﻿ G-Brown运动驱动的非线性随机泛函微分方程解的存在唯一性 The Existence and Uniqueness of Solutions to Nonlinear Stochastic Functional Differential Equations Driven by G-Brownian Motion

Vol. 10  No. 11 ( 2021 ), Article ID: 46329 , 6 pages
10.12677/AAM.2021.1011389

G-Brown运动驱动的非线性随机泛函微分方程解的存在唯一性

The Existence and Uniqueness of Solutions to Nonlinear Stochastic Functional Differential Equations Driven by G-Brownian Motion

Weisheng Liang, Huayan Su, Guangjie Li*

School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou Guangdong

Received: Oct. 5th, 2021; accepted: Oct. 26th, 2021; published: Nov. 8th, 2021

ABSTRACT

There are not so many results on the existence and uniqueness of solutions to nonlinear stochastic functional differential equations driven by G-Brownian motion (G-SFDEs). By G-Lyapunov function technique, the existence and uniqueness of the global solution to a G-SFDE is obtained. Finally, an example is presented to illustrate the obtained theory.

Keywords:Nonlinear Stochastic Functional Differential Equations, G-Brownian Motion, The Existence and Uniqueness

1. 引言

2. 预备知识

${R}^{n}$ 表示n-维的欧式空间。若 $x\in {R}^{n}$$|x|$ 表示欧式范数。记 $R=\left(-\infty ,+\infty \right)$${R}^{+}=\left[0,+\infty \right)$。对任意的 $a,b\in R$$a\vee b$ 表示二者中的最大值。令 $\tau >0$$C\left(\left[-\tau ,0\right];R\right)$ 表示所有定义在 $\left[-\tau ,0\right]$ 上的实值连续函数 $\phi$ 的全体，且其上的范数为 $‖\phi ‖={\mathrm{sup}}_{-\tau \le \theta \le 0}|\phi \left(\theta \right)|$$U\left(\left[-\tau ,0\right];{R}^{+}\right)$ 表示所有定义在 $\left[-\tau ,0\right]$ 上Borel可测的有界非负函数 $\eta \left(s\right)$ 且满足 ${\int }_{-\tau }^{0}\eta \left(s\right)\text{d}s=1$ 的全体。 ${I}_{B}$ 表示集合B的示性函数。

${\left(\omega \left(t\right)\right)}_{t\ge 0}$，且对任意的 $a\in R$$G\left(a\right)=\frac{1}{2}\stackrel{^}{E}\left[a{\omega }^{2}\left(1\right)\right]=\frac{1}{2}\left({\stackrel{¯}{\sigma }}^{2}{a}^{+}-{\underset{_}{\sigma }}^{2}{a}^{-}\right)$，其中 $\stackrel{^}{E}\left[{\omega }^{2}\left(1\right)\right]={\stackrel{¯}{\sigma }}^{2}$$-\stackrel{^}{E}\left[-{\omega }^{2}\left(1\right)\right]={\underset{_}{\sigma }}^{2}$ ( $0\le \underset{_}{\sigma }\le \stackrel{¯}{\sigma }<\infty$ )。 ${\mathcal{F}}_{t}$ 表示由G-Brown运动 ${\left(\omega \left(t\right)\right)}_{t\ge 0}$ 生成的滤子。对 $\forall T\in {R}^{n}$$\left[0,T\right]$ 上的一个分割 ${\pi }_{T}=\left\{{t}_{0},{t}_{1},\cdots ,{t}_{N}\right\}$ 满足 $0={t}_{0}<{t}_{1}<{t}_{2}<\cdots <{t}_{N}=T$$\mu \left({\pi }_{T}\right)=\mathrm{max}\left\{|{t}_{i+1}-{t}_{i}|:i=0,1,\cdots ,N-1\right\}$。给定 $p\ge 1$，定义

${M}_{G}^{p,0}\left(\left[0,T\right]\right)=\left\{{\eta }_{t}=\underset{j=0}{\overset{N-1}{\sum }}{\xi }_{j}{I}_{\left[{t}_{j},{t}_{j+1}\right)}\left(t\right):{\xi }_{j}\in {L}_{G}^{p}\left({\Omega }_{{t}_{j}}\right)\right\}$.

