南宁师范大学，数学与统计学院，广西 南宁

收稿日期：2022年4月21日；录用日期：2022年5月23日；发布日期：2022年5月30日

摘要

本文主要研究了G-布朗运动驱动的金融风险系统模型的渐近行为。利用李雅普诺夫函数和Gronwall不等式，证明了G-布朗运动驱动的随机金融风险系统模型解的存在唯一性。运用G-伊藤公式和G期望不等式等相关知识，研究了G-布朗运动驱动的随机金融风险系统模型解在不同平衡点的有界性与全局指数吸引集。

关键词

金融风险系统模型，G-布朗运动，存在唯一性，有界性，全局指数吸引集

Asymptotic Behavior of Stochastic Financial Risk System Models Driven by G-Brownian Motion

Yitian Gao, Shijia Liu, Qi Li, Zaitang Huang^*

School of Mathematics and Statistics, Nanning Normal University, Nanning Guangxi

Received: Apr. 21^st, 2022; accepted: May 23^rd, 2022; published: May 30^th, 2022

ABSTRACT

This paper mainly studies the asymptotic behavior of the financial risk system model driven by G-Brownian motion. Using Lyapunov function and Gronwall inequality, the existence and uniqueness of the solution of the stochastic financial risk system model driven by G-Brownian motion are proved. Using G-Itô formula and G-expectation inequality, the boundedness and global exponential attraction set of the solution of the stochastic financial risk system model driven by G-Brownian motion at different equilibrium points are discussed.

Keywords:Financial Risk System Model, G-Brownian Motion, Existence and Uniqueness, Boundedness, Global Exponential Attractive Set

Copyright © 2022 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

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

1. 引言

人们试图使用混沌理论的知识和方法来探究金融模型和经济相关问题，是因为诸如股价、金融风险、量化数据等从数据表面看是极不规律的，而混沌正是系统极不规律的一种表现形式 [1] [2] [3]。2016年，徐玉华等人提出金融系统风险的演化分为三个阶段：第一阶段，金融系统受到来自外部或内部的冲击；第二阶段，金融系统间的传染效应，使得系统风险再一次改变，从而破坏系统稳定性；第三阶段，面对系统风险，金融机构、监管部门和货币政策作出调整和管控。因此，他们提出了金融风险系统模型 [4]，

$\{\begin{array}{l}\stackrel{\dot{}}{x}=yz-\left(a+1\right)x,\hfill \\ \stackrel{\dot{}}{y}=xz-\left(b+1\right)y,\hfill \\ \stackrel{\dot{}}{z}=\left(c-1\right)z-xy,\hfill \end{array}$ (1.1)

其中，x表示第一阶段来自金融系统外部或内部对系统冲击的总风险值，y表示第二阶段经过金融系统传染效应后的总风险值，z表示第三阶段金融系统受到调控后的总风险值。a，b表示风险程度，c表示控制强度， $a,b\in R^{+}$，$c>1$。三个阶段的风险相互作用，相互影响。金融风险系统模型(1.1)存在一个无风险均衡点 $E_0=\left(0,0,0\right)$ 和四个有风险均衡点。

非线性期望理论发展非常迅速，与概率理论所推导出的线性期望理论不同的是，线性期望理论主要擅长处理那些其相应的概率模型能够通过数理统计方法和数据分析予以确定的情形，而真实世界与人有关的各项数据并不能确定其分布情况 [5]。将非线性期望理论运用在金融系统具有现实意义和应用基础 [6]，如次线性期望的平移不变性，意味着为了降低风险，可以在风险位置增加现金流；次线性期望的次可加性，意味着投资组合可以在很大程度上降低风险，这是符合人们认知的。且由G-布朗运动驱动的随机动力系统的动力学性质研究甚少，包括随机吸引子、随机有界性、随机分岔和随机混沌等。

因此，本文不再局限于经典的标准布朗运动，反之引入由次线性期望的体系中建立的非线性布朗运动，即G-布朗运动，从而进一步将金融风险系统模型(1.1)转化成如下随机金融风险系统模型：

