# Convergence of the Euler Method of Stochastic Differential Equations with Piecewise Continuous Arguments

Authors: Ling Zhang; Minghui Song
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The main purpose of this paper is to investigate the strong convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). Firstly, it is proved that the Euler approximation solution converges to the analytic solution under local Lipschitz condition and the boundedpth moment condition. Secondly, the Euler approximation solution converge to the analytic solution is given under local Lipschitz condition and the linear growth condition. Then an example is provided to show which is satisfied with the monotone condition without the linear growth condition. Finally, the convergence of numerical solutions to SEPCAs under local Lipschitz condition and the monotone condition is established.


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## Body

## 1. Introduction

Recently, differential equations with piecewise continuous arguments (EPCAs) have attracted much attention, and many useful conclusions have been obtained. These systems have applications in certain biomedical models, control systems with feedback delay in the work of Cooke and Wiener [1]. The general theory and basic results for EPCAs have by now been thoroughly investigated in the book of Wiener [2]. Song et al. [3] deal with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments. A typical EPCA contains arguments that are constant on certain intervals. The solutions are determined by a finite set of initial data, rather than by an initial function, as in the case of general functional differential equation. A solution is defined as a continuous, sectionally smooth function that satisfies the equation within these intervals. Continuity of a solution at a point joining any two consecutive intervals leads to recursion relations for the solution at such points. Hence, EPCAs represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations.However, up to now there are few people who have considered the influence of noise to EPCAs. Actually, the environment and accidental events may greatly influence the systems. Thus analyzing SEPCAs is an interesting topic both in theory and applications. There is in general no explicit solution to an SEPCA, hence numerical solutions are required in practice. Numerical solutions to stochastic differential equations (SDEs) have been discussed under the local Lipschitz condition and the linear growth condition by many authors (see [4, 5]). Mao [6] discusses numerical solutions to stochastic differential delay equations (SDDEs) under the local Lipschitz condition and the linear growth condition. Mao and Sabanis [7] discuss numerical solutions to SDDEs with variable delay under the local Lipschitz condition and the linear growth condition. Mao discusses numerical solutions to SDEs and SDDEs under the local Lipschitz condition and the monotone condition (see [8]). Dai and Liu [9] give the mean-square stability of the numerical solutions of linear SEPCAs. However, SEPCAs do not have the convergence results. The main aim of this paper is to establish convergence of numerical solution for SEPCAs under the differential conditions.The paper is organized as follows. In Section2, we introduce necessary notations and the Euler method. In Section 3, the strong convergence of the Euler-Maruyama method to SEPCAs under local Lipschitz condition and the bounded pth moment condition will be given. In Section 4, the strong convergence of the Euler-Maruyama method to SEPCAs under local Lipschitz condition and the linear growth condition will be presented. In Section 5, an example is provided to show which is satisfied with the monotone condition without the linear growth condition. In Section 6, we obtain the convergence of numerical solutions to SEPCAs under local Lipschitz condition and the monotone condition is established.


