Pure Mathematics
Vol. 09  No. 10 ( 2019 ), Article ID: 33564 , 8 pages
10.12677/PM.2019.910145

Stability of Solutions for Stochastic Volterra-Levin Equations Driven by α-Stable Noise

Weiya Rao, Huanquan Lin, Tong Jiang

School of Science, Changchun University, Changchun Jilin

Received: Dec. 3rd, 2019; accepted: Dec. 16th, 2019; published: Dec. 23rd, 2019

ABSTRACT

In this paper, we study stochastic Volterra-Levin equations driven by α-stable noise. We have a try to deal with the stability conditions in distribution of the segment process of the solutions to the stochastic systems.

Keywords:α-Stable Noise, Stochastic Volterra-Levin Equation, Stability

α-稳定噪声驱动的随机Volterra-Levin方程解的稳定性

饶维亚,蔺焕泉,姜童

长春大学理学院,吉林 长春

收稿日期:2019年12月3日;录用日期:2019年12月16日;发布日期:2019年12月23日

摘 要

本文研究了α-稳定噪声驱动的随机Volterra-Levin方程。在一定条件下,得到了该类方程的解部分过程的依分布稳定性。

关键词 :α-稳定噪声,随机Volterra-Levin方程,依分布稳定性

Copyright © 2019 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

1928年,Volterra [1] 研究了如下微分方程

d x ( t ) d t = t L t q ( s t ) f ( x ( s ) ) d s ,

其中 q , f , L 满足下文中的假设。Volterra尝试利用Laypunov方法研究该类微分方程的稳定性。1963年,Levin [2] 利用Laypunov方程得到该类微分方程的稳定性,所以该类方程被称为Volterra-Levin方程。该方面更一步的结果可以参考MacCamy,Wong [3] 和Burton [4] 的工作及参考文献。最近许多学者开始研究随机型Volterra-Levin方程(参见Gushchin和Küchler [5] ;Liu [6] ;Reiß [7],Li和Xu [8] 等及相关文献)。许多学者研究了随机Volterra-Levin方程的稳定性。Appleby [9] 和Burton [10] 在一定条件下,利用不动点定理研究了随机Volterra-Levin方程在概率1意义下的稳定性。Luo [11] 和Zhao等 [12] 进一步得到了均方意义下的指数稳定性,在更弱的条件下得到概率1意义下的指数稳定性。Guo和Zhu [13] 研究了带Poisson跳的Volterra-Levin方程解的存在唯一性,并得到p-阶矩意义下的稳定性。Yin等 [14] 研究了带Poisson跳和变时滞的Volterra-Levin方程p-阶矩意义下的稳定性。

许多学者尝试着讨论比概率1更弱的稳定性,随机微分方程的解收敛到某一分布,一个值得研究的课题。该种稳定称为解依分布渐近稳定。1996年,Basak等 [15] 首次研究了漂移项为线性的随机微分方程的依分布稳定性。在此基础上 [15],Bao等 [16] [17] [18] [19],Hu和Wang [20] 以及Yuan和Mao [21],Li和Zhang [22] 研究了随机Volterra-Levin方程解的依分布稳定性。

近年来,Lévy噪声驱动的随机微分方程受到学者们的广泛关注。α-稳定噪声是特殊的Lévy噪声,它可以展现重尾现象,因此研究α-稳定噪声驱动的随机微分方程非常有意义,成为一个重要的研究课题,Priola和Zabczyk [23] 研究了α-稳定噪声驱动的随机微分方程解的渐近行为,Zang和Li [24] 研究了α-稳定噪声驱动的随机微分方程解的依分布稳定性。