$\{\begin{array}{l}\text{d}x\left(t\right)=\left(y\left(t\right)z\left(t\right)-\left(a+1\right)x\left(t\right)\right)\text{d}t+c_{1}x\left(t\right)\text{d}{\langle B_1,B_1\rangle }_t+{\sigma }_{1}x\left(t\right)\text{d}{B}_1\left(t\right),\hfill \\ \text{d}y\left(t\right)=\left(x\left(t\right)z\left(t\right)-\left(b+1\right)y\left(t\right)\right)\text{d}t+c_{2}y\left(t\right)\text{d}{\langle B_2,B_2\rangle }_t+{\sigma }_{2}y\left(t\right)\text{d}{B}_2\left(t\right),\hfill \\ \text{d}z\left(t\right)=\left(\left(c-1\right)z\left(t\right)-x\left(t\right)y\left(t\right)\right)\text{d}t+c_{3}z\left(t\right)\text{d}{\langle B_3,B_3\rangle }_t+{\sigma }_{3}z\left(t\right)\text{d}{B}_3\left(t\right),\hfill \end{array}$ (1.2)

其中， $B_i$ 是两两相互独立的G-布朗运动，它们都满足 $\frac{B_i\left(t\right)}{\sqrt{t}}\stackrel{d}{=}N\left(0,\left[{\underset{\_}{\sigma }}^2,{\bar{\sigma }}^2\right]\right),i=1,2,3$，$\langle B_i,B_i\rangle$ 是G-布朗运动二次变差过程。 ${\sigma }_1,{\sigma }_2,{\sigma }_3$ 是G-布朗运动二次变差过程对应的扰动强度， $c_1,c_2,c_3$ 是G-布朗运动对应的扰动强度。

本文的组织结构如下：在第2节中，给出一些准备知识，供后续章节使用。在第3节中，利用李雅普诺夫函数，证明了G-布朗运动驱动的随机金融风险系统模型(1.2)解的存在唯一性。在第4节中，运用G-伊藤公式和G期望不等式等相关知识，研究了G-布朗运动驱动的随机金融风险系统模型(1.2)解在不同平衡点的有界性与全局指数吸引集。

2. 准备知识

本节，我们介绍一些关于次线性期望和G-布朗运动的基本概念和性质 [7] - [18]。

定义2.1 [7] 一个非线性期望 $\hat{\mathbb{E}}$ 是定义在 $\mathcal{H}$ 上的实值函数， $\mathcal{H}$ 是定义在 $\Omega$ 上的实质函数所组成的一个线性空间， $\hat{\mathbb{E}}:\mathcal{H}\to ℝ$，并满足以下四个条件：

1) 单调性：对于所有的 $X,Y\in \mathcal{H}$，且满足 $X\left(\omega \right)\ge Y\left(\omega \right)$，那么 $\hat{\mathbb{E}}\left(X\right)\ge \hat{\mathbb{E}}\left(Y\right)$ ；

2) 保常数性：对于常数 $c\in ℝ$，则有 $\hat{\mathbb{E}}\left(X+c\right)=\hat{\mathbb{E}}\left(X\right)+c$ ；

3) 次可加性：对任意的 $X,Y\in \mathcal{H}$，有 $\hat{\mathbb{E}}\left(X+Y\right)\le \hat{\mathbb{E}}\left(X\right)+\hat{\mathbb{E}}\left(Y\right)$ ；

4) 正齐次性：对于任意的 $X\in \mathcal{H}$ 和数 $\lambda >0$，有 $\hat{\mathbb{E}}\left(\lambda X\right)=\lambda \hat{\mathbb{E}}\left(X\right)$，

称二元组 $\left(\Omega ,\mathcal{H},\hat{\mathbb{E}}\right)$ 为次线性期望空间。

定义2.2 [7] (G-布朗运动的定义)定义在一个次线性期望 $\left(\Omega ,\mathcal{H},\hat{\mathbb{E}}\right)$ 空间中的随机过程 $B_t\left(\omega \right)\left(t\ge 0\right)$ 为关于 $\hat{\mathbb{E}}$ 的布朗运动，如果对于每一个 $n\in \mathbb{N}$ 和 $0\le t_1,\cdots ,t_n\le \infty$，都有

1) $B_0\left(\omega \right)=0$ ；

2) $B_t$ 的增量平稳且独立，即对于每一个 $t,s\ge 0$，$B_{t+s}-B_t$ 服从G分布；

3) 对于所有的 $n\in \mathbb{N}$ 和 $t_1,\cdots ,t_n\in \left[0,t\right]$，$B_{t+s}-B_t$ 独立于 $B_{t_1},B_{t_2},\cdots ,B_{t_n}$。