## 2. Preliminary Notation and Euler Method

In this paper, unless otherwise specified, let|x| be the Euclidean norm in x∈Rn. If A is a vector or matrix, its transpose is defined by AT. If A is a matrix, its trace norm is defined by |A|=trace(ATA). For simplicity, we also have to denote by a∧b=min{a,b},a∨b=max{a,b}.Let(Ω,ℱ,P) be a complete probability space with a filtration {ℱt}t≥0, satisfying the usual conditions. ℒ1([0,∞),Rn) and ℒ2([0,∞),Rn) denote the family of all real valued ℱt-adapted process f(t)t≥0, such that for every T>0, ∫0T‍|f(t)|dt<∞a.s. and ∫0T‍|f(t)|2dt<∞a.s., respectively. For any a,b∈R with a<b, denote C([a,b];Rn) the family of continuous functions ϕ from [a,b] to Rn with the norm ∥ϕ∥=supa≤θ≤b|ϕ(θ)|. Denote Cℱtb([a,b];Rn) the family of all bounded ℱt-measurable C([a,b];Rn)-valued random variables. Let B(t)=(B1(t),…,Bd(t))T be a d-dimensional Brownian motion defined on the probability space.Throughout this paper, we consider stochastic differential equations with piecewise continuous arguments:(2.1)dx(t)=f(x(t),x([t]))dt+g(x(t),x([t]))dB(t)∀t≥0,
with initial data x(0)=x0, where f:Rn×Rn→Rn,g:Rn×Rn→Rn×d,x0 is a vector, and [·] denotes the greatest-integer function. By the definition of stochastic differential, this equation is equivalent to the following stochastic integral equation:
(2.2)x(t)=x(0)+∫0t‍f(x(s),x([s]))ds+∫0t‍g(x(s),x([s]))dB(s)∀t≥0.
Moreover, we also require the coefficients f and g to be sufficiently smooth.To be precise, let us state the following conditions.(H1) The local Lipschitz condition: for every integeri≥1, there exists a positive constant Li such that
(2.3)|f(x,y)-f(x-,y-)|2∨|g(x,y)-g(x-,y-)|2≤Li(|x-x-|2+|y-y-|2),
for those x,x-,y,y-∈Rn with |x|∨|x-|∨|y|∨|y-|≤i.(H2) Linear growth condition: there exists a positive constantK such that
(2.4)|f(x,y)|2∨|g(x,y)|2≤K(1+|x|2+|y|2),
for all (x,y)∈Rn×Rn.(H3) Monotone condition: there exists a positive constantK1 such that
(2.5)xTf(x,y)+p-12|g(x,y)|2≤K1(1+|x|2+|y|2),
for all (x,y)∈Rn×Rn.(H4) The boundedpth moment condition: there exists a pair of constants p>2 and K2>0 such that
(2.6)E[sup0≤t≤T|x(t)|p]∨E[sup0≤t≤T|y(t)|p]≤K2.Let us first give the definition of the solution.Definition 2.1 (see [10]).
AnRn-valued stochastic process {x(t)} is called a solution of (2.1) on [0,∞), if it has the following properties: (1)
{x(t)} is continuous on [0,∞) and ℱt adapted;(2)
{f(x(t),x([t]))}∈ℒ1([0,∞),Rn) and {g(x(t),x([t]))}∈ℒ2([0,∞),Rn×d);(3)
Equation (2.2) is satisfied on each interval [n,n+1)⊂[0,∞) with integral end points almost surely. A solution {x(t)} is said to be unique if any other solution {x-(t)} is indistinguishable from {x(t)}, that is,
(2.7)P{x(t)=x-(t)∀t∈[0,∞)}=1.Let h=1/m be a given stepsize with integer m≥1 and the grid points tn defined by tn=nh(n=0,1,2,…). For simplicity, we assume T=Nh. We consider the Euler-Maruyama method to (2.1),
(2.8)yn+1=yn+f(yn,yh([nh]))h+g(yn,yh([nh]))ΔBn,
for n=0,1,2,…, where ΔBn=B(tn)-B(tn-1),yh([nh]) is approximation to the exact solution x([nh]). Let n=km+l(k=0,1,2,…,l=0,1,2,…,m-1). The adaptation of the Euler method to (2.1) leads to a numerical process of the following type:
(2.9)ykm+l+1=ykm+l+f(ykm+l,ykm)h+g(ykm+l,ykm)ΔBkm+l,
where ΔBkm+l=B(tkm+l)-B(tkm+l-1),ykm+l and ykm are approximations to the exact solution x(tkm+l) and x([tkm+l]), respectively. The continuous Euler-Maruyama approximate solution is defined by
(2.10)y(t)=y(0)+∫0t‍f(z(s),z([s]))ds+∫0t‍g(z(s),z([s]))dB(s),
where z(t)=ykm+l and z([t])=ykm for t∈[tkm+l,tkm+l+1). It is not difficult to see that y(tkm+l)=z(tkm+l)=ykm+l for k=0,1,2,…,l=0,1,2,…,m-1. For sufficiently large integer i, define the stopping times ηi=inf{t≥0:|x(t)|≥i},θi=inf{t≥0:|y(t)|≥i},τi=ηi∧θi.