${M}_{G}^{p}\left(\left[0,T\right]\right)$ 表示 ${M}_{G}^{p,0}\left(\left[0,T\right]\right)$ 在范数 ${‖\eta ‖}_{{M}_{G}^{p,0}\left(\left[0,T\right]\right)}={\left(\frac{1}{T}{\int }_{0}^{T}\stackrel{^}{E}\left[{|{\eta }_{t}|}^{p}\right]\text{d}t\right)}^{1/p}$ 下的完备空间。接下来给出一个命题

(见文献 [8] )。

$\text{d}x\left(t\right)=f\left(x\left(t\right),{x}_{t},t\right)\text{d}t+g\left(x\left(t\right),{x}_{t},t\right)\text{d}〈\omega 〉\left(t\right)+h\left(x\left(t\right),{x}_{t},t\right)\text{d}\omega \left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 0$ (1)

${x}_{0}=\phi =\left\{\phi \left(\theta \right):-\tau \le \theta \le 0\right\}$${\mathcal{F}}_{0}$ -可测的， $C\left(\left[-\tau ,0\right];R\right)$ -值随机变量，且满足 $\phi \in {M}_{G}^{2}\left(\left[-\tau ,0\right];R\right)$(2)

(A1) 对任意的 $k>0$，存在一个常数 ${L}_{k}>0$ 使得对 $\forall t\ge 0$$\phi ,\varphi \in C\left(\left[-\tau ,0\right];R\right)$，有

$\begin{array}{l}|f\left(\phi \left(0\right),\phi ,t\right)-f\left(\varphi \left(0\right),\varphi ,t\right)|\vee |g\left(\phi \left(0\right),\phi ,t\right)-g\left(\varphi \left(0\right),\varphi ,t\right)|\\ \vee |h\left(\phi \left(0\right),\phi ,t\right)-h\left(\varphi \left(0\right),\varphi ,t\right)|\le {L}_{k}‖\phi -\varphi ‖\end{array}$,

${C}^{2,1}\left(R×{R}^{+};{R}^{+}\right)$ 表示关于变量x二阶连续可导，关于变量t一阶连续可导的全体非负函数 $V\left(x,t\right)$ 的全体。给定 $\forall V\left(x,t\right)\in {C}^{2,1}\left(R×{R}^{+};{R}^{+}\right)$，定义算子

$LV\left(x,y,t\right)={V}_{t}\left(x,t\right)+{V}_{x}\left(x,t\right)f\left(x,y,t\right)+G\left(2g\left(x,y,t\right){V}_{x}\left(x,t\right)+{V}_{xx}\left(x,t\right){h}^{2}\left(x,y,t\right)\right)$,

3. 主要结果

$\mathrm{lim}{\mathrm{inf}}_{|x|\to \infty ,0\le t<\infty }V\left(x,t\right)=\infty$, (3)

$LV\left(\phi \left(0\right),\phi ,t\right)\le C-{\alpha }_{1}W\left(\phi \left(0\right),t\right)+{\alpha }_{2}{\int }_{-\tau }^{0}\eta \left(\theta \right)W\left(\phi \left(\theta \right),t+\theta \right)\text{d}\theta$ (4)

$\begin{array}{c}V\left(x\left(t\right),t\right)=V\left(x\left(0\right),0\right)+{\int }_{0}^{t}LV\left(x\left(s\right),{x}_{s},s\right)\text{d}s+{\int }_{0}^{t}{V}_{x}\left(x\left(s\right),s\right)h\left(x\left(s\right),{x}_{s},s\right)\text{d}\omega \left(s\right)\\ \text{\hspace{0.17em}}+{\int }_{0}^{t}{V}_{x}\left(x\left(s\right),s\right)g\left(x\left(s\right),{x}_{s},s\right)\text{d}〈\omega 〉\left(s\right)+\frac{1}{2}{\int }_{0}^{t}{V}_{xx}\left(x\left(s\right),s\right){h}^{2}\left(x\left(s\right),{x}_{s},s\right)\text{d}〈\omega 〉\left(s\right)\\ \text{\hspace{0.17em}}-{\int }_{0}^{t}G\left(2g\left(x\left(s\right),{x}_{s},s\right){V}_{x}\left(x\left(s\right),s\right)+{V}_{xx}\left(x\left(s\right),s\right){h}^{2}\left(x\left(s\right),{x}_{s},s\right)\right)\text{d}s\\ =V\left(x\left(0\right),0\right)+{\int }_{0}^{t}LV\left(x\left(s\right),{x}_{s},s\right)\text{d}s+{G}_{t}\end{array}$, (5)