定理2.3 [7] 在 $\left(\Omega ,L_G^P\left(\Omega \right),\hat{\mathbb{E}}\right)$ 中的一个完备的伊藤过程为

$X_t^{\mu }=X_0^{\mu }+{\displaystyle {\int }_0^t{\alpha }_s^{\mu }}\text{d}s+{\displaystyle {\int }_0^t{\eta }_s^{\mu }}\text{d}{\langle B\rangle }_s+{\displaystyle {\int }_0^t{\beta }_s^{\mu }}\text{d}{B}_s$

设 ${\alpha }^{\mu },{\eta }^{\mu }\in M_G^1\left(0,T\right)$ 和 ${\beta }^{\mu }\in M_G^2\left(0,T\right)$，$\mu =1,\cdots ,n$，则对于每一个 $t\in \left[0,T\right]$ 和函数 $\Phi \in C^{1,2}\left(\left[0,T\right]\times {ℝ}^n\right)$，有

$\begin{array}{c}\Phi \left(t,X_t\right)-\Phi \left(s,X_s\right)={\displaystyle \underset{\mu =1}{\overset{n}{\sum }}{\displaystyle {\int }_s^t{\partial }_{x^{\mu }}\Phi \left(u,X_u\right){\beta }_u^{\mu }\text{d}{B}_u}+{\displaystyle {\int }_s^t\left[{\partial }_u\Phi \left(u,X_u\right)+{\partial }_{x^{\mu }}\Phi \left(u,X_u\right){\alpha }_u^{\mu }\right]\text{d}u}}\\ +{\displaystyle {\int }_s^t\left\{{\displaystyle \underset{\mu =1}{\overset{n}{\sum }}{\partial }_{x^{\mu }}\Phi \left(u,X_u\right){\eta }_u^{\mu }}+\frac{1}{2}{\displaystyle \underset{\mu =1}{\overset{n}{\sum }}{\partial }_{x^{\mu }{x}^{\mu }}^2\Phi \left(u,X_u\right){\beta }_u^{\mu }{\beta }_u^{\mu }}\right\}\text{d}{\langle B\rangle }_u}.\end{array}$

3. 全局解的存在唯一性

在本小节文，主要证了G-布朗运动驱动的明随机金融风险系统模型(1.2)解的存在唯一性。

定理3.1对任意给定的初值 $\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)\in {ℝ}^3$，则G-布朗运动驱动的随机金融风险系统模型(1.2)存在唯一解 $\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)$。

证明：由于G-布朗运动驱动的随机金融风险系统模型(1.2)的系数是局部Lipschitz连续的，因此对给定的初值 $\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)\in {ℝ}^3$，存在一个局部解 $\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)$，$t\in \left[0,{\tau }_e\right]$，其中 ${\tau }_e$ 为爆破时刻。为了证明解是全局的，我们只需要证明 ${\tau }_e=\infty$，q.s.。

令 $k_0\ge 1$ 充分大，使得 $x_0,y_0,z_0\in \left[-k_0,k_0\right]$，对于每个整数 $k\ge k_0$，定义停时：

${\tau }_k=\inf \left\{t\in \left[0,{\tau }_e\right);\min \left\{x\left(t\right),y\left(t\right),z\left(t\right)\right\}\le -k\&\max \left\{x\left(t\right),y\left(t\right),z\left(t\right)\right\}\ge k\right\}.$

显然，随着 $k\to \infty$，${\tau }_k$ 不断递增。令 ${\tau }_{\infty }=\underset{k\to \infty }{\lim }{\tau }_k$，有 ${\tau }_{\infty }\le {\tau }_e$，q.s.如果能证明 ${\tau }_e=\infty$，q.s.，则有 ${\tau }_e=\infty$，q.s.，并且对所有的 $t\ge 0$，$\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)\in {ℝ}^3$，q.s.。

