)2-h2,I2=∫tntn+1‍∫tns‍dw(pr)dw(s),
where Δwn≔∫tntn+1dw(s)=w(tn+1)-w(tn), denoting independent N(0,h)-distributed Gaussian random variables. Moreover, on space Rd, we define an inner product 〈·,·〉 and the corresponding induced norm |·| as follows:
(13)〈U,V〉=∑i=1d‍uivi,|U|=∑i=1d‍ui2,
where U=(u1,u2,…,ud)T, V=(v1,v2,…,vd)T∈Rd.Applying the Milstein method to (2) derives the following numerical scheme:
(14)xn+1=xn+hf(tn,xn,x-n)+g(tn,xn,x-n)Δwn+g1′(tn,xn,x-n)g(tn,xn,x-n)I1+g2′(tn,xn,x-n)g(tn,x-n,x^n)I2,n≥0,x0=ξ,
where h>0 is the computational stepsize, Δwn=w(tn+1)-w(tn), and xn, x¯n, and x^n are approximations to x(tn), x(ptn), and x(p2tn), respectively. When set
(15)ptn=(n-νn)h+δnh,p2tn=(n-ν¯n)h+δ¯nh,
where νn,ν¯n∈ℕ and δn,δ¯n∈[0,1), the approximations of x(ptn) and x(p2tn) can be defined as follows:
(16)x¯n=δnxn-νn+1+(1-δn)xn-νn,x^n=δ¯nxn-ν¯n+1+(1-δ¯n)xn-ν¯n,n≥0.
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
(17)limn→+∞E|xn|2=0,h∈(0,h0].Lemma 4.
The Itô-type double integralsI1, I2 have the following properties:
(18)E[I1]=E[I2]=0,E|I1|2=h22,E|I2|2=ph22.Proof.
The equalitiesE[I1]=E[I2]=0 can be derived directly from the properties of martingales. Moreover, by the equality I1=[(Δw)2-h]/2, we have
(19)E|I1|2=14E[(Δwn)2-h]2=h22.
Also, it follows from the properties of Itô integral that
(20)E|I2|2=E[∫tntn+1‍∫tns‍dw(pr)dw(s)]2=∫tntn+1‍E[∫tns‍dw(pr)]2ds=∫tntn+1∫tns‍d(pr)ds=ph22.‍
This completes the proof.Letq=1-p. Then, we have the following lemma.Lemma 5.
Assume that there exist positive integersr, νi and δi∈[0,1) such that
(21)r≤1q<r+1,iq=νi-δi,i=0,1,2,….
Then, the sequence {νi} is monoincreasing and has at most r+1 equal components.Proof.
It follows fromiq=νi-δi that
(22)νi+1+δi=νi+δi+1+q,i=0,1,2,….
Let ⌊·⌋ denote the integer part of a real number. Then, by q,δi∈[0,1) and νi∈ℕ, we have for all i that
(23)⌊νi+1+δi⌋=νi+1,⌊νi+δi+1+q⌋≥νi.
Hence, it holds that
(24)νi+1≥νi,i=0,1,2,…,
This shows that the sequence {νi} is monoincreasing.
For proving the second part of this lemma, we use reduction to absurdity. If the sequence{νi} has r+2 components which satisfy that
(25)νi0=νi1=⋯=νir+1,where0≤i0<i1<⋯<ir+1,
then, by νir+1-νi0=0, δir+1∈[0,1), and q>1/(r+1), we have
(26)δi0=δir+1+(ir+1-i0)q≥ir+1-i0r+1≥1.
This is contrary to δi0∈[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
(27)|f(t,x,u)|2≤κ(|x|2+|u|2),x,u∈Rd,(28)|g1′(t,x,u)|≤M,|g2′(t,x,u)|≤N,x,u∈Rd.
Then, the extended Milstein method (14)–(16) is MS-stable whenever
(29)c1+2c2(r+1)+2c3(r+1)<0,
where
(30)c1=-2(α-2γ-M2γ),c2=2(β+2γ+M2γ+N2γp),c3=2N2γp.Proof.
By (14), we have
(31)|xn+1|2≤|xn|2+2〈g2′(tn,xn,x-n)g(tn,x-n,x^n)I2]xn,hf(tn,xn,x-n)+g(tn,xn,x-n)Δwnfffff+g1′(tn,xn,x-n)g(tn,xn,x-n)I1fffff+g2′(tn,xn,x-n)g(tn,x-n,x^n)I2〉+|hf(tn,xn,x-n)+g(tn,xn,x-n)Δwnffff+g1′(tn,xn,x-n)g(tn,xn,x-n)I1ffff+g2′(tn,xn,x-n)g(tn,x-n,x^n)I2|2≤|xn|2+2hxnTf(tn,xn,x-n)+2xnT[g(tn,xn,x-n)Δwn]+2xnT[g1′(tn,xn,x-n)g(tn,xn,x-n)I1]+2xnT[g2′(tn,xn,x-n)g(tn,x-n,x^n)I2]+4h2|f(tn,xn,x-n)|2+4|g(tn,xn,x-n)|2|Δwn|2+4|g1′(tn,xn,x-n)|2|g(tn,xn,x-n)|2|I1|2+4|g2′(tn,xn,x-n)|2|g(tn,x-n,x^n)|2|I2|2.
