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

1. 引言

1928年，Volterra [1] 研究了如下微分方程

$\frac{\text{d}x\left(t\right)}{\text{d}t}=-{\int }_{t-L}^{t}q\left(s-t\right)f\left(x\left(s\right)\right)\text{d}s,$

2. 预备知识

$\left\{\Omega ,F,{\left\{{F}_{t}\right\}}_{t\ge 0},P\right\}$ 是完备的概率空间，其中 ${\left\{{F}_{t}\right\}}_{t\ge 0}$ 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 $\left\{Z\left(t\right),t\ge 0\right\}$ 为定义在 $\left\{\Omega ,F,{\left\{{F}_{t}\right\}}_{t\ge 0},P\right\}$ 上的α-稳定过程。对于给定的常数 $L>0$$C:=C\left(\left[-L,0\right];ℝ\right)$ 记为连续函数 $\phi :\left[-L,0\right]\to ℝ$ 构成的空间，其范数为 $‖\phi ‖={\mathrm{sup}}_{t\in \left[-L,0\right]}|\phi \left(t\right)|$

$\text{d}x\left(t\right)=-\left({\int }_{t-L}^{t}q\left(s-t\right)f\left(x\left(s\right)\right)\text{d}s\right)\text{d}t+\text{d}{Z}_{t},$ (1)

$x\left(\cdot \right)=\psi \left(\cdot \right)\in C\left(\left[-L,0\right];R\right),-L\le s\le 0$ (2)

(H1) $f\left(0\right)=0$，且存在一个常数 $\lambda >0$，使得 $\frac{f\left(x\right)}{x}\ge 2\lambda$

(H2) $\mu ={\mathrm{lim}}_{x\to 0}\frac{f\left(x\right)}{x}$

(H3) 存在一个常数 $K>0$，使得对于任意的 $x,y\in R$$|f\left(x\right)-f\left(y\right)|\le K|x-y|$

(H4) 存在一个常数 $m>0$，使得 ${\int }_{-L}^{0}q\left(s\right)\text{d}s=m$

(H5) $2K{\int }_{-L}^{0}|q\left(s\right)|\text{d}s<1$

3. 主要结果

$x\left(t,\psi \right)$ 为方程(1)满足初始条件 $\psi \left(\cdot \right)\in C\left(\left[-L,0\right];R\right)$ 的解，则(1)的相应部分解过程为 ${x}_{t}\left(\psi \right)=x\left(t+\theta ;\psi \right),-L\le \theta \le 0,t\ge 0$。于是 ${x}_{t}\left(\psi \right),t\ge 0$ 的转移概率 $P\left(\psi ,t,\cdot \right),\psi \in C\left(\left[-L,0\right];R\right)$ 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 ${x}_{t}\left(\psi \right)$ 的分布稳定性。

$\underset{0\le t<\infty }{\mathrm{sup}}E‖{x}_{t}\left(\psi \right)‖<\infty .$ (3)

$a\left(t\right)=\left\{\begin{array}{l}\frac{f\left(x\left(t\right)\right)}{x\left(t\right)},x\left(t\right)\ne 0,\\ \mu ,\text{ }\text{ }\text{ }x\left(t\right)=0.\end{array}$

$\text{d}x\left(t\right)=-ma\left(t\right)x\left(t\right)\text{d}t+\text{d}\left({\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}f\left(x\left(u\right)\right)\text{d}u\text{d}s\right)+\text{d}{Z}_{t},t\ge 0.$ (4)

$\begin{array}{c}x\left(t\right)={\text{e}}^{-{\int }_{0}^{t}ma\left(u\right)\text{d}u}\left(\psi \left(0\right)-{\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}f\left(\psi \left(u\right)\right)\text{d}u\text{d}s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}f\left(x\left(u\right)\right)\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right){\int }_{-L}^{0}q\left(s\right){\int }_{v+s}^{v}f\left(x\left(u\right)\right)\text{d}u\text{d}s\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{t}{\text{e}}^{-{\int }_{s}^{t}ma\left(u\right)\text{d}u}\text{d}Z\left(s\right)\text{ }\text{ }.\end{array}$