为了证明该结论，我们先定义一个 $C^3$ 上的函数 $V:R^3\to R$，如下：

$V\left(x,y,z\right)=m{x}^2+n{y}^2+\left(m+n\right)z^2,$

$m,n,l$ 为常数。令 $k>k_0,T>0$ 为任意数，当 $0\le t\le {\tau }_k\wedge T$，对函数V运用G-伊藤公式，得到

$\begin{array}{l}V\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)\\ =V\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)+{\displaystyle {\int }_0^{T\wedge {\tau }_k}2}mx\left(t\right)\cdot \left(y\left(t\right)z\left(t\right)-\left(a+1\right)x\left(t\right)\right)\\ +2ny\left(t\right)\cdot \left(x\left(t\right)z\left(t\right)-\left(b+1\right)y\left(t\right)\right)+2\left(m+n\right)z\left(t\right)\cdot \left(\left(c-1\right)z\left(t\right)-x\left(t\right)y\left(t\right)\right)\text{d}t\\ +{\displaystyle {\int }_0^{T\wedge {\tau }_k}2m{\sigma }_1\cdot x^2\left(t\right)\text{d}{B}_1\left(t\right)}+2n{\sigma }_2\cdot y^2\left(t\right)\text{d}{B}_2\left(t\right)+2\left(m+n\right){\sigma }_3\cdot z^2\left(t\right)\text{d}{B}_3\left(t\right)\\ +{\displaystyle {\int }_0^{T\wedge {\tau }_k}2m{c}_1{x}^2\left(t\right)}+m{\sigma }_1^2{x}^2\left(t\right)\text{d}{\langle B_1,B_1\rangle }_t+2n{c}_2{y}^2\left(t\right)+n{\sigma }_2^2{y}^2\left(t\right)\text{d}{\langle B_2,B_2\rangle }_t\\ +2\left(m+n\right)c_3{z}^2\left(t\right)+\left(m+n\right){\sigma }_3^2{z}^2\left(t\right)\text{d}{\langle B_3,B_3\rangle }_t.\end{array}$ (3.1)

因此，上述等式(3.1)可以写成：

$\begin{array}{l}V\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)-V\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)\\ ={\displaystyle {\int }_0^{T\wedge {\tau }_k}2}mx\left(t\right)\cdot \left(y\left(t\right)z\left(t\right)-\left(a+1\right)x\left(t\right)\right)+2ny\left(t\right)\cdot \left(x\left(t\right)z\left(t\right)-\left(b+1\right)y\left(t\right)\right)\\ +2\left(m+n\right)z\left(t\right)\cdot \left(\left(c-1\right)z\left(t\right)-x\left(t\right)y\left(t\right)\right)\text{d}t+{\displaystyle {\int }_0^{T\wedge {\tau }_k}2}m{\sigma }_1\cdot x^2\left(t\right)\text{d}{B}_1\left(t\right)\\ +2n{\sigma }_2\cdot y^2\left(t\right)\text{d}{B}_2\left(t\right)+2\left(m+n\right){\sigma }_3\cdot z^2\left(t\right)\text{d}{B}_3\left(t\right)+{\displaystyle {\int }_0^{T\wedge {\tau }_k}{\bar{\sigma }}_1^2}\left(2m{c}_1{x}^2\left(t\right)+m{\sigma }_1^2{x}^2\left(t\right)\right)\text{d}t\\ +{\bar{\sigma }}_2^2\left(2n{c}_2{y}^2\left(t\right)+n{\sigma }_2^2{y}^2\left(t\right)\right)\text{d}t+{\bar{\sigma }}_3^2\left(2\left(m+n\right)c_3{z}^2\left(t\right)+\left(m+n\right){\sigma }_3^2{z}^2\left(t\right)\right)\text{d}t\end{array}$

$\begin{array}{l}={\displaystyle {\int }_0^{T\wedge {\tau }_k}\left(2m{c}_1{\bar{\sigma }}_1^2+m{\sigma }_1^2{\bar{\sigma }}_1^2-2m\left(a+1\right)\right)x^2\left(t\right)1_{0,{\tau }_k}\text{d}t}\\ +\left(2n{c}_2{\bar{\sigma }}_2^2+n{\sigma }_2^2{\bar{\sigma }}_2^2-2n\left(b+1\right)\right)y^2\left(t\right)1_{0,{\tau }_k}\text{d}t\\ +\left(2\left(m+n\right)c_3{\bar{\sigma }}_3^2+\left(m+n\right){\sigma }_3^2{\bar{\sigma }}_3^2+2\left(m+n\right)\left(c-1\right)\right)z^2\left(t\right)1_{0,{\tau }_k}\text{d}t\\ +{\displaystyle {\int }_0^{T\wedge {\tau }_k}2}m{\sigma }_1\cdot x^2\left(t\right)1_{0,{\tau }_k}\text{d}{B}_1\left(t\right)+2n{\sigma }_2\cdot y^2\left(t\right)1_{0,{\tau }_k}\text{d}{B}_2\left(t\right)\\ +2\left(m+n\right){\sigma }_3\cdot z^2\left(t\right)1_{0,{\tau }_k}\text{d}{B}_3\left(t\right).\end{array}$ (3.2)