## 3. Convergence of the Euler-Maruyama Method under the Boundedpth Moment

We will show the strong convergence of the EM method on (2.1) under local Lipschitz condition and the bounded pth moment condition. The following lemma shows that both y(t) and z(t) are close to each other.Lemma 3.1.
Under the condition (H1), letT>0 be arbitrary. Then
(3.1)Esup0≤t≤T|y(t)-z(t)|2≤C1(x0,i)h,
where C1(x0,i)=4Li(T+4)K2.Proof.
Fort∈[0,T), there are two integers k and l such that t∈[tkm+l,tkm+l+1). By the Hölder inequality, we compute
(3.2)|y(t)-z(t)|2=|∫tkm+lt‍f(z(s),z([s]))ds+∫tkm+lt‍g(z(s),z([s]))dB(s)|2≤2|∫tkm+lt‍f(z(s),z([s]))ds|2+2|∫tkm+lt‍g(z(s),z([s]))dB(s)|2≤2T∫tkm+lt‍|f(z(s),z([s]))|2ds+2|∫tkm+lt‍g(z(s),z([s]))dB(s)|2.
This implies that, for any 0≤t1≤T,
(3.3)Esup0≤t≤t1|y(t)-z(t)|2≤2TEsup0≤t≤t1∫tkm+lt‍|f(z(s),z([s]))|2ds+2Esup0≤t≤t1|∫tkm+lt‍g(z(s),z([s]))dB(s)|2.
By the Doob martingale inequality, we have
(3.4)Esup0≤t≤t1|y(t)-z(t)|2≤2TE∫tkm+lt1‍|f(z(s),z([s]))|2ds+8E∫tkm+lt1‍|g(z(s),z([s]))|2ds.
Using the local Lipschitz conditions
(3.5)Esup0≤t≤t1|y(t)-z(t)|2≤2Li(T+4)E∫tkm+lt1‍(|z(s)|2+|z([s])|2)ds≤4Li(T+4)∫tkm+lt1‍(Esup0≤u≤s|y(u)|2)ds≤C1(x0,i)h,
where C1(x0,i)=4Li(T+4)K2. The proof is completed.Theorem 3.2.
Under the conditions (H1) and (H4), the EM approximate solution converges to the exact solution of (2.1) in the sense that
(3.6)limh→0E[sup0≤t≤T|y(t)-x(t)|2]=0.Proof.
Fix ap>2; let e(t)=x(t)-y(t); it is easy to see that
(3.7)E[sup0≤t≤T|e(t)|2]=E[sup0≤t≤T|e(t)|21{θi>Tandηi>T}]+E[sup0≤t≤T|e(t)|21{θi≤Torηi≤T}]=E[sup0≤t≤T|e(t)|21{τi>T}]+E[sup0≤t≤T|e(t)|21{θi≤Torηi≤T}]≤E[sup0≤t≤T|e(t∧τi)|2]+E[sup0≤t≤T|e(t)|21{θi≤Torηi≤T}].
By the Young inequality xy≤(xp/p)+(yq/q), for any a,b,p,q,δ>0,(1/p)+(1/q)=1, we have
(3.8)ab≤aδ1/pbδ1/p≤(aδ1/p)pp+bqqδq/p=apδp+bqqδq/p.
Thus for any δ>0, we have
(3.9)E[sup0≤t≤T|e(t)|21{θi≤Torηi≤T}]≤2δpE[sup0≤t≤T|e(t)|p]+1-(2/p)δ2/(p-2)P{θi≤Torηi≤T}.
By condition (H4), we have
(3.10)P(θi≤T)=E[1{θi≤T}|y(θi)|pip]≤1ipE[sup0≤t≤T|y(t)|p]≤K2ip.
