具有Markov切换Poisson跳的随机微分方程的均方指数稳定性
Mean Square Exponential Stability of Stochastic Differential Equations with Markovian Switching and Poisson Jumps

作者: 王吉平 , 李光洁 :广东外语外贸大学数学与统计学院,广东 广州;

关键词: 随机微分方程均方指数稳定性Markov切换Poisson跳Lyapunov函数Stochastic Differential Equations Mean Square Exponential Stability Markovian Switching and Poisson Jumps Lyapunov Functions

摘要:
研究一类带Markov切换Poisson跳的随机微分方程的均方指数稳定性。运用Lyapunov稳定性理论、随机分析以及不等式技巧获得了该方程的平凡解是均方指数稳定性的充分条件。最后,给出一个例子说明所得的结果。

Abstract: This paper investigates the mean square exponential stability of stochastic differential equations with Markovian switching and Poisson jumps. By using Lyapunov stability theory, stochastic anal-ysis and inequality techniques, some sufficient conditions are derived to obtain the mean square exponential stability of the trivial solution. At last, an example is presented to illustrate the ob-tained results.

1. 引言

随机微分方程常被用来刻画随机动力系统,且已广泛地应用于物理、生物、医药、社会科学、经济、金融等诸多领域。在随机微分方程的研究中,稳定性是其研究的一个重要课题,关于随机微分方程稳定性的研究已取得一定的成果,见文献 [1] [2] [3] [4] 及其中的参考文献。实际中,许多物理系统在遭遇突变现象(如分支和内部联系紊乱、参数转移等)时,结构会发生随机改变。对于这样的物理系统,人们常用包含连续系统状态和离散系统状态的Markov切换系统来描述,关于带Markov切换的随机微分方程的稳定性研究也引起了学者们的极大兴趣,参阅文献 [5] [6] [7] [8]。

Brown运动是连续的随机过程,然而许多实际系统会遭受跳跃形式的随机突发扰动,如随机故障、地震、海啸等。在这些情形下,不能用Brown运动来刻画这些系统,因此将带跳过程引入到系统中来处理这些实际情形是合理的,见文献 [9] [10] [11]。近年来,关于带跳的随机微分方程的稳定性研究也获得了相应的成果,参阅文献 [12] [13] 及其中的参考文献。

本文研究一类不仅带有Markov切换还带有Poisson跳的随机微分方程的均方指数稳定性。该文结构如下:第2节给出所需的预备知识;第3节利用Lyapunov函数方法证明该类方程的平凡解是均方指数稳定的。第4节通过一个例子说明所得的结果。

2. 预备知识

记一个完备的概率空间为 ( Ω , F , P ) ,其滤子 F t 满足通常的条件,即 { F t } t 0 是右连续的且 F 0 包含所

有的零测集。 { r ( t ) } t 0 代表的是定义在此完备概率空间上的一个取值为 S = { 1 , 2 , , N } 的右连续时齐Markov链,其生成元(密度矩阵) Γ = ( γ i j ) N × N 由转移概率矩阵确定,即

P { r ( t + Δ ) = j | r ( t ) = i } = { γ i j Δ + o ( Δ ) , i j , 1 + γ i i Δ + o ( Δ ) , i = j ,

这里, Δ > 0 lim Δ 0 o ( Δ ) / Δ = 0 ,对于 i j γ i j 0 表示从状态 i 到状态 j 的转移速率且 γ i i = j i γ i j

作为一个事实, r ( ) 的每个样本轨道是一个右连续的阶梯函数,且在 R + = [ 0 , + ) 上的任何一个有限区间里至多存在有限个跳跃点(参阅 [14] )。

考虑如下形式的带Markov切换Poisson跳的随机微分方程:

d x ( t ) = f ( x ( t ) , t , r ( t ) ) d t + g ( x ( t ) , t , r ( t ) ) d B ( t ) + h ( x ( t ) , t , r ( t ) ) d N ( t ) , t 0 , (1)

初始值 x ( 0 ) = x 0 R m r ( 0 ) = i 0 S ,其中 f , g , h : R m × R + × S R m B ( t ) N ( t ) 分别是定义在该概率空间上的1-维Brown运动和强度为 λ 的Poisson过程,这里假设Markov链 r ( t ) 与Brown运动 B ( t ) 和Poisson过程 N ( t ) 是相互独立的。 N ˜ ( t ) = N ( t ) λ t 表示 N ( t ) 的补偿Poisson过程。本文假设函数 f , g , h 满足方程(1)解的全局存在唯一性的必需条件。而且,对 t 0 i S ,假设 f ( 0 , t , i ) = g ( 0 , t , i ) = h ( 0 , t , i ) = 0 。因此,当方程(1)的初始值 x 0 = 0 时,该方程存在平凡解 x ( t ) 0

定义 若存在正常数 γ K 使得对于任意给定的初始值 x ( 0 ) = x 0

E | x ( t ) | 2 K e γ t E | x 0 | 2 , t 0

成立,则称方程(1)的平凡解是均方指数稳定的。

C 2 , 1 ( R m × R + ; R + ) 表示关于变量 x 二阶连续可导且关于变量 t 一阶连续可导的全体非负函数 V ( x , t ) 的集合。给定任意的 V ( x , t ) C 2 , 1 ( R m × R + ; R + ) ,定义算子

