耿曉晶

(暨南大學(xué)經(jīng)濟(jì)學(xué)院統(tǒng)計學(xué)系,廣東廣州 510632)

多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的數(shù)值解

耿曉晶

(暨南大學(xué)經(jīng)濟(jì)學(xué)院統(tǒng)計學(xué)系,廣東廣州 510632)

由于多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程不存在解析解,利用Euler-Maruyama方法給出多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的漸進(jìn)數(shù)值解,并證明了此數(shù)值解收斂到方程的解析解.將單一馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的數(shù)值解問題延伸到多維馬爾科夫轉(zhuǎn)制情形,增強(qiáng)了馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的適用性.

多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程;Euler-Maruyama數(shù)值解;收斂性

1 引言

在馬爾科夫轉(zhuǎn)制隨機(jī)微分方程領(lǐng)域,文獻(xiàn)[1-3]做了大量研究.諸如研究方程的平穩(wěn)性[1]、Euler-Maruyama(EM)數(shù)值解的收斂性[2],以及平穩(wěn)分布的數(shù)值方法[3]等.但上述研究成果均針對單一馬爾科夫轉(zhuǎn)制隨機(jī)微分方程.

文獻(xiàn)[4]于2013年創(chuàng)新性的提出多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的概念.相較單一馬爾科夫轉(zhuǎn)制,多維馬爾科夫轉(zhuǎn)制能細(xì)致刻畫不同隨機(jī)因素對各個系數(shù)的影響.文獻(xiàn)[4]詳述了多維轉(zhuǎn)制的優(yōu)點(diǎn),并給出隨機(jī)微分方程解的存在性、唯一性證明與解的p階矩估計.本文將進(jìn)一步探討多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的數(shù)值解及其收斂性.

2 基本設(shè)定

3 預(yù)備知識

4 EM數(shù)值解

5 數(shù)值解的收斂性

6 結(jié)論

本文借鑒文獻(xiàn)[6]研究單一馬爾科夫轉(zhuǎn)制隨機(jī)微分方程數(shù)值解的方法,在文獻(xiàn)[4]提出的多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的基礎(chǔ)上,證明了EM法得出的多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程的數(shù)值解收斂到真實(shí)解.

后續(xù)可進(jìn)一步對多維馬爾科夫轉(zhuǎn)制隨機(jī)微分方程解的有界性與平穩(wěn)性等做出研究.

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[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.

耿曉晶(1990-),碩士生,研究方向：數(shù)理金融與精算學(xué).

2010 MSC:65C20