Similarly, the result is
(3.11)P(ηi≤T)≤K2ip.
So that
(3.12)P(θi≤Torηi≤T)≤P(θi≤T)+P(ηi≤T)≤2K2ip.
Using these bounds, then
(3.13)E[sup0≤t≤T|e(t)|p]≤2p-1(E[sup0≤t≤T|x(t)|p]+E[sup0≤t≤T|y(t)|p])≤2pK2.E[sup0≤t≤T|e(t)|21{θi≤Torηi≤T}]≤2p+1δK2p+2(p-2)K2pδ2/(p-2)ip.
By the definitions of x(t) and y(t), we have
(3.14)|x(t∧τi)-y(t∧τi)|2≤2|∫0t∧τi‍[f(x(s),x([s])-f(z(s),z([s]))]ds|2+2|∫0t∧τi‍[g(x(s),x([s])-g(z(s),z([s]))]dB(s)|2.
Thus, for any t1∈[0,T](3.15)Esup0≤t≤t1|x(t∧τi)-y(t∧τi)|2≤2Esup0≤t≤t1|∫0t∧τi‍[f(x(s),x([s])-f(z(s),z([s]))]ds|2+2Esup0≤t≤t1|∫0t∧τi‍[g(x(s),x([s])-g(z(s),z([s]))]dB(s)|2.
By the Hölder inequality, condition (H1), and Lemma 3.1, one gets
(3.16)Esup0≤t≤t1|∫0t∧τi‍[f(x(s),x([s])-f(z(s),z([s]))]ds|2≤TE∫0t1∧τi‍|f(x(s),x([s])-f(z(s),z([s]))|2ds=TE∫0t1‍|f(x(s∧τi),x([s∧τi])-f(z(s∧τi),z([s∧τi]))|2ds≤TLiE∫0t1‍(|x(s∧τi)-z(s∧τi)|2+|x([s∧τi])-z([s∧τi])|2)ds≤2TLi∫0t1‍(E|x(s∧τi)-y(s∧τi)|2+E|x([s∧τi])-y([s∧τi])|2)ds+2TLi∫0t1‍(E|y(s∧τi)-z(s∧τi)|2+E|y([s∧τi])-z([s∧τi])|2)ds≤2TLi∫0t1‍(E|x(s∧τi)-y(s∧τi)|2+E|x([s∧τi])-y([s∧τi])|2)ds+4T2LiC1(x0,i)h≤4TLi∫0t1‍Esup0≤ν≤s∧τi|x(ν)-y(ν)|2ds+4T2LiC1(x0,i)h.
Similarly, by the Burkhôlder-Davis-Gundy inequality, then
(3.17)Esup0≤t≤t1|∫0t∧τi‍[g(x(s),x([s]))-g(z(s),z([s]))]dB(s)|2≤4E∫0t∧τi‍|g(x(s),x([s]))-g(z(s),z([s]))|2ds≤16Li∫0t1‍Esup0≤ν≤s∧τi|x(ν)-y(ν)|2ds+16TLiC1(x0,i)h.
Substituting (3.16) and (3.17) into (3.15) gives
(3.18)Esup0≤t≤t1|x(t∧τi)-y(t∧τi)|2≤8(T+4)Li∫0t1‍Esup0≤ν≤s∧τi|x(ν)-y(ν)|2ds+8T(T+4)LiC1(x0,i)h.
By the Gronwall inequality, we must get
(3.19)Esup0≤t≤T|x(t∧τi)-y(t∧τi)|2≤C2(x0,i)h,
where C2(x0,i)=8T(T+4)LiC1(x0,i)e8T(T+4)Li. So we have
(3.20)E[sup0≤t≤T|e(t)|2]≤E[sup0≤t≤T|e(t∧τi)|2]+E[sup0≤t≤T|e(t)|21{θi<Torηi<T}]≤C2(x0,i)h+2p+1δK2p+2(p-2)K2pδ2/(p-2)ip.
Given any ϵ>0, we can choose δ sufficiently small for
(3.21)2p+1δK2p<ϵ3,
then choose i sufficiently large for
(3.22)2(p-2)K2pδ2/(p-2)ip<ϵ3,
and finally choose h sufficiently small, so that
(3.23)C2(x0,i)h<ϵ3.
Thus E[sup0≤t≤T|e(t)|2]<ϵ. The proof is completed.


