＊ 帶Poisson跳和M arkovian調制的年齡相關隨機種群方程數值解的收斂性

馬維軍,張啟敏

(寧夏大學數學計算機學院,寧夏銀川 750021)

＊帶Poisson跳和M arkovian調制的年齡相關隨機種群方程數值解的收斂性

馬維軍,張啟敏

(寧夏大學數學計算機學院,寧夏銀川 750021)

研究了帶Poisson跳和Markovian調制的年齡相關隨機種群方程數值解的收斂性,在給定條件下證明了數值解收斂到解析解,并給出了Euler逼近的階數.

與年齡相關隨機種群方程;Markovian調制;Poisson跳;數值解;收斂性

0 引言

考慮如下與年齡相關的隨機種群系統(tǒng):

近年來,很多學者更加關注隨機微分方程數值解的研究.例如,Wang討論了帶Poisson跳和Markovian調制的隨機時滯微分方程數值解的收斂性[1],Svishchuk和 Kazmerchuk研究了帶Poisson跳和Markovian調制的隨機時滯微分方程的指數穩(wěn)定性[2].Luo給出了帶Poisson跳和M arkovian調制的It^o隨機時滯微分方程的比較原則和指數穩(wěn)定性[3].然而對與年齡相關的隨機種群方程,當h=0時,Zhang證明了系統(tǒng)(1)強解的存在性、唯一性,指數穩(wěn)定性和數值分析[4-5],Gu討論了與年齡相關的具有擴散的隨機種群系統(tǒng)數值解的收斂性[6],Li研究了與年齡相關的隨機種群方程數值解的收斂性[7].

一般情況下,形如(1)的種群方程的解析解很難求得,故通常利用數值方法求得其數值逼近解,因而研究種群系統(tǒng)數值解的收斂性問題具有重大意義.而目前關于帶Poisson跳和Markovian調制的年齡相關隨機種群系統(tǒng)數值解的收斂性研究很少見到.本文研究了方程(1)數值解的收斂性,并給出數值解收斂的階數,所得到的結論是文獻[7-9]的擴展.

1 預備知識

設(Ω,F,{Ft}t≥0,P)是一完備概率空間,{Ft}t≥0是其上的一個濾子且滿足一般性條件(即左極限右連續(xù)的,且F0包含所有的零測集).

設N(t)是強度為λ的數值 Poisson過程,假設N(t)與Brow n運動W t相互獨立.設{r(t),t≥0}是定義在概率空間(Ω,F,P)上取值于有限狀態(tài)S={1,2,…,N}的右連續(xù)Markov鏈,其生成元Γ=(γij)N×N如下給定:

其中Δ＞0.若i≠j,γij≥0表示從狀態(tài)i轉移到狀態(tài)j的概率,γii=- ∑i≠jγij.假設r(·)與Brow n運動W t和Poisson過程N(t)相互獨立.易知,r(t)的每一個樣本軌道是右連續(xù)的階梯函數,且在R+上的任何一個有限子區(qū)間上至多含有限多個跳躍點.

我們假設滿足以下條件:

2 引理

P(h)=(Pij(h))N×N=ehΓ.

引理2.2 若條件(ii)-(iii)成立,則存在常數k≥2,C1＞0使得

證明 對|Pt|k應用 It^o公式,可得

3 主要結果

定理3.1 若條件(i)-(iii)成立,則

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Convergence of Numerical Solutions to Stochastic Age-dependent Population Equations with Poisson Jumps and Markovian Switching

MA Wei-jun,ZHANG Qi-min
(School of Mathematics and Computer Science,Ningxia University,Yinchuan750021,China)

The convergence of numerical approximation of stochastic age-dependent population equations with Poisson jumps and Markovian switching is studied.It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions.The order of the Euler approximation is also provided.

stochastic age-dependent population equations;Markovian switching;Poisson jump s;numerical solutions;convergence

O211.63

A

0253-2395(2011)01-0051-09＊

2010-04-18;

2010-08-17

國家自然科學基金(11061024)

馬維軍(1981-),男,寧夏人,碩士,從事隨機計算及其應用的研究.E-mail:maweijun_2008@163.com