对式子(3.2)，两边取G-期望后，有

$\hat{\mathbb{E}}{\displaystyle {\int }_0^{T\wedge {\tau }_k}2}m{\sigma }_1\cdot x^2\left(t\right)1_{0,{\tau }_k}\text{d}{B}_1\left(t\right)=0,$

$\hat{\mathbb{E}}{\displaystyle {\int }_0^{T\wedge {\tau }_k}2}n{\sigma }_2\cdot y^2\left(t\right)1_{0,{\tau }_k}\text{d}{B}_2\left(t\right)=0,$

$\hat{\mathbb{E}}{\displaystyle {\int }_0^{T\wedge {\tau }_k}2}l{\sigma }_3\cdot z^2\left(t\right)1_{0,{\tau }_k}\text{d}{B}_3\left(t\right)=0.$

因此，可以得到

$\begin{array}{c}\hat{\mathbb{E}}V\left(X_{T\wedge {\tau }_k},Y_{T\wedge {\tau }_k},Z_{T\wedge {\tau }_k}\right)\le V\left(x_0,y_0,z_0\right)+\hat{\mathbb{E}}{\displaystyle {\int }_0^{T\wedge {\tau }_k}C}V\left(t,x,y,z\right)\text{d}t\\ \le V\left(x_0,y_0,z_0\right)+\hat{\mathbb{E}}{\displaystyle {\int }_0^{T\wedge {\tau }_k}C}V\left(X_{T\wedge {\tau }_k},Y_{T\wedge {\tau }_k},Z_{T\wedge {\tau }_k}\right)\text{d}t.\end{array}$

其中， $C=\max \{2m{c}_1{\bar{\sigma }}_1^2+m{\sigma }_1^2{\bar{\sigma }}_1^2-2m\left(a+1\right)$，$2n{c}_2{\bar{\sigma }}_2^2+n{\sigma }_2^2{\bar{\sigma }}_2^2-2n\left(b+1\right)$，$2\left(m+n\right)c_3{\bar{\sigma }}_3^2+\left(m+n\right){\sigma }_3^2{\bar{\sigma }}_3^2+2\left(m+n\right)\left(c-1\right)\}$，因此，由Gronwall不等式可以得到：

$\hat{\mathbb{E}}V\left(X_{T\wedge {\tau }_k},Y_{T\wedge {\tau }_k},Z_{T\wedge {\tau }_k}\right)\le V\left(x_0,y_0,z_0\right){\text{e}}^{CT}.$ (3.3)

对于每一个 $\omega \in {\tau }_i\le T$，至少存在 $X_{0,{\tau }_k}$ 、 $Y_{0,{\tau }_k}$ 、 $Z_{0,{\tau }_k}$ 中的一项等于 $-k$ 或k。记 $\hat{c}=\min \left\{m,n,m+n\right\}$，于是

$V\left(X_{T\wedge {\tau }_k},Y_{T\wedge {\tau }_k},Z_{T\wedge {\tau }_k}\right)\le \hat{c}{k}^2.$ (3.4)

因此，可以得到：

$\begin{array}{c}V\left(x_0,y_0,z_0\right){\text{e}}^{CT}\ge \hat{\mathbb{E}}\left[1_{{\tau }_k\le T}V\left(X_{T\wedge {\tau }_k},Y_{T\wedge {\tau }_k},Z_{T\wedge {\tau }_k}\right)\right]\\ \ge \bar{C}\left(\left\{\omega :{\tau }_k\le T\right\}\right)V\left(X_{T\wedge {\tau }_k},Y_{T\wedge {\tau }_k},Z_{T\wedge {\tau }_k}\right)\\ \ge \bar{C}\left(\left\{\omega :{\tau }_k\le T\right\}\right)\hat{c}{k}^2,\end{array}$ (3.5)