## 4. Convergence of the Euler-Maruyama Method under Linear Growth Condition

We will show the strong convergence of the EM method on (2.1) under local Lipschitz condition and the linear growth condition. In the following we will show that the linear growth condition (H2) implies the bounded pth moment condition (H4).Lemma 4.1.
Under the linear growth conditions (H2), there exists a positive constantC3 such that the solution of (2.1) satisfies
(4.1)Esup0≤t≤T|x(t)|p≤C3(1+|x0|p),
where C3=C3(p,T,K) is a constant independent of h.Proof.
It follows from (2.2) that
(4.2)|x(t)|p=|x(0)+∫0t‍f(x(s),x([s]))ds+∫0t‍g(x(s),x([s]))dB(s)|p≤3p-1[|x(0)|p+|∫0t‍f(x(s),x([s]))ds|p+|∫0t‍g(x(s),x([s]))dB(s)|p].
By the Hölder inequality, we obtain
(4.3)|x(t)|p≤3p-1[|x(0)|p+Tp-1∫0t‍|f(x(s),x([s]))|pds+|∫0t‍g(x(s),x([s]))dB(s)|p].
This implies that, for any 0≤t1≤T,
(4.4)Esup0≤t≤t1|x(t)|p≤3p-1[|x(0)|p+Tp-1Esup0≤t≤t1∫0t‍|f(x(s),x([s]))|pds+Esup0≤t≤t1|∫0t‍g(x(s),x([s]))dB(s)|p].
By the Burkholder-Davis-Gundy inequality and the Hölder inequality, it is not difficult to show that
(4.5)Esup0≤t≤t1|x(t)|p≤3p-1[|x(0)|p+Tp-1E∫0t1‍|f(x(s),x([s]))|pds+CpTp/2-1E∫0t1‍|g(x(s),x([s]))|pds],
where Cp is a constant. Note from the linear growth conditions that
(4.6)Esup0≤t≤t1|x(t)|p≤3p-1[|x(0)|p+Tp-1E∫0t1‍Kp/2(1+|x(s)|2+|x([s])|2)p/2ds+CpTp/2-1E∫0t1‍Kp/2(1+|x(s)|2+|x([s])|2)p/2ds]≤3p-1|x(0)|p+3(3p/2)-2Kp/2(Tp-1+CpTp/2-1)×E∫0t1‍(1+|x(s)|p+|x([s])|p)ds≤3p-1|x(0)|p+3(3p/2)-2Kp/2(Tp+CpTp/2)+2×3(3p/2)-2Kp/2(Tp-1+CpTp/2-1)Esup0≤u≤s|x(u)|pds.
By the Gronwall inequality, we must get
(4.7)Esup0≤t≤T|x(t)|p≤C3(1+|x0|p),
where C3=C3(p,T,K) is a constant independent of h.The following lemma shows that the continuous Euler-Maruyama approximate solution has boundedpth moments.Lemma 4.2.
Under the linear growth conditions (H2), there exists a positive constantC4 such that the continuous approximate solution of the Euler-Maruyama (2.10) satisfies
(4.8)Esup0≤t≤T|y(t)|p≤C4(1+|x0|p),
where C4=C4(p,T,K) is a constant independent of h.Proof.
By the inequality|a+b+c|2≤3|a|2+3|b|2+3|c|2 and (2.10), in the same way as Lemma 4.1, for any 0≤t1≤T, we can obtain
(4.9)Esup0≤t≤t1|y(t)|p≤3p-1[|y(0)|p+Tp-1E∫0t1‍|f(z(s),z([s]))|pds+CpT(p/2)-1E∫0t1‍|g(z(s),z([s]))|pds],
where Cp is a constant. Note from the linear growth conditions that
(4.10)Esup0≤t≤t1|y(t)|p≤3p-1[|y(0)|p+Tp-1E∫0t1‍Kp/2(1+|z(s)|2+|z([s])|2)p/2ds+CpTp/2-1E∫0t1‍Kp/2(1+|z(s)|2+|z([s])|2)p/2ds]≤3p-1|y(0)|p+3(3p/2)-2Kp/2(Tp-1+CpTp/2-1)×E∫0t1‍(1+|z(s)|p+|z([s])|p)ds≤3p-1|y(0)|p+3(3p/2)-2Kp/2(Tp+CpTp/2)+2×3(3p/2)-2Kp/2(Tp-1+CpTp/2-1)Esup0≤u≤s|y(u)|pds.
By the Gronwall inequality, we must get
(4.11)Esup0≤t≤T|y(t)|p≤C4(1+|x0|p),
where C4=C4(p,T,K) is a constant independent of h.According to Theorem3.2, we have the following theorem.Theorem 4.3.
Under the conditions (H1) and (H2), the EM approximate solution converges to the exact solution of (2.1) in the sense that
(4.12)limh→0E[sup0≤t≤T|y(t)-x(t)|2]=0.


