中立型隨機(jī)延遲微分方程θ-方法的均方穩(wěn)定性*

王文強(qiáng)

(湘潭大學(xué)數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,湖南 湘潭 411105)

中立型隨機(jī)延遲微分方程θ-方法的均方穩(wěn)定性*

王文強(qiáng)

(湘潭大學(xué)數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,湖南 湘潭 411105)

討論θ-方法用于求解非線性中立型隨機(jī)延遲微分方程初值問題時(shí)數(shù)值解的穩(wěn)定性，給出了θ-方法均方穩(wěn)定的一個(gè)充分條件.

中立型隨機(jī)延遲微分方程;θ-方法；均方穩(wěn)定

隨機(jī)延遲微分方程數(shù)值方法的穩(wěn)定性研究是一件很有意義的工作,近年來(lái)已經(jīng)開始受到越來(lái)越多的學(xué)者關(guān)注,相關(guān)的研究成果逐漸多起來(lái).文獻(xiàn)[1]提出了隨機(jī)延遲微分方程Milstein方法.文獻(xiàn)[2]建立了數(shù)值方法的均方穩(wěn)定性(MS-穩(wěn)定性)概念，證明了當(dāng)線性標(biāo)量系統(tǒng)的真解是均方穩(wěn)定時(shí)，Euler-Maruyama方法的數(shù)值解是MS-穩(wěn)定的.文獻(xiàn)[3]研究了帶有延遲項(xiàng)的隨機(jī)微分方程半隱式Milstein數(shù)值方法的穩(wěn)定性,通過對(duì)數(shù)值方法應(yīng)用到線性試驗(yàn)方程上得到的差分方程進(jìn)行討論，給出了半隱式Milstein方法MS-穩(wěn)定與GMS-穩(wěn)定的條件.文獻(xiàn)[4]運(yùn)用Halanay-type理論，對(duì)常系數(shù)線性隨機(jī)延遲微分方程給出了Euler-Maruyama方法均方穩(wěn)定的判別準(zhǔn)則.文獻(xiàn)[5]研究了一類帶有延遲項(xiàng)的線性隨機(jī)延遲微分方程Milstein數(shù)值方法的穩(wěn)定性,通過對(duì)數(shù)值方法應(yīng)用到線性試驗(yàn)方程上得到的差分方程進(jìn)行討論，給出了Milstein方法MS-穩(wěn)定的條件.文獻(xiàn)[6]研究了改進(jìn)的Milstein方法在有限區(qū)間上對(duì)隨機(jī)延遲微分方程的分片近似的相關(guān)結(jié)論.文獻(xiàn)[7-12]研究了隨機(jī)延遲微分方程不同數(shù)值方法的均方穩(wěn)定性與收斂性.文獻(xiàn)[13]研究了中立型非線性隨機(jī)延遲微分方程單步方法的均方收斂性.文獻(xiàn)[14]進(jìn)一步研究了中立型非線性隨機(jī)延遲微分方程半隱式Euler方法的均方穩(wěn)定性.

筆者主要討論非線性中立型隨機(jī)延遲微分方程初值問題，給出了θ-方法均方穩(wěn)定的一個(gè)充分條件.

1 中立型隨機(jī)延遲微分方程

設(shè)(Ω,F,{Ft}t≥0,P)是完備的概率空間，濾子{Ft}t≥0滿足通常條件,即它們是右連續(xù)的且每一個(gè)Ft都包含所有的零概率集.考慮下列中立型隨機(jī)延遲微分方程初值問題：

其中：實(shí)常數(shù)τ>0；W(t)是一維標(biāo)準(zhǔn)Wiener過程；初始函數(shù)φ是H?lder連續(xù)的，即存在常數(shù)γ>0,L>0,使當(dāng)t,s∈[-τ,0]時(shí)，有E(|φ(t)-φ(s)|p)≤L|t-s|pγ,p=1,2；映射f:[0,+∞)×R×R→R和g:[0,+∞)×R×R→R充分光滑且滿足

和

(2)

其中L,K1,K2均為常數(shù)，x∨y=max(x,y),且存在常數(shù)λ∈(0,1),對(duì)任意x,y1,y2∈R,有|N(y1)-N(y2)|≤λ|y1-y2|,

|N(x)|≤λ|x|.

(3)

此時(shí)方程(1)存在唯一強(qiáng)解X(t).

2 θ-方法的均方穩(wěn)定性

將θ-方法用于數(shù)值求解初值問題(1)，得到

Xk+1-N(Xk+1-m)=Xk-N(Xk-m)+(θf(wàn)(tk+1,Xk+1,Xk+1-m)+(1-θ)f(tk,Xk,Xk-m))h+

g(tk,Xk,Xk-m)ΔWk2∈k=0,1,2,....

(4)

引理1 用θ-方法求解初值問題(1)所得的數(shù)值解{Xk}滿足下列不等式：

證明由Yk=Xk-N(Xk-m)和(3)式,可得

|Xk|=|Yk+N(Xk-m)|≤|Yk|+|N(Xk-m)|≤|Yk|+λ|Xk-m|.