$\begin{array}{c}E|x\left(t\right)|\le E|{\text{e}}^{-{\int }_{0}^{t}ma\left(u\right)\text{d}u}\left(\psi \left(0\right)-{\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}f\left(\psi \left(u\right)\right)\text{d}u\text{d}s\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E|{\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}f\left(x\left(u\right)\right)\text{d}u\text{d}s|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E|{\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right){\int }_{-L}^{0}q\left(s\right){\int }_{v+s}^{v}f\left(x\left(u\right)\right)\text{d}u\text{d}s\text{d}v|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{t}{\text{e}}^{-{\int }_{s}^{t}ma\left(u\right)\text{d}u}\text{d}Z\left(s\right)\\ :={I}_{1}\left(t\right)+{I}_{2}\left(t\right)+{I}_{3}\left(t\right)+{I}_{4}\left(t\right).\end{array}$ (5)

${I}_{1}\left(t\right)\le {\text{e}}^{-{\int }_{0}^{t}ma\left(u\right)\text{d}u}\left(1+KLm\right)‖{\psi }_{0}‖,$ (6)

${I}_{2}\left(t\right)\le K|{\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}E|x\left(u\right)|\text{d}u\text{d}s|\le \left(K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s\right)\cdot \underset{-L\le \theta \le 0}{\mathrm{sup}}E|x\left(t+\theta \right)|,$ (7)

$\begin{array}{c}{I}_{3}\left(t\right)=E|{\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right){\int }_{-L}^{0}q\left(s\right){\int }_{v+s}^{v}f\left(x\left(u\right)\right)\text{d}u\text{d}s\text{d}v|\\ \le KE\left({\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right){\int }_{-L}^{0}|q\left(s\right)|{\int }_{v+s}^{v}E|x\left(u\right)|\text{d}u\text{d}s\text{d}v\right)\\ \le \left(K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s\right)\cdot {\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right)\underset{-L\le \theta \le 0}{\mathrm{sup}}E|x\left(v+\theta \right)|\text{d}v.\end{array}$ (8)

${E}^{\prime }{|\underset{k=1}{\overset{\infty }{\sum }}\text{ }{C}_{k}{\xi }_{k}|}^{p}={A}_{p}{\left(\underset{k=1}{\overset{\infty }{\sum }}{|{C}_{k}|}^{2}\right)}^{\frac{p}{2}},{A}_{p}:={\int }_{R}\frac{{|x|}^{p}}{\sqrt{2\pi }{\text{e}}^{-\frac{{x}^{2}}{2}\text{d}x}}.$

$\begin{array}{c}{I}_{4}\left(t\right)=E|{\int }_{0}^{t}{\text{e}}^{-{\int }_{s}^{t}ma\left(u\right)\text{d}u}\text{d}Z\left(s\right)|={\left(E{E}^{\prime }{|\underset{k=1}{\overset{\infty }{\sum }}\text{ }\text{ }{\xi }_{k}{\int }_{0}^{t}{\text{e}}^{-{\int }_{s}^{t}ma\left(u\right)\text{d}u}\text{d}{W}_{{S}_{s}}|}^{p}\right)}^{\frac{1}{p}}\\ ={\left({E}^{\prime }E{|\underset{k=1}{\overset{\infty }{\sum }}\text{ }\text{ }{\xi }_{k}{\int }_{0}^{t}{\text{e}}^{-{\int }_{s}^{t}ma\left(u\right)\text{d}u}\text{d}{W}_{{S}_{s}}|}^{p}\right)}^{\frac{1}{p}}\le {\left(C{E}^{\prime }{\int }_{0}^{t}{\left(\underset{k=1}{\overset{\infty }{\sum }}\text{ }\text{ }{\xi }_{k}^{2}{\text{e}}^{-2{\int }_{s}^{t}ma\left(u\right)\text{d}u}\right)}^{\frac{\alpha }{2}}\text{d}s\right)}^{\frac{p}{\alpha }}\\ \le {\left(C{\int }_{0}^{t}{\text{e}}^{-\alpha {\int }_{s}^{t}ma\left(u\right)\text{d}u}\text{d}s\right)}^{\frac{1}{\alpha }}\le {\left(C{\int }_{0}^{t}{\text{e}}^{-\alpha \lambda \left(t-s\right)}\text{d}s\right)}^{\frac{1}{\alpha }}\le {\left[\frac{C}{\alpha \lambda }\left(1-{\text{e}}^{-\alpha \lambda t}\right)\right]}^{\frac{1}{\alpha }}.\end{array}$ (9)

