多维马尔科夫转制随机微分方程的数值解

耿晓晶

(暨南大学经济学院统计学系,广东广州 510632)

多维马尔科夫转制随机微分方程的数值解

耿晓晶

(暨南大学经济学院统计学系,广东广州 510632)

由于多维马尔科夫转制随机微分方程不存在解析解,利用Euler-Maruyama方法给出多维马尔科夫转制随机微分方程的渐进数值解,并证明了此数值解收敛到方程的解析解.将单一马尔科夫转制随机微分方程的数值解问题延伸到多维马尔科夫转制情形,增强了马尔科夫转制随机微分方程的适用性.

多维马尔科夫转制随机微分方程;Euler-Maruyama数值解;收敛性

1 引言

在马尔科夫转制随机微分方程领域,文献[1-3]做了大量研究.诸如研究方程的平稳性[1]、Euler-Maruyama(EM)数值解的收敛性[2],以及平稳分布的数值方法[3]等.但上述研究成果均针对单一马尔科夫转制随机微分方程.

文献[4]于2013年创新性的提出多维马尔科夫转制随机微分方程的概念.相较单一马尔科夫转制,多维马尔科夫转制能细致刻画不同随机因素对各个系数的影响.文献[4]详述了多维转制的优点,并给出随机微分方程解的存在性、唯一性证明与解的p阶矩估计.本文将进一步探讨多维马尔科夫转制随机微分方程的数值解及其收敛性.

2 基本设定

3 预备知识

4 EM数值解

5 数值解的收敛性

6 结论

本文借鉴文献[6]研究单一马尔科夫转制随机微分方程数值解的方法,在文献[4]提出的多维马尔科夫转制随机微分方程的基础上,证明了EM法得出的多维马尔科夫转制随机微分方程的数值解收敛到真实解.

后续可进一步对多维马尔科夫转制随机微分方程解的有界性与平稳性等做出研究.

[1]Mao X R.Stability of stochastic diferential equations with Markovian switching[J].Stochastic Processes and Their Applications,1999,79:45-67.

[2]Yuan C G,Mao X R.Convergence of the Euler-Maruyama method for stochastic diferential equations with Markovian switching[J].Mathematics and Computers in Simulation,2004,64:223-235.

[3]Mao X R,Yuan C G,Yin G.Numerical method for stationary distribution of stochastic diferential equations with Markovian switching[J].Journal of Computational and Applied Mathematics,2005,174:1-27.

[4]Liu M,Wang K.Stochastic diferential equations with Multi-Markovian switching[J].Journal of Applied Mathematics,2013,2013:1-11.

[5]Mao X R,Yuan C G.Stochastic Diferential Equations with Markovian Switching[M].London:Imperial College Press,2006.

[6]Mao X R.Numerical solutions of stochastic diferential delay equations under the generalized Khasminskiitype conditions[J].Applied Mathematics and Computation,2011,217:5512-5524.

Numerical solutions of stochastic diferential equations with Multi-Markovian switching

Geng Xiaojing

(Department of Statistics,Ji′nan University,Guangzhou510632,China)

Since stochastic diferential equations with Multi-Markovian switching do not have explicit solutions, the Euler-Maruyama numerical solutions are obtained according to the Euler-Maruyama scheme.And it is proved that the approximate solutions will converge to the exact solutions.In this paper,the numerical theory of stochastic diferential equations with single Markovian switching has been extended to the case of Multi-Markovian switching,which will lead to better applicability of stochastic diferential equations with Markovian switching.

SDEs with Multi-Markovian switching,Euler-Maruyama scheme,convergence

O211.63

A

1008-5513(2013)06-0646-08

10.3969/j.issn.1008-5513.2013.06.015

2013-09-14.

耿晓晶(1990-),硕士生,研究方向：数理金融与精算学.

2010 MSC:65C20