2. 预备知识

{ Ω , F , { F t } t 0 , P } 是完备的概率空间,其中 { F t } t 0 是滤流,满足通有条件,即滤流是右连续的并且 F 0 包含所有零集。令 { Z ( t ) , t 0 } 为定义在 { Ω , F , { F t } t 0 , P } 上的α-稳定过程。对于给定的常数 L > 0 C : = C ( [ L ,0 ] ; ) 记为连续函数 φ : [ L ,0 ] 构成的空间,其范数为 φ = sup t [ L ,0 ] | φ ( t ) |

本文将研究下述α-稳定噪声驱动的随机Volterra-Levin方程解的依分布稳定性。

d x ( t ) = ( t L t q ( s t ) f ( x ( s ) ) d s ) d t + d Z t , (1)

初始条件,

x ( ) = ψ ( ) C ( [ L ,0 ] ; R ) , L s 0 (2)

其中,映射 q C ( [ L ,0 ] ; R ) , f C ( R ; R ) Z t 是概率空间 { Ω , F , { F t } t 0 , P } 上的α-稳定噪声。对任意 ψ C ( [ L ,0 ] ; R ) ,定义 ψ t ( s ) = ψ ( t + s ) , s [ L ,0 ]

参见Appleby [9] 和Burton [10],本文给出如下假设:

(H1) f ( 0 ) = 0 ,且存在一个常数 λ > 0 ,使得 f ( x ) x 2 λ

(H2) μ = lim x 0 f ( x ) x

(H3) 存在一个常数 K > 0 ,使得对于任意的 x , y R | f ( x ) f ( y ) | K | x y |

(H4) 存在一个常数 m > 0 ,使得 L 0 q ( s ) d s = m

(H5) 2 K L 0 | q ( s ) | d s < 1

3. 主要结果

x ( t , ψ ) 为方程(1)满足初始条件 ψ ( ) C ( [ L ,0 ] ; R ) 的解,则(1)的相应部分解过程为 x t ( ψ ) = x ( t + θ ; ψ ) , L θ 0 , t 0 。于是 x t ( ψ ) , t 0 的转移概率 P ( ψ , t , ) , ψ C ( [ L ,0 ] ; R ) 是一个齐次的马尔科夫过程(参考Mohammed Mohammed [25] )。在本节中,将研究方程(1)的部分解过程 x t ( ψ ) 的分布稳定性。

定义1. 如果存在一个 C ( [ L ,0 ] ; R ) 上的概率测度 π ( ) ,使得对任意的 ψ C ( [ L ,0 ] ; R ) ,当 t 时, x t ( ψ ) 的概率转移函数 P ( ψ , t , ) 弱收敛到 π ( ) 。则方程(1)的部分解过程 x t ( ψ ) , t 0 为依分布意义下渐近稳定的分布。

引理 1 假设(H1)-(H5)成立,则对于任意 ψ C ( [ L ,0 ] ; R ) ,有

sup 0 t < E x t ( ψ ) < . (3)

证明:利用文献 [22] 的方法,定义连续函数 a ( t ) : [ 0, ] [ 0, ]

a ( t ) = { f ( x ( t ) ) x ( t ) , x ( t ) 0 , μ , x ( t ) = 0.

由假设(H4),则方程(1)满足:

d x ( t ) = m a ( t ) x ( t ) d t + d ( L 0 q ( s ) t + s t f ( x ( u ) ) d u d s ) + d Z t , t 0. (4)

利用变量替换以及分步积分,则方程(1)可写成如下形式:

x ( t ) = e 0 t m a ( u ) d u ( ψ ( 0 ) L 0 q ( s ) t + s t f ( ψ ( u ) ) d u d s ) + L 0 q ( s ) t + s t f ( x ( u ) ) d u d s 0 t e v t m a ( s ) d s m a ( v ) L 0 q ( s ) v + s v f ( x ( u ) ) d u d s d v + 0 t e s t m a ( u ) d u d Z ( s ) .