$\begin{array}{c}E|x\left(t\right)|\le {\text{e}}^{-{\int }_{0}^{t}ma\left(u\right)\text{d}u}\left(1+KLm\right)‖{\psi }_{0}‖+\left(K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s\right)\cdot \underset{-L\le \theta \le 0}{\mathrm{sup}}E|x\left(t+\theta \right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s\right)\cdot {\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right)\underset{-L\le \theta \le 0}{\mathrm{sup}}E|x\left(v+\theta \right)|\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left[\frac{C}{\alpha \lambda }\left(1-{\text{e}}^{-\alpha \lambda t}\right)\right]}^{\frac{1}{\alpha }}.\end{array}$ (10)

${\eta }_{1}:=\left(1+KLm\right)‖{\psi }_{0}‖,{\eta }_{2}={\eta }_{3}:=K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s,{\eta }_{4}={\left(C{\int }_{0}^{t}{\text{e}}^{-\alpha {\int }_{s}^{t}ma\left(u\right)\text{d}u}\text{d}s\right)}^{\frac{1}{\alpha }}$

$E|x\left(t\right)|\le N{\text{e}}^{-{\lambda }_{1}{\int }_{0}^{t}ma\left(v\right)\text{d}v}+{\left(1-\rho \right)}^{-1}{\eta }_{4},t\ge 0.$ (11)

$\begin{array}{c}E‖{x}_{nL}\left(\psi \right)‖\le E\underset{-L\le \theta \le 0}{\mathrm{sup}}|{\text{e}}^{-{\int }_{\left(n-1\right)L}^{nL+\theta }ma\left(u\right)\text{d}u}\left(x\left(n-1\right)L\right)-{\int }_{-L}^{0}q\left(s\right){\int }_{s+\left(n-1\right)L}^{\left(n-1\right)L}f\left(x\left(u\right)\right)\text{d}u\text{d}s|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\underset{-L\le \theta \le 0}{\mathrm{sup}}|{\int }_{-L}^{0}q\left(s\right){\int }_{nL+\theta +s}^{nL+\theta }f\left(x\left(u\right)\right)\text{d}u\text{d}s|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\underset{-L\le \theta \le 0}{\mathrm{sup}}|{\int }_{\left(n-1\right)L}^{nL+\theta }{\text{e}}^{-{\int }_{v}^{nL+\theta }ma\left(s\right)\text{d}s}ma\left(v\right){\int }_{-L}^{0}q\left(s\right){\int }_{v+s}^{v}f\left(x\left(u\right)\right)\text{d}u\text{d}s\text{d}v|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\underset{-L\le \theta \le 0}{\mathrm{sup}}|{\int }_{\left(n-1\right)L}^{nL+\theta }{\text{e}}^{-{\int }_{s}^{nL+\theta }ma\left(u\right)\text{d}u}\text{d}Z\left(s\right)|\\ \le \left(K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s\right)E‖{x}_{\left(n-1\right)L}\left(\psi \right)‖+C\left(1+\underset{\left(n-2\right)L\le t\le nL}{\mathrm{sup}}E|x\left(t,\psi \right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\underset{\left(n-2\right)L\le t\le nL}{\mathrm{sup}}|{\int }_{0}^{t}{\text{e}}^{-{\int }_{s+\left(n-1\right)L}^{t}ma\left(u\right)\text{d}u}\text{d}\stackrel{˜}{Z}\left(s\right)|\right),\end{array}$ (12)