其中 $1_{{\tau }_k\le T}$ 是 ${\tau }_k\le T$ 的示性函数。令 $k\to \infty$ 有

$\underset{k\to \infty }{\lim }\bar{C}\left\{\omega :{\tau }_k\le T\right\}=0.$

从而，有

$\bar{C}\left\{{\tau }_{\infty }\le T\right\}=0.$

因为 $T>0$ 是任意的，我们可以得出：

$\bar{C}\left\{{\tau }_{\infty }=\infty \right\}=1.\text{q}\text{.s}.$

则在 ${ℝ}^3$ 上，随机金融风险系统模型(1.2)存在全局唯一解。证明完毕。

4. 随机吸引子与有界性

本小节主要研究G-布朗运动驱动的随机金融风险系统模型(1.2)的渐近行为，包括随机吸引子和有界性。

定理4.1如果 $2\left(a+1\right)>{\bar{\sigma }}_1^2\left(2c_1+{\sigma }_1^2\right)$，$2\left(b+1\right)>{\bar{\sigma }}_2^2\left(2c_2+{\sigma }_2^2\right)$，$K_1>0$ 成立。则对给定初解 $E_0=\left(0,0,0\right)\in {ℝ}^3$，G-布朗运动驱动的随机金融风险系统模型(1.2)的解在无风险均衡点有以下性质：

$\begin{array}{l}\underset{t\to \infty }{\lim \sup }\frac{1}{t}\hat{\mathbb{E}}{\displaystyle {\int }_0^t{x}^2}\left(s\right)+y^2\left(s\right)+{\left(z\left(s\right)+\frac{1}{K_1}\right)}^2\text{d}t\\ \le \frac{1}{K_2}\left(a+b+2+\left(\frac{{\sigma }_1^2}{2}-c_1\right){\bar{\sigma }}_1^2+\left(\frac{{\sigma }_2^2}{2}-c_2\right){\bar{\sigma }}_2^2+\frac{1}{K_1}\right).\end{array}$

其中，

$K_1=4\left(1-c\right)-{\bar{\sigma }}_3^2\left(4c_3+2{\sigma }_3^2\right),$

$K_2=\min \left\{2\left(a+1\right)-{\bar{\sigma }}_1^2\left(2c_1+{\sigma }_1^2\right),2\left(b+1\right)-{\bar{\sigma }}_2^2\left(2c_2+{\sigma }_2^2\right),K_1\right\}.$

证明：构造以下Lyapunov函数：

$V\left(x,y,z\right)=x^2-\ln x+y^2-\ln y+2z^2$

根据G-伊藤公式，可以得到：

$\begin{array}{c}\text{d}V=\left(\left(2x-\frac{1}{x}\right)\frac{\text{d}x}{\text{d}t}+\left(2y-\frac{1}{y}\right)\frac{\text{d}y}{\text{d}t}+4z\frac{\text{d}z}{\text{d}t}\right)\text{d}t\\ +\left(\left(2x-\frac{1}{x}\right)\frac{\text{d}x}{\text{d}{B}_t}+\left(2y-\frac{1}{y}\right)\frac{\text{d}y}{\text{d}{B}_t}+4z\frac{\text{d}z}{\text{d}{B}_t}\right)\text{d}{B}_t\\ +\left(\left(2x-\frac{1}{x}\right)\frac{\text{d}x}{\text{d}{\langle B_1,B_1\rangle }_t}+\left(2y-\frac{1}{y}\right)\frac{\text{d}y}{\text{d}{\langle B_2,B_2\rangle }_t}+4z\frac{\text{d}z}{\text{d}{\langle B_3,B_3\rangle }_t}\right)\text{d}{\langle B,B\rangle }_t\\ +\left(\left(2+\frac{1}{x^2}\right)\frac{{\text{d}}^{2}x}{2\text{d}{B}_t^2}+\left(2+\frac{1}{y^2}\right)\frac{{\text{d}}^{2}y}{2\text{d}{B}_t^2}+4\frac{{\text{d}}^{2}z}{2\text{d}{B}_t^2}\right)\text{d}{\langle B,B\rangle }_t.\end{array}$ (4.1)