于是

E | x ( t ) | E | e 0 t m a ( u ) d u ( ψ ( 0 ) L 0 q ( s ) t + s t f ( ψ ( u ) ) d u d s ) | + E | L 0 q ( s ) t + s t f ( x ( u ) ) d u d s | + E | 0 t e v t m a ( s ) d s m a ( v ) L 0 q ( s ) v + s v f ( x ( u ) ) d u d s d v | + 0 t e s t m a ( u ) d u d Z ( s ) : = I 1 ( t ) + I 2 ( t ) + I 3 ( t ) + I 4 ( t ) . (5)

对于 I 1 ( t ) 有:

I 1 ( t ) e 0 t m a ( u ) d u ( 1 + K L m ) ψ 0 , (6)

由(H3)得:

I 2 ( t ) K | L 0 q ( s ) t + s t E | x ( u ) | d u d s | ( K L 0 | s q ( s ) | d s ) sup L θ 0 E | x ( t + θ ) | , (7)

类似地,

I 3 ( t ) = E | 0 t e v t m a ( s ) d s m a ( v ) L 0 q ( s ) v + s v f ( x ( u ) ) d u d s d v | K E ( 0 t e v t m a ( s ) d s m a ( v ) L 0 | q ( s ) | v + s v E | x ( u ) | d u d s d v ) ( K L 0 | s q ( s ) | d s ) 0 t e v t m a ( s ) d s m a ( v ) sup L θ 0 E | x ( v + θ ) | d v . (8)

对于 I 4 ,令 { ξ k } k N 为一个定义在某一个正态分布 N ( 0,1 ) 的概率空间 { Ω , F , P } 上的独立随机变量序列, { C k } k N 是一个实数序列,则

E | k = 1 C k ξ k | p = A p ( k = 1 | C k | 2 ) p 2 , A p : = R | x | p 2 π e x 2 2 d x .

I 4 ( t ) = E | 0 t e s t m a ( u ) d u d Z ( s ) | = ( E E | k = 1 ξ k 0 t e s t m a ( u ) d u d W S s | p ) 1 p = ( E E | k = 1 ξ k 0 t e s t m a ( u ) d u d W S s | p ) 1 p ( C E 0 t ( k = 1 ξ k 2 e 2 s t m a ( u ) d u ) α 2 d s ) p α ( C 0 t e α s t m a ( u ) d u d s ) 1 α ( C 0 t e α λ ( t s ) d s ) 1 α [ C α λ ( 1 e α λ t ) ] 1 α . (9)

对于 t 0 ,将(6)~(9)代入(5),可以得到:

E | x ( t ) | e 0 t m a ( u ) d u ( 1 + K L m ) ψ 0 + ( K L 0 | s q ( s ) | d s ) sup L θ 0 E | x ( t + θ ) | + ( K L 0 | s q ( s ) | d s ) 0 t e v t m a ( s ) d s m a ( v ) sup L θ 0 E | x ( v + θ ) | d v + [ C α λ ( 1 e α λ t ) ] 1 α . (10)

η 1 : = ( 1 + K L m ) ψ 0 , η 2 = η 3 : = K L 0 | s q ( s ) | d s , η 4 = ( C 0 t e α s t m a ( u ) d u d s ) 1 α

由假设(H5)可知 ρ = η 2 + η 3 = K L 0 | s q ( s ) | d s < 1

另一方面,由(H1)和(H3)有, H : = sup t 0 t L t a ( s ) d s 2 λ L 存在,并且 lim t 0 t a ( s ) d s = 。则对 ψ C ( [ L ,0 ] ; R ) ,存在 N > 0 , λ 1 ( 0 , 1 ) ,使得 ψ 0 < N 以及 η 1 N + e η 1 2 λ L η 2 + e 2 λ 2 L η 3 1 λ 1 < 1 成立。

由文献 [22] 中的引理3.1,可得

E | x ( t ) | N e λ 1 0 t m a ( v ) d v + ( 1 ρ ) 1 η 4 , t 0. (11)