Using conditions (4) and (27) generates
(32)E[xnTf(tn,xn,x¯n)]≤-αE|xn|2+βE|x¯n|2,(33)E|f(tn,xn,x¯n)|2≤κ(E|xn|2+E|x¯n|2),
respectively. Moreover, the 𝒜tn-measurability implies that
(34)E[xnT[g(tn,xn,x¯n)Δwn]]=E[xnT[g(tn,xn,x¯n)E(Δwn∣𝒜tn)]]=0,E[|g(tn,xn,x¯n)|2|Δwn|2]=E[|g(tn,xn,x¯n)|2E(|Δwn|2∣𝒜tn)]≤γh(E|xn|2+E|x¯n|2),E[xnT[g1′(tn,xn,x¯n)g(tn,xn,x¯n)I1]]=E[xnT[g1′(tn,xn,x¯n)g(tn,xn,x¯n)E(I1∣𝒜tn)]]=0,E[xnT[g2′(tn,xn,x¯n)g(tn,x¯n,x^n)I2]]=E[xnT[g2′(tn,xn,x¯n)g(tn,x¯n,x^n)E(I2∣𝒜tn)]]=0,
and a combination of Lemma 4, (5), and (28) gives
(35)E[|g1′(tn,xn,x¯n)|2|g(tn,xn,x¯n)|2|I1|2]=E[|g1′(tn,xn,x¯n)|2|g(tn,xn,x¯n)|2E(|I1|2∣𝒜tn)]≤12M2γh(E|xn|2+E|x¯n|2),(36)E[|g2′(tn,xn,x¯n)|2|g(tn,x¯n,x^n)|2|I2|2]=E[|g2′(tn,xn,x¯n)|2|g(tn,x¯n,x^n)|2E(|I2|2∣𝒜tn)]≤12N2γph(E|x¯n|2+E|x^n|2).
Taking expectation on both sides of (31) and then substituting (32)–(36) into the obtained inequality yield
(37)E|xn+1|2≤E|xn|2+(c1+4κh)hE|xn|2+(c2+4κh)hE|x¯n|2+c3hE|x^n|2.
Combining (16) and (37) derives
(38)E|xn+1|2≤E|xn|2+(c1+4κh)hE|xn|2+(c2+4κh)hδnE|xn-νn+1|2+(c2+4κh)×h(1-δn)E|xn-νn|2+c3hδ¯nE|xn-ν¯n+1|2+c3h(1-δ¯n)E|xn-ν¯n|2.
An induction to (38) yields
(39)E|xn+1|2≤E|ξ|2+(c1+4κh)h∑i=1n‍E|xi|2+(c2+4κh)h∑i=1n‍δiE|xi-νi+1|2+(c2+4κh)h×∑i=1n‍(1-δi)E|xi-νi|2+c3h∑i=1n‍δ¯iE|xi-ν¯i+1|2+c3h∑i=1n‍(1-δ¯i)E|xi-ν¯i|2.
Applying Lemma 5 to (39), it follows that
(40)E|xn+1|2≤E|ξ|2+(c1+4κh)h∑i=1n‍E|xi|2+(c2+4κh)(r+1)h∑i=1n‍E|xi|2+(c2+4κh)(r+1)h∑i=0n‍E|xi|2+c3(r+1)h∑i=1n‍E|xi|2+c3h(r+1)∑i=0n‍E|xi|2≤[1+(c2+4κ)(r+1)+c3(r+1)]E|ξ|2+h[c1+2c2(r+1)+2c3(r+1)+4κhgggggg+8κ(r+1)h2c3(r+1)]∑i=1n‍E|xi|2.
This shows that the positive series ∑i=1n‍E|xi|2 is bounded when (29) holds and h∈(0,h0), where
(41)h0=min{1,-c1-2c2(r+1)-2c3(r+1)4κ[1+2(r+1)]}.
Therefore, it holds that limn→∞E|xn|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:(42)dx(t)=-14x(t)[1+cos2x(t2)]dt+15x(t)x(t2)dw(t),t>0,x(0)=2.
It is easy to verify that the conditions of Theorems 2 and 6 can be satisfied with parameters
(43)α=14,β=0,γ=150,κ=116,r=1,M=N=15,h0=4286250>124.
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.Numerical solutions with stepsizesh=1/24, 1/25, 1/26, 1/27.
(a)
(b)
(c)
(d)Numerical solutions with stepsizesh=1/22, 1/23.
(a)
(b)


## 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 conditionh≤h0, where h0 is given by (41), then the Milstein methods preserve the mean-square stability. How does one obtain an exact critical stepsize h~0 such that the method is stable for h∈(0,h~0] and unstable for h∈(h~0,+∞)? This is a difficult problem which keeps open at present. We will work on it in the future research.



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