(5)

同理可得

(6)

將(6)式代入(5)式，有

|Xk|≤|Yk|+λ|Yk-m|+λ2|Yk-2m|+...+λc(k)|Yk-c(k)m|+λc(k)+1|Xk-c(k)m-m|.

(7)

(7)式兩邊平方，利用Cauchy不等式得

(8)

(8)式兩邊取數(shù)學(xué)期望，并注意到當(dāng)l≤0時(shí)，有Xl=φ(lh)，則引理1的結(jié)論得證.

作為一種特殊情形，根據(jù)文獻(xiàn)[15]中推論6.8容易得到下面的結(jié)論：

定理1 如果方程(1a)滿足下列條件：

(ⅰ) 存在2個(gè)正數(shù)λ1,λ2，使得對(duì)任意的x,y∈R,有

2(x-N(y))f(t,x,y)+g2(t,x,y)≤-λ1|x-N(y)|2+λ2y2；

那么方程(1)的零解是均方漸近穩(wěn)定的.

將定理1稍加修改，可以得到下面的結(jié)論：

引理2 如果方程(1a)滿足下列條件：

(ⅰ) 存在2個(gè)常數(shù)μ1>0,μ2≥0，使得對(duì)任意的x,y∈R,有

2(x-N(y))f(t,x,y)≤-μ1|x-N(y)|2+μ2y2；

(9)

(ⅱ)

(10)

那么方程(1)的零解是均方漸近穩(wěn)定的.

證明根據(jù)三角不等式知|x|2=|x-N(y)+N(y)|2≤(|x-N(y)|+|N(y)|)2≤(|x-N(y)|+λ|y|)2.根據(jù)Cauchy不等式知

|x|2≤(|x-N(y)|+λ|y|)2≤(1+λ2)(|x-N(y)|2+|y|2).

(11)

因此聯(lián)立(2),(9)，(11)式可得

2(x-N(y))f(t,x,y)+g2(t,x,y)≤ -(μ1-(1+λ2)K2)|x-N(y)|2+

(μ2+(1+λ2)K2)y2.

(12)

根據(jù)定理1聯(lián)立(10)和(12)式即知結(jié)論成立.

(13)

(14)

-7x2(1+x)+4x4(2x-1)<0,

(15)

下面給出關(guān)于數(shù)值方法穩(wěn)定性分析的結(jié)論.首先記

證明由格式(4)得

Yk+1-θf(wàn)(tk+1,Xk+1,Xk+1-m)h=Yk+(1-θ)f(tk,Xk,Xk-m)h+g(tk,Xk,Xk-m)ΔWk,

(16)

(16)式兩邊同時(shí)平方,移項(xiàng)整理得

(1-θ)2f2(tk,Xk,Xk-m)h2+g2(tk,Xk,Xk-m)(ΔWk)2+

2(1-θ)Ykf(tk,Xk,Xk-m)h+2Ykg(tk,Xk,Xk-m)ΔWk+

2(1-θ)hf(tk,Xk,Xk-m)g(tk,Xk,Xk-m)ΔWk.

因此

2(1-θ)hf(tk,Xk,Xk-m)g(tk,Xk,Xk-m)ΔWk.

(17)

注意到E(ΔWk)=0,E[(ΔWk)2]=h,而且Xk,Xk-m都是Ftk可測(cè)的，因此容易得到

(18)

又根據(jù)已知條件(9)得

(19)

根據(jù)數(shù)學(xué)期望的性質(zhì)和(2)式知

(20)

將(18),(19)和(20)式代入(15)式取數(shù)學(xué)期望得

或

(21)

根據(jù)引理1的結(jié)論整理(21)式可得

+(1+λ2)μ2θhλ2c(k+1-m)S,

(22)

(23)

其中σ1(h;θ,λ,μ2,K1,K2,S)=(1+λ2)S(2(1-θ)2K1h+μ2+2K2).

(24)

記M=max(ρ,λ)<1,則由(24)式進(jìn)一步可得

定理3 當(dāng)步長(zhǎng)h

證明由Xk=Yk+N(Xk-m)和Cauchy不等式,可得

(25)

(25)式兩邊取數(shù)學(xué)期望有

(26)

又根據(jù)定理2的結(jié)論，對(duì)(26)式兩邊同時(shí)取極限得

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(責(zé)任編輯 向陽(yáng)潔)

Mean-SquareStabilityofθ-MethodsforNeutralNonlinearStochasticDelayDifferentialEquations

WANG Wenqiang

(School of Mathematics and Computational Science,Xiangtan University,Xiangtan 411105,Hunan China)

The mean-square stability of Euler method is investigated for nonlinear neutral stochastic delay differential equations.It is proved that the numerical method is mean-square stable(MS-stable) under a sufficient condition.

neutral stochastic delay differential equations;θ-methods;mean-square stable

1007-2985(2014)02-0010-05

2013-11-20

國(guó)家自然科學(xué)基金資助項(xiàng)目(11271311,11171352)

王文強(qiáng)(1971-)，男(苗族)，湖南邵陽(yáng)人,湘潭大學(xué)數(shù)學(xué)與計(jì)算科學(xué)學(xué)院教授，博士后，主要從事常微分方程數(shù)值解研究.

O175.13

A

10.3969/j.issn.1007-2985.2014.02.004