$\begin{array}{c}{G}_{t}={\int }_{0}^{t}{V}_{x}\left(x\left(s\right),s\right)h\left(x\left(s\right),{x}_{s},s\right)\text{d}\omega \left(s\right)+{\int }_{0}^{t}{V}_{x}\left(x\left(s\right),s\right)g\left(x\left(s\right),{x}_{s},s\right)\text{d}〈\omega 〉\left(s\right)\\ \text{\hspace{0.17em}}+\frac{1}{2}{\int }_{0}^{t}{V}_{xx}\left(x\left(s\right),s\right){h}^{2}\left(x\left(s\right),{x}_{s},s\right)\text{d}〈\omega 〉\left(s\right)\\ \text{\hspace{0.17em}}-{\int }_{0}^{t}G\left(2g\left(x\left(s\right),{x}_{s},s\right){V}_{x}\left(x\left(s\right),s\right)+{V}_{xx}\left(x\left(s\right),s\right){h}^{2}\left(x\left(s\right),{x}_{s},s\right)\right)\text{d}s\end{array}$

$\stackrel{^}{E}V\left(x\left(t\wedge {\tau }_{k}\right),t\wedge {\tau }_{k}\right)=V\left(x\left(0\right),0\right)+\stackrel{^}{E}{\int }_{0}^{t\wedge {\tau }_{k}}LV\left(x\left(s\right),{x}_{s},s\right)\text{d}s$.

$\begin{array}{c}\stackrel{^}{E}V\left(x\left(t\wedge {\tau }_{k}\right),t\wedge {\tau }_{k}\right)\le \stackrel{^}{E}V\left(x\left(0\right),0\right)+Ct-{\alpha }_{1}\stackrel{^}{E}{\int }_{0}^{t\wedge {\tau }_{k}}W\left(x\left(s\right),s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{ }+{\alpha }_{2}\stackrel{^}{E}{\int }_{0}^{t\wedge {\tau }_{k}}{\int }_{-\tau }^{0}\eta \left(\theta \right)W\left(x\left(s+\theta \right),s+\theta \right)\text{d}\theta \text{d}s\end{array}$. (6)

$\begin{array}{l}{\int }_{0}^{t\wedge {\tau }_{k}}{\int }_{-\tau }^{0}\eta \left(\theta \right)W\left(x\left(s+\theta \right),s+\theta \right)\text{d}\theta \text{d}s\\ ={\int }_{0}^{t\wedge {\tau }_{k}}{\int }_{s-\tau }^{s}\eta \left(u-s\right)W\left(x\left(u\right),u\right)\text{d}u\text{d}s\\ ={\int }_{-\tau }^{t\wedge {\tau }_{k}}\left({\int }_{u\vee 0}^{\left(u+\tau \right)\wedge \left(t\wedge {\tau }_{k}\right)}\eta \left(u-s\right)\text{d}s\right)W\left(x\left(u\right),u\right)\text{d}u\\ \le {\int }_{-\tau }^{t\wedge {\tau }_{k}}\left({\int }_{u}^{u+\tau }\eta \left(u-s\right)\text{d}s\right)W\left(x\left(u\right),u\right)\text{d}u\\ ={\int }_{-\tau }^{t\wedge {\tau }_{k}}\left({\int }_{-\tau }^{0}\eta \left(r\right)\text{d}r\right)W\left(x\left(u\right),u\right)\text{d}u\\ ={\int }_{-\tau }^{t\wedge {\tau }_{k}}W\left(x\left(s\right),s\right)\text{d}s\end{array}$.(7)

$\begin{array}{c}\stackrel{^}{E}V\left(x\left(t\wedge {\tau }_{k}\right),t\wedge {\tau }_{k}\right)\le \stackrel{^}{E}V\left(x\left(0\right),0\right)+Ct-{\alpha }_{1}\stackrel{^}{E}{\int }_{0}^{t\wedge {\tau }_{k}}W\left(x\left(s\right),s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{ }+{\alpha }_{2}\stackrel{^}{E}{\int }_{-\tau }^{0}W\left(x\left(s\right),s\right)\text{d}s+{\alpha }_{2}E{\int }_{0}^{t\wedge {\tau }_{k}}W\left(x\left(s\right),s\right)\text{d}s\end{array}$.