## 5. A Motivating Example

In the above section, we give the strong convergence numerical solution of SEPCAs under the local Lipschitz condition (H1) and the linear growth condition (H2). However, there are many SEPCAs that do not satisfy the linear growth condition, consider the following SEPCA:(5.1)dx(t)=[-x3(t)+x([t])]dt+[sinx2(t)+x([t])]dB(t)∀t≥0.
Clearly, the equation do not satisfy the linear growth condition (H2). But the example is analyzed under condition (H3) which covers many nonlinear SEPCAs. On the other hand, we have
(5.2)x(-x3+y)+12(sinx2+y)2≤-x4+xy+(sinx2)2+y2≤2(1+x2+y2).
In other words, the equation satisfies condition (H3). Moreover, we also have
(5.3)2xTf(x,y)+|g(x,y)|2≤|x|2+2K(1+|x|2+|y|2)≤(1+2K)(1+|x|2+|y|2).
We see clearly that (H3) follows from (H2). Therefore, The following result is more general than Theorem 3.2. Let us now turn to establish the convergence of the Euler-Maruyama method to (2.1) under the conditions (H1) and (H3).


## 6. Convergence of the Euler-Maruyama Method under Monotone Condition

In this section, we give the convergence of the EM method to (2.1) under the local Lipschitz condition (H1) and the monotone condition (H3). We prove the bounded pth moment property of the EM approximate solution and the exact solution to (2.1) under the monotone condition (H3).Lemma 6.1.
Under the monotone condition (H3), there exists a positive constantC5 such that the solution of (2.1) satisfies
(6.1)Esup0≤t≤T|x(t)|p∨Esup0≤t≤T|y(t)|p≤C5,
where C5=C5(p,T,K1,x0) is a constant independent of h.Proof.
By Itô formula, for allt≥0, we have
(6.2)(1+|x(t)|2)p/2=(1+|x(0)|2)p/2+∫0t‍p(1+|x(t)|2)(p-2)/2×[x(t)Tf(x(s),x([s]))+p-12|g(x(s),x([s]))|2]ds+∫0t‍p(1+|x(t)|2)(p-2)/2x(s)Tg(x(s),x([s]))dB(s)≤(1+|x(0)|2)p/2+∫0t‍p(1+|x(t)|2)(p-2)/2[K1(1+|x(s)|2+|x([s])|2)]ds+∫0t‍p(1+|x(t)|2)(p-2)/2x(s)Tg(x(s),x([s]))dB(s).
For any t1∈[0,T], we have
