多維馬爾科夫轉制隨機微分方程的數值解

耿曉晶

(暨南大學經濟學院統計學系,廣東廣州 510632)

多維馬爾科夫轉制隨機微分方程的數值解

耿曉晶

(暨南大學經濟學院統計學系,廣東廣州 510632)

由于多維馬爾科夫轉制隨機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制隨機微分方程的漸進數值解,并證明了此數值解收斂到方程的解析解.將單一馬爾科夫轉制隨機微分方程的數值解問題延伸到多維馬爾科夫轉制情形,增強了馬爾科夫轉制隨機微分方程的適用性.

多維馬爾科夫轉制隨機微分方程;Euler-Maruyama數值解;收斂性

1 引言

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

文獻[4]于2013年創新性的提出多維馬爾科夫轉制隨機微分方程的概念.相較單一馬爾科夫轉制,多維馬爾科夫轉制能細致刻畫不同隨機因素對各個系數的影響.文獻[4]詳述了多維轉制的優點,并給出隨機微分方程解的存在性、唯一性證明與解的p階矩估計.本文將進一步探討多維馬爾科夫轉制隨機微分方程的數值解及其收斂性.

2 基本設定

3 預備知識

4 EM數值解

5 數值解的收斂性

6 結論

本文借鑒文獻[6]研究單一馬爾科夫轉制隨機微分方程數值解的方法,在文獻[4]提出的多維馬爾科夫轉制隨機微分方程的基礎上,證明了EM法得出的多維馬爾科夫轉制隨機微分方程的數值解收斂到真實解.

后續可進一步對多維馬爾科夫轉制隨機微分方程解的有界性與平穩性等做出研究.

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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：數理金融與精算學.

2010 MSC:65C20