$\underset{\left(n-2\right)L\le t\le nL}{\mathrm{sup}}E|x\left(t,\psi \right)|\le {\stackrel{˜}{K}}_{1}<\infty ,$

$E\underset{0\le t\le L}{\mathrm{sup}}|{\int }_{0}^{t}{\text{e}}^{-{\int }_{s+\left(n-1\right)L}^{t}ma\left(u\right)\text{d}u}\text{d}\stackrel{˜}{Z}\left(s\right)|\le C\left(\alpha \right)={C}_{2}<\infty .$

$E‖{x}_{nL}\left(\psi \right)‖\le \left(K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s\right)E‖{x}_{\left(n-1\right)L}\left(\psi \right)‖+{\stackrel{˜}{K}}_{2},$

${\stackrel{˜}{K}}_{2}=C\left(1+\underset{\left(n-2\right)L\le t\le nL}{\mathrm{sup}}E|x\left(t,\psi \right)|\right)+{C}_{2}.$

$K{\int }_{-L}^{0}|sq\left(s\right)|\text{d}s=:\mu <1.$

$\begin{array}{c}E‖{x}_{nL}\left(\psi \right)‖\le \mu E‖{x}_{\left(n-1\right)L}\left(\psi \right)‖+{\stackrel{˜}{K}}_{2}\\ \le \mu \left(\mu E‖{x}_{\left(n-1\right)L}\left(\psi \right)‖+{\stackrel{˜}{K}}_{2}\right)\\ \le {\mu }^{n}\underset{-L\le \theta \le 0}{\mathrm{sup}}|\psi \left(\theta \right)|+{\stackrel{˜}{K}}_{2}\left(1+\mu +{\mu }^{2}+\cdot \cdot \cdot +{\mu }^{n-1}\right)\\ \le \underset{-L\le \theta \le 0}{\mathrm{sup}}|\psi \left(\theta \right)|+\frac{{\stackrel{˜}{K}}_{2}}{1-\mu }.\end{array}$ (13)

$E‖{x}_{t}\left(\psi \right)‖\le E‖{x}_{\left(n+1\right)L}\left(\psi \right)‖+E‖{x}_{nL}\left(\psi \right)‖.$

$\underset{0\le t<\infty }{\mathrm{sup}}E‖{x}_{t}\left(\psi \right)‖<\infty .$

$\underset{t\to \infty }{\mathrm{lim}}E|x\left(t,\varphi \right)-x\left(t,\psi \right)|=0.$ (14)

$\underset{t\to \infty }{\mathrm{lim}}E|{x}_{t}\left(\varphi \right)-{x}_{t}\left(\psi \right)|=0,$ (15)

$\begin{array}{l}E|x\left(t,\varphi \right)-x\left(t,\psi \right)|\\ \le E|{\int }_{0}^{t}{\text{e}}^{-{\int }_{0}^{t}ma\left(u\right)\text{d}u}\left(\left(\varphi \left(0\right)-\psi \left(0\right)\right)-{\int }_{-L}^{0}q\left(s\right){\int }_{s}^{0}\left[f\left(\varphi \left(u\right)\right)-f\left(\psi \left(u\right)\right)\right]\text{d}u\text{d}s\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E|{\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}\left[f\left(x\left(u,\varphi \right)\right)-f\left(x\left(u,\psi \right)\right)\right]\text{d}u\text{d}s|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E|{\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right){\int }_{-L}^{0}q\left(s\right){\int }_{t+s}^{t}\left[f\left(x\left(u,\varphi \right)\right)-f\left(x\left(u,\psi \right)\right)\right]\text{d}u\text{d}s\text{d}v|\\ \le {\stackrel{˜}{b}}_{1}{\text{e}}^{-{\int }_{0}^{t}ma\left(u\right)\text{d}u}+{\stackrel{˜}{b}}_{2}\underset{-L\le \theta \le 0}{\mathrm{sup}}E|x\left(t+\theta ,\varphi \right)-x\left(t+\theta ,\psi \right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\stackrel{˜}{b}}_{3}{\int }_{0}^{t}{\text{e}}^{-{\int }_{v}^{t}ma\left(s\right)\text{d}s}ma\left(v\right)\underset{-L\le \theta \le 0}{\mathrm{sup}}E|x\left(v+\theta ,\varphi \right)-x\left(v+\theta ,\psi \right)|\text{d}v,\end{array}$