这说明 x ( t ) 是有界的。

接下来我们还需要证明部分解过程 x t ( ψ ) 的有界性。由α-稳定过程的自相似性,对于任意 n 1

E x n L ( ψ ) E sup L θ 0 | e ( n 1 ) L n L + θ m a ( u ) d u ( x ( n 1 ) L ) L 0 q ( s ) s + ( n 1 ) L ( n 1 ) L f ( x ( u ) ) d u d s | + E sup L θ 0 | L 0 q ( s ) n L + θ + s n L + θ f ( x ( u ) ) d u d s | + E sup L θ 0 | ( n 1 ) L n L + θ e v n L + θ m a ( s ) d s m a ( v ) L 0 q ( s ) v + s v f ( x ( u ) ) d u d s d v | + E sup L θ 0 | ( n 1 ) L n L + θ e s n L + θ m a ( u ) d u d Z ( s ) | ( K L 0 | s q ( s ) | d s ) E x ( n 1 ) L ( ψ ) + C ( 1 + sup ( n 2 ) L t n L E | x ( t , ψ ) | + E sup ( n 2 ) L t n L | 0 t e s + ( n 1 ) L t m a ( u ) d u d Z ˜ ( s ) | ) , (12)

其中 C > 0 时,由自相似性知 Z ˜ = Z ( s + ( n 1 ) L ) Z ( ( n 1 ) L ) 仍然是一个α-稳定过程。

对于 n N ,由(11)式,对于某些常数 K 1 > 0 ,都有

sup ( n 2 ) L t n L E | x ( t , ψ ) | K ˜ 1 < ,

并且对于某些常数 K ˜ 2 > 0 ,都有

E sup 0 t L | 0 t e s + ( n 1 ) L t m a ( u ) d u d Z ˜ ( s ) | C ( α ) = C 2 < .

从而有

E x n L ( ψ ) ( K L 0 | s q ( s ) | d s ) E x ( n 1 ) L ( ψ ) + K ˜ 2 ,

K ˜ 2 = C ( 1 + sup ( n 2 ) L t n L E | x ( t , ψ ) | ) + C 2 .

所以,由条件(H5),有

K L 0 | s q ( s ) | d s = : μ < 1.

于是,对任意的整数 n 1 ,根据迭代法可得

E x n L ( ψ ) μ E x ( n 1 ) L ( ψ ) + K ˜ 2 μ ( μ E x ( n 1 ) L ( ψ ) + K ˜ 2 ) μ n sup L θ 0 | ψ ( θ ) | + K ˜ 2 ( 1 + μ + μ 2 + + μ n 1 ) sup L θ 0 | ψ ( θ ) | + K ˜ 2 1 μ . (13)

对于任意的 t 0 ,存在 n 0 ,使得,当 t [ n L , ( n + 1 ) L ] 时,

E x t ( ψ ) E x ( n + 1 ) L ( ψ ) + E x n L ( ψ ) .

这样,根据(13)可以得到如下结论:

sup 0 t < E x t ( ψ ) < .

引理2. 假设(H1)-(H5)成立,则对于任意有界子集 S C ( [ L ,0 ] ; R ) ,并且 ϕ , ψ S ,下述结论成立

lim t E | x ( t , ϕ ) x ( t , ψ ) | = 0. (14)

证明. 类似于引理1的证明,首先证明:

lim t E | x t ( ϕ ) x t ( ψ ) | = 0 , (15)

其中 ϕ , ψ S

对任意的 t 0 ,可以得到

E | x ( t , ϕ ) x ( t , ψ ) | E | 0 t e 0 t m a ( u ) d u ( ( ϕ ( 0 ) ψ ( 0 ) ) L 0 q ( s ) s 0 [ f ( ϕ ( u ) ) f ( ψ ( u ) ) ] d u d s ) | + E | L 0 q ( s ) t + s t [ f ( x ( u , ϕ ) ) f ( x ( u , ψ ) ) ] d u d s | + E | 0 t e v t m a ( s ) d s m a ( v ) L 0 q ( s ) t + s t [ f ( x ( u , ϕ ) ) f ( x ( u , ψ ) ) ] d u d s d v | b ˜ 1 e 0 t m a ( u ) d u + b ˜ 2 sup L θ 0 E | x ( t + θ , ϕ ) x ( t + θ , ψ ) | + b ˜ 3 0 t e v t m a ( s ) d s m a ( v ) sup L θ 0 E | x ( v + θ , ϕ ) x ( v + θ , ψ ) | d v ,