L V ( x , t , i ) = V t ( x , t ) + V x ( x , t ) f ( x , t , i ) + 1 2 trace [ g T ( x , t , i ) V x x ( x , t ) g ( x , t , i ) ] + λ [ V ( x + h ( x , t , i ) , t ) V ( x , t ) ] ,

其中, V t ( x , t ) = v ( x , t ) t V x x ( x , t ) = ( 2 v ( x , t ) x i x j ) m × m V x ( x , t ) = ( v ( x , t ) x 1 , v ( x , t ) x 2 , , v ( x , t ) x m )

3. 主要结果

本节利用Lyapunov函数证明方程(1)的平凡解是均方指数稳定的。为了讨论方程(1)的平凡解的稳定性,对 i S ,假设存在常数 α i 1 α i 2 ,以及正常数 α i 3 满足如下条件:

x T ( t ) f ( x ( t ) , t , i ) + 1 2 | g ( x ( t ) , t , i ) | 2 α i 1 | x ( t ) | 2 , (2)

x T ( t ) h ( x ( t ) , t , i ) α i 2 | x ( t ) | 2 , (3)

| h ( x ( t ) , t , i ) | 2 α i 3 | x ( t ) | 2 . (4)

定理 若条件(2)~(4)成立,且存在正常数 α 0 使得

α 0 + 2 α i 1 + λ ( 2 a i 2 + α i 3 ) < 0 (5)

成立,则方程(1)的平凡解是均方指数稳定的。

证明 取Lyapunov函数 V ( x ( t ) , t ) = e α 0 t | x ( t ) | 2 。则对 i S

L V ( x ( t ) , t , i ) = α 0 e α 0 t | x ( t ) | 2 + 2 e α 0 t x T ( t ) f ( x ( t ) , t , i ) + e α 0 t | g ( x ( t ) , t , i ) | 2 + e α 0 t λ ( 2 x T ( t ) h ( x ( t ) , t , i ) + | h ( x ( t ) , t , i ) | 2 ) = e α 0 t [ α 0 | x ( t ) | 2 + 2 x T ( t ) f ( x ( t ) , t , i ) + | g ( x ( t ) , t , i ) | 2 + λ ( 2 x T ( t ) h ( x ( t ) , t , i ) + | h ( x ( t ) , t , i ) | 2 ) ] .

由条件(2)~(4)可得

L V ( x ( t ) , t , i ) e α 0 t [ α 0 + 2 α i 1 + λ ( 2 a i 2 + α i 3 ) ] | x ( t ) | 2 .

利用条件(5)进一步可得

E L V ( x ( t ) , t , i ) 0.

E V ( x ( t ) , t ) E V ( x ( 0 ) , 0 ) 0 t E L V ( x ( s ) , s , r ( s ) ) d s 0 ,

所以

E V ( x ( t ) , t ) E V ( x ( 0 ) , 0 ) .

注意 V ( x ( t ) , t ) = e α 0 t | x ( t ) | 2 ,从而

e α 0 t E | x ( t ) | 2 E | x 0 | 2 ,

E | x ( t ) | 2 e α 0 t E | x 0 | 2 .

证毕。

4. 例子

考虑如下带Markov切换Poisson跳的随机微分方程:

d x ( t ) = f ( x ( t ) , t , r ( t ) ) d t + g ( x ( t ) , t , r ( t ) ) d B ( t ) + h ( x ( t ) , t , r ( t ) ) d N ( t ) , (6)

其中, B ( t ) N ( t ) 分别是1-维Brown运动和强度 λ = 0.5 的Poisson过程, r ( t ) 是取值为 S = { 1 , 2 } 的Markov链且其生成元如下:

Γ = ( 1 1 2 2 ) .

方程(6)中的系数参数如下:

i = 1 时, f ( x , t , 1 ) = 2 x , g ( x , t , 1 ) = x , h ( x , t , 1 ) = x

i = 2 时, f ( x , t , 1 ) = 3 x , g ( x , t , 1 ) = 2 x , h ( x , t , 1 ) = 1 2 x

i S ,可知 f ( 0 , t , i ) = g ( 0 , t , i ) = h ( 0 , t , i ) = 0 。经计算得,当 i = 1 时,

x T ( t ) f ( x ( t ) , t , 1 ) + 1 2 | g ( x ( t ) , t , 1 ) | 2 = 3 2 | x ( t ) | 2 ,

x T ( t ) h ( x ( t ) , t , 1 ) = | x ( t ) | 2 ,

| h ( x ( t ) , t , 1 ) | 2 = | x ( t ) | 2 .

i = 2 时,

x T ( t ) f ( x ( t ) , t , 2 ) + 1 2 | g ( x ( t ) , t , 2 ) | 2 = | x ( t ) | 2 ,

x T ( t ) h ( x ( t ) , t , 2 ) = 1 2 | x ( t ) | 2 ,

| h ( x ( t ) , t , 1 ) | 2 = 1 4 | x ( t ) | 2 .

α 0 = 1 ,由此可知对 i S ,有 α 0 + 2 α i 1 + λ ( 2 a i 2 + α i 3 ) < 0 成立。根据第三节中的定理可得方程(6)的平凡解是均方指数稳定的。

NOTES

*通讯作者。

文章引用: 王吉平 , 李光洁 (2021) 具有Markov切换Poisson跳的随机微分方程的均方指数稳定性。 理论数学, 11, 1276-1280. doi: 10.12677/PM.2021.117142

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