(6.3)Esup0≤t≤t1(1+|x(t)|2)p/2≤(1+|x(0)|2)p/2+Esup0≤t≤t1∫0t‍p(1+|x(t)|2)(p-2)/2[K1(1+|x(s)|2+|x([s])|2)]ds+Esup0≤t≤t1|∫0t‍p(1+|x(t)|2)(p-2)/2x(s)Tg(x(s),x([s]))dB(s)|.
By the Burkholder-Davis-Gundy inequality and Exercise 2.5 in [8], it is not difficult to show that
(6.4)Esup0≤t≤t1|∫0t‍p(1+|x(t)|2)(p-2)/2x(s)Tg(x(s),x([s]))dB(s)|≤3E(∫0t1‍p2(1+|x(t)|2)p-2|x(s)|2|g(x(s),x([s]))|2ds)1/2≤3E(sup0≤t≤t1(1+|x(t)|2)p/2∫0t1‍p2(1+|x(t)|2)(p-2)/2|g(x(s),x([s]))|2ds)1/2≤3E(sup0≤t≤t1(1+|x(t)|2)p/2∫0t1‍p2K1(1+|x(t)|2)(p-2)/2(1+|x(s)|2+|x([s])|2)ds)1/2≤0.5Esup0≤t≤t1(1+|x(t)|2)p/2+4.5E∫0t1‍p2K1(1+|x(t)|2)(p-2)/2(1+|x(s)|2+|x([s])|2)ds.
Substituting (6.4) into (6.3) and using the Hölder inequality
(6.5)Esup0≤t≤t1(1+|x(t)|2)p/2≤2(1+|x(0)|2)p/2+2pK1(1+4.5p)E∫0t1‍(1+|x(t)|2)(p-2)/2(1+|x(s)|2+|x([s])|2)ds≤2(1+|x(0)|2)p/2+2pK1(1+4.5p)(E∫0t1‍(1+x(t)|2)p/2ds)(p-2)/p×(E∫0t1‍(1+|x(s)|2+|x([s])|2)p/2ds)2/p≤2(1+|x(0)|2)p/2+2pK1(1+4.5p)(TEsup0≤t≤t1(1+|x(t)|2)p/2)(p-2)/p×(E∫0t1‍(1+|x(s)|2+|x([s])|2)p/2ds)2/p.
So we obtain
(6.6)Esup0≤t≤t1(1+|x(t)|2)p/2≤[2(1+|x(0)|2)p/2+2pK1(1+4.5p)T(p-2)/p(E∫0t1‍(1+|x(s)|2+|x([s])|2)p/2ds)2/p]p/2≤2(p/2)-1[2(1+|x(0)|2)p/2]p/2+2(p/2)-1[2pK1(1+4.5p)T(p-2)/p(E∫0t1‍(1+|x(s)|2+|x([s])|2)p/2ds)2/p]p/2≤2p-1(1+|x(0)|2)p2/4+2p-1[pK1(1+4.5p)T(p-2)/p]p/2E∫0t1‍(1+|x(s)|2+|x([s])|2)p/2ds≤2p-1(1+|x(0)|2)p2/4+2(3p/2)-1[pK1(1+4.5p)T(p-2)/p]p/2∫0t1‍Esup0≤u≤s(1+|x(u)|2)p/2ds.
By the Gronwall inequality, we must get
(6.7)Esup0≤t≤T(|x(t)|p)≤C5,
where C5=C5(p,T,K,x0) is a constant independent of h. Similarly, we can show that
(6.8)Esup0≤t≤T(|y(t)|p)≤C5.
The proof is completed.According to Theorem3.2, we obtain the following.Theorem 6.2.
Under the conditions (H1) and (H3), the EM approximate solution converges to the exact solution of (2.1) in the sense that
(6.9)limh→0E(sup0≤t≤T|y(t)-x(t)|2)=0.



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*Source: 643783-2012-12-05.xml*