其中 b ˜ 1 = ( 1 + K L m ) | ϕ ( 0 ) ψ ( 0 ) | b ˜ 2 = b ˜ 3 = K L 0 | s q ( s ) | d s 。因此,根据引理1,即可得到(15)。

t 2 L ,根据假设(H1)-(H3)得:

E sup L θ 0 | x ( t + θ , ϕ ) x ( t + θ , ψ ) | E sup L θ 0 | e t L t + θ m a ( u ) d u ( x ( t L , ϕ ) x ( t L , ψ ) ) L 0 q ( s ) t L + s t L [ f ( x ( u , ϕ ) ) f ( x ( u , ψ ) ) ] d u d s | + E sup L θ 0 | t L t + θ t + θ + s t + θ [ f ( x ( u , ϕ ) ) f ( x ( u , ψ ) ) ] d u d s | + E sup L θ 0 | t L t + θ e v t + θ m a ( s ) d s m a ( v ) L 0 q ( s ) v + s v [ f ( x ( u , ϕ ) ) f ( x ( u , ψ ) ) ] d u dsd v |

E | x ( t L , ϕ ) x ( t L , ψ ) | + m K t 2 L t L E | x ( s , ϕ ) x ( s , ψ ) | d s + m K t 2 L t L E | x ( s , ϕ ) x ( s , ψ ) | d s + K m L t 2 L t L E | x ( u , ϕ ) x ( u , ψ ) | d u .

于是利用(15),可得

lim t sup L θ 0 | x ( t + θ , ϕ ) x ( t + θ , ψ ) | = 0. (16)

因此, (14)式必定成立,引理证毕。

接下来,讨论方程(1)的部分解过程的分布稳定性。

P ( C ( [ L ,0 ] ; R ) ) 为在 C ( [ L ,0 ] ; R ) 上的概率测度空间。当 p 1 , p 2 P ( C ( [ L ,0 ] ; R ) ) 时,定义:

d L ( p 1 , p 2 ) = sup f L | e f ( Φ ) p 1 ( d ϕ ) e f ( ψ ) p 2 ( d Ψ ) | ,

其中,对于任意的 ϕ , ψ C ( [ L ,0 ] ; R ) ,有

L = { f | C ( [ L ,0 ] ; R ) R : | f ( ϕ ) f ( ψ ) | ϕ ψ , | f ( ) | 1 } .

参考文献 [22],有如下的结果:

引理 3 ( [22] ) 假设(H1)-(H5)成立,则对于任意的初值 ϕ C ( [ L ,0 ] ; R ) P ( ϕ , t , ) : t 0 是空间 P ( C ( [ L ,0 ] ; R ) ) 上具有度量 d L 的柯西列。

本文的主要结果如下:

定理1 假设(H1)-(H5)成立,则α-稳定噪声驱动的随机方程(1)的部分解过程 x t ( ψ ) , t 0 是分布稳定的。

证明. 类似于文献 [22] 中定理3.1的证明,可以很类似地得到该定理的证明。证明从略。

文章引用

饶维亚,蔺焕泉,姜 童. α-稳定噪声驱动的随机Volterra-Levin方程解的稳定性
Stability of Solutions for Stochastic Volterra-Levin Equations Driven by α-Stable Noise[J]. 理论数学, 2019, 09(10): 1187-1194. https://doi.org/10.12677/PM.2019.910145

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