$t\ge 2L$，根据假设(H1)-(H3)得：

$\begin{array}{l}E\underset{-L\le \theta \le 0}{\mathrm{sup}}|x\left(t+\theta ,\varphi \right)-x\left(t+\theta ,\psi \right)|\\ \le E\underset{-L\le \theta \le 0}{\mathrm{sup}}|{\text{e}}^{-{\int }_{t-L}^{t+\theta }ma\left(u\right)\text{d}u}\left(x\left(t-L,\varphi \right)-x\left(t-L,\psi \right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-{\int }_{-L}^{0}q\left(s\right){\int }_{t-L+s}^{t-L}\left[f\left(x\left(u,\varphi \right)\right)-f\left(x\left(u,\psi \right)\right)\right]\text{d}u\text{d}s|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E\underset{-L\le \theta \le 0}{\mathrm{sup}}|-{\int }_{t-L}^{t+\theta }{\int }_{t+\theta +s}^{t+\theta }\left[f\left(x\left(u,\varphi \right)\right)-f\left(x\left(u,\psi \right)\right)\right]\text{d}u\text{d}s|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E\underset{-L\le \theta \le 0}{\mathrm{sup}}|{\int }_{t-L}^{t+\theta }{\text{e}}^{-{\int }_{v}^{t+\theta }ma\left(s\right)\text{d}s}ma\left(v\right)\cdot {\int }_{-L}^{0}q\left(s\right){\int }_{v+s}^{v}\left[f\left(x\left(u,\varphi \right)\right)-f\left(x\left(u,\psi \right)\right)\right]\text{d}u\text{dsd}v|\end{array}$

$\begin{array}{l}\le E|x\left(t-L,\varphi \right)-x\left(t-L,\psi \right)|+mK{\int }_{t-2L}^{t-L}E|x\left(s,\varphi \right)-x\left(s,\psi \right)|\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+mK{\int }_{t-2L}^{t-L}E|x\left(s,\varphi \right)-x\left(s,\psi \right)|\text{d}s+KmL{\int }_{t-2L}^{t-L}E|x\left(u,\varphi \right)-x\left(u,\psi \right)|\text{d}u\text{ }\text{ }.\end{array}$

$\underset{t\to \infty }{\mathrm{lim}}\underset{-L\le \theta \le 0}{\mathrm{sup}}|x\left(t+\theta ,\varphi \right)-x\left(t+\theta ,\psi \right)|=0.$ (16)

$P\left(C\left(\left[-L,0\right];R\right)\right)$ 为在 $C\left(\left[-L,0\right];R\right)$ 上的概率测度空间。当 ${p}_{1},{p}_{2}\in P\left(C\left(\left[-L,0\right];R\right)\right)$ 时，定义：

${d}_{L}\left({p}_{1},{p}_{2}\right)=\underset{f\in L}{\mathrm{sup}}|{\int }_{e}f\left(\Phi \right){p}_{1}\left(d\varphi \right)-{\int }_{e}f\left(\psi \right){p}_{2}\left(\text{d}\Psi \right)|,$

$L=\left\{f|C\left(\left[-L,0\right];R\right)\to R:|f\left(\varphi \right)-f\left(\psi \right)|\le ‖\varphi -\psi ‖,|f\left(\cdot \right)|\le 1\right\}.$

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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