求解隨機(jī)微分方程split-step歐拉方法的收斂性

賈俊梅

(內(nèi)蒙古工業(yè)大學(xué)理學(xué)院,內(nèi)蒙古 呼和浩特 010051)

隨機(jī)微分方程在描述現(xiàn)象中起著越來越重要的作用,其理論廣泛應(yīng)用于金融、生物、物理、微電子、機(jī)械等學(xué)科和工程領(lǐng)域.但是除了少數(shù)隨機(jī)微分方程,一般的隨機(jī)微分方程很難求其理論解,因而數(shù)值方法的構(gòu)造顯得尤為重要. 多數(shù)情況下是將隨機(jī)微分方程離散化為差分方程,然后利用隨機(jī)差分進(jìn)行計(jì)算或模擬.在所有的離散化方法中,歐拉格式是最基本且最重要的一種[1-13]. 在文獻(xiàn)[1]中Wang和Li給出了自治標(biāo)量split-step歐拉方法的數(shù)值格式,并且求其收斂性和穩(wěn)定性,本文將文獻(xiàn)[1]中提出的split-step歐拉方法推廣到求解一般的伊藤型隨機(jī)微分方程,并且求其收斂性.

1 split-step歐拉方法

考慮一維伊藤型隨機(jī)微分方程(SDE)

dx(t)=f(x(t),t)dt+g(x(t),t)dw(t),t∈[0,T],x(0)=x0.

(1)

式中:f,g為R×[0,T]上的連續(xù)可測函數(shù),分別稱為偏移系數(shù)和擴(kuò)散系數(shù);w(t)是標(biāo)準(zhǔn)的Wiener過程, 其增量Δw(t)=w(t+h)-w(t)服從正態(tài)分布N(0,h).00,K2>0,使得

|f(x,t)-f(y,t)|2∨|g(x,t)-g(y,t)|2≤K1|x-y|2.

(2)

|f(x,t)|2∨|g(x,t)|2≤K2(1+|x|2) .

(3)

方程(2)、(3)保證方程(1)解的存在并且唯一. 對于方程(1) split-step歐拉方法,即擴(kuò)散項(xiàng)split-step歐拉(DISSE)方法:

(4)

(5)

和偏移項(xiàng)split-step歐拉(DRSSE)方法:

(6)

(7)

當(dāng)t∈[tk,tk+1)時(shí),定義

(8)

(9)

(10)

(11)

由方程(10)、(11)方程(8)可以寫成如下的形式

(12)

由方程(10)、(11)方程(9)可以寫成如下的形式

(13)

2 split-step歐拉方法的均方收斂性

在這部分,來證明split-step歐拉方法的均方收斂性.主要證明擴(kuò)散項(xiàng)split-step歐拉(DISSE)方法的均方收斂性,偏移項(xiàng)split-step歐拉(DRSSE)方法的均方收斂性的證明過程類似.此證明類似于文獻(xiàn)[2].為了證明主要的定理,將使用如下幾個(gè)引理.

引理1 設(shè)h<1并且方程(3)成立,那么存在2個(gè)正常數(shù)A=1+K2,B=K2,使得

對以上方程兩邊求數(shù)學(xué)期望并由h<1,得到

(14)

引理2 設(shè)h<1并且方程(3)成立,那么存在2個(gè)正常數(shù)F,G,使得

對上式兩邊求數(shù)學(xué)期望,由h<1和引理1,得.

E|yk+1|2≤E|yk|2+(1+K2+2AK2)hE|yk|2+(3K2+2k2B)h=(1+Ch)E|yk|2+Dh,

其中:C=(1+K2+2AK2),D=(3K2+2K2B).由Gronwall不等式得

引理3 在引理2成立的條件下,那么存在一個(gè)正常數(shù)H(H不依賴h)使得

E|y(t)-z1(t)|2∨E|y(t)-z2(t)|2≤Hh.

所以 |y(t)-z2(t)|2≤K2h(10+6F+4G).

E|y(t)-z1(t)|2∨E|y(t)-z2(t)|2≤Hh,這里H=K2(10+6F+4G).

定理設(shè)x(t)是方程(1)的解析解,f、g滿足方程(2)、(3)且h≤1.假設(shè)存在一個(gè)正常數(shù)K3使得

|f(x,s)-f(x,t)|2∨|g(x,s)-g(x,t)|2≤K3(1+|x|2)|s-t|

(15)

證明由方程(1)、(12), 當(dāng)t∈[0,T]

由H?der不等式,

由方程(2)、(16)和基本不等式(a+b)2≤2a2+2b2,

由殃矩不等式,

由引理1、2、3得

Th(K1H(8T+32)+K3(4T+4TG+16F+16)).

其中:M=T(K1H(8T+32)+K3(4T+4TG+16F+16))e(8T+32)K1T.

證畢.

DRSSE方法的均方收斂性的證明與DISSE方法的類似,所以在這里省略.

參考文獻(xiàn):

[1]Wang Peng, Li Yong. Split-step forward methods for stochastic differential equations [J]. Journal of Computational and Applied Mathematics, 2010, 233: 2641-2651.

[2]Ding Xiaohua, Ma Qiang, Zhang Lei. Convergence and stability of the split-stepθ-method for stochastic differential equations[J].Computers and Mathematics with Applications, 2010, 60: 1310-1321.

[3]Fan Zhengcheng,Liu Mingzhu,Cao Wanrong. Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equation[J]. J math Anal Appl, 2007, 325: 1142-1159.

[4]Marion G,Mao Xuerong, Renshaw E. Convergence of the Euler shceme for a class of stochastic differential equations[J]. International Mathematical Journal, 2002, 1(1):1-14.

[5]Zhang Haomin, Gan Siqing, Hu Lin. The split-step backward Euler method for linear stochastic delay differential equations[J]. Journal of Computational and Applied Mathematics, 2009, 225: 558-568.

[6]Cao Wanrong.T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise[J]. Applied Mathematics and Computation, 2010,216: 999-1006.

[7]Zhang Xicheng. Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications[J]. J Math Anal Appl, 2006, 316: 447-458.

[8]Raul F,Soledad T. The Euler scheme for Hilbert space valued stochastic differential equations[J]. Statistics and probability Letters, 2001, 51: 207-213.

[9]Martin A S, Soledad T. Euler scheme for solutions of a countable system of stochastic differential equations[J]. Statistics and Probability Letters, 2001, 54: 251-259.

[10]Lepingle D.Euler scheme for reflected stochastic differential equations[J]. Mathematics and Computers in Simulation, 1995, 38: 119-126.

[11]Wang Wenqiang, Chen Yanping. Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations[J]. Applied Numerical Mathematics, 2001, 61: 696-701.

[12]Cao Wanrong, Zhu Mingzhu, Fan Zhencheng. Ms-stability of the Euler-Maruyama method for stochastic differential equations[J]. Applied Mathematics and Computation, 2004, 159: 127-135.

[13]Higham D J, Mao X, Stuart A M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations[J].SIAM J Numer Anal, 2002, 40(3): 1041-1063.