${\alpha }_{1}\ge {\alpha }_{2}$，所以

$\stackrel{^}{E}V\left(x\left(t\wedge {\tau }_{k}\right),t\wedge {\tau }_{k}\right)\le \stackrel{^}{E}V\left(x\left(0\right),0\right)+Ct+{\alpha }_{2}\stackrel{^}{E}{\int }_{-\tau }^{0}W\left(x\left(s\right),s\right)\text{d}s$.

${V}_{k}=\mathrm{lim}{\mathrm{inf}}_{|x|\ge k,0\le t<\infty }V\left(x,t\right)$ ( $\forall k\ge {k}_{0}$ )。从而可得对 $\forall P\in \mathcal{P}$，有

$\begin{array}{c}P\left({\tau }_{k}\le t\right){V}_{k}\le \stackrel{^}{E}\left({I}_{\left\{{\tau }_{k}\le t\right\}}V\left(x\left({\tau }_{k}\right),{\tau }_{k}\right)\right)\\ \le \stackrel{^}{E}V\left(x\left(0\right),0\right)+Ct+{\alpha }_{2}\stackrel{^}{E}{\int }_{-\tau }^{0}W\left(x\left(s\right),s\right)\text{d}s\end{array}$.

$k\to \infty$ 时，

$\begin{array}{c}P\left({\tau }_{\infty }\le t\right)=\underset{k\to \infty }{\mathrm{lim}}P\left({\tau }_{k}\le t\right)\\ \le \underset{k\to \infty }{\mathrm{lim}}\frac{\stackrel{^}{E}V\left(x\left(0\right),0\right)+Ct+{\alpha }_{2}\stackrel{^}{E}{\int }_{-\tau }^{0}W\left(x\left(s\right),s\right)\text{d}s}{{V}_{k}}\\ =0\end{array}$.

$P\left({\tau }_{\infty }>t\right)=1$。由t的任意性知 $P\left({\tau }_{\infty }=\infty \right)=1$ a.s.进一步可得

$\stackrel{^}{C}\left({\tau }_{\infty }=\infty \right)={\mathrm{sup}}_{P\in \mathcal{P}}P\left({\tau }_{\infty }=\infty \right)=1$.

${\tau }_{\infty }=\infty$ q.s.

4. 例子

$\tau =1$$\eta \left(\theta \right)=1,\theta \in \left[-1,0\right]$。考虑如下形式的G-Brown运动驱动的随机泛函微分方程：

$\text{d}x\left(t\right)=\left(-3x\left(t\right)+{\int }_{-1}^{0}|x\left(t+\theta \right)|\text{d}\theta \right)\text{d}t+x\left(t\right)\text{d}〈\omega 〉\left(t\right)+\mathrm{sin}\left(x\left(t\right)\right)\text{d}\omega \left(t\right)$ (8)

$\forall t\ge 0$，这里 $\omega \left(t\right)$ 是1-维的G-Brown运动且 $\omega \left(1\right)~N\left(0,\left[\frac{1}{2},1\right]\right)$。易验证方程(8)满足条件(A1)。取 $V\left(x,t\right)={x}^{2}$。由

$LV\left(x,y,t\right)={V}_{t}\left(x,t\right)+{V}_{x}\left(x,t\right)f\left(x,y,t\right)+G\left(2g\left(x,y,t\right){V}_{x}\left(x,t\right)+{V}_{xx}\left(x,t\right){h}^{2}\left(x,y,t\right)\right)$,

$\begin{array}{c}LV\left(x\left(t\right),{x}_{t},t\right)=2x\left(t\right)\left(-3x\left(t\right)+{\int }_{-1}^{0}|x\left(t+\theta \right)|\text{d}\theta \right)+G\left(4{x}^{2}\left(t\right)+2{\mathrm{sin}}^{2}\left(x\left(t\right)\right)\right)\\ \le -2{x}^{2}\left(t\right)+{\int }_{-1}^{0}{|x\left(t+\theta \right)|}^{2}\text{d}\theta \end{array}$.

The Existence and Uniqueness of Solutions to Nonlinear Stochastic Functional Differential Equations Driven by G-Brownian Motion[J]. 应用数学进展, 2021, 10(11): 3673-3678. https://doi.org/10.12677/AAM.2021.1011389

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13. NOTES

*通讯作者。