因此，获得

$\begin{array}{l}V\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)\\ =V\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)+{\displaystyle {\int }_0^t\left(2x\left(s\right)-\frac{1}{x\left(s\right)}\right)}\left(y\left(s\right)z\left(s\right)-\left(a+1\right)x\left(s\right)\right)\\ +\left(2y\left(s\right)-\frac{1}{y\left(s\right)}\right)\left(x\left(s\right)z\left(s\right)-\left(b+1\right)y\left(s\right)\right)+4z\left(s\right)\left(\left(c-1\right)z\left(s\right)-x\left(s\right)y\left(s\right)\right)\text{d}s\\ +{\displaystyle {\int }_0^t\left(2x\left(s\right)-\frac{1}{x\left(s\right)}\right)c_{1}x\left(s\right)}+\frac{1}{2}\left(2+\frac{1}{x{\left(s\right)}^2}\right){\sigma }_1^2{x}^2\left(s\right)\text{d}{\langle B_1,B_1\rangle }_s\\ +\left(2y\left(s\right)-\frac{1}{y\left(s\right)}\right)c_{2}y\left(s\right)+\frac{1}{2}\left(2+\frac{1}{y{\left(s\right)}^2}\right){\sigma }_2^2{y}^2\left(s\right)\text{d}{\langle B_2,B_2\rangle }_s\\ +4z\left(s\right)c_{3}z\left(s\right)+4\frac{1}{2}{\sigma }_3^2{z}^2\left(s\right)\text{d}{\langle B_3,B_3\rangle }_s.\end{array}$ (4.2)

对于等式(4.2)，两边取G-期望后，得到：

$\begin{array}{c}0\le \hat{\mathbb{E}}V\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)\\ \le \hat{\mathbb{E}}V\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)+\hat{\mathbb{E}}{\displaystyle {\int }_0^t-}2\left(a+1\right)x^2\left(s\right)-\frac{y\left(t\right)z\left(t\right)}{x\left(t\right)}+a+1-2\left(b+1\right)y^2\left(s\right)\\ -\frac{z\left(t\right)z\left(t\right)}{y\left(t\right)}+b+1-4\left(1-c\right)z^2\left(s\right)\text{d}t+\hat{\mathbb{E}}{\displaystyle {\int }_0^{t}2c_1{x}^2\left(s\right)}-c_1+{\sigma }_1^2{x}^2\left(s\right)+\frac{{\sigma }_1^2}{2}\text{d}{\langle B_1,B_1\rangle }_s\\ +\hat{\mathbb{E}}{\displaystyle {\int }_0^{t}2c_2{y}^2\left(s\right)}-c_2+{\sigma }_2^2{y}^2\left(s\right)+\frac{{\sigma }_2^2}{2}\text{d}{\langle B_2,B_2\rangle }_s+\hat{\mathbb{E}}{\displaystyle {\int }_0^{t}4c_3{z}^2\left(s\right)}+2{\sigma }_3^2{z}^2\left(s\right)\text{d}{\langle B_3,B_3\rangle }_s,\end{array}$ (4.3)

对不等式(4.3)，通过基本不等式，可以得到：

$\begin{array}{c}0\le \hat{\mathbb{E}}V\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)-\hat{\mathbb{E}}V\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)\\ \le \hat{\mathbb{E}}{\displaystyle {\int }_0^t-}2\left(a+1\right)x^2\left(s\right)+{\bar{\sigma }}_1^2{x}^2\left(s\right)\left(2c_1+{\sigma }_1^2\right)-2\left(b+1\right)y^2\left(s\right)+{\bar{\sigma }}_2^2{y}^2\left(s\right)\left(2c_2+{\sigma }_2^2\right)\\ -4\left(1-c\right)z^2\left(s\right)+{\bar{\sigma }}_3^2{z}^2\left(s\right)\left(4c_3+2{\sigma }_3^2\right)-2z\left(s\right)\text{d}s\\ +\left(a+b+2\right)t+\left(\frac{{\sigma }_1^2}{2}-c_1\right){\bar{\sigma }}_1^{2}t+\left(\frac{{\sigma }_2^2}{2}-c_2\right){\bar{\sigma }}_2^{2}t.\end{array}$ (4.4)

从而，有

$\begin{array}{l}\hat{\mathbb{E}}{\displaystyle {\int }_0^t\left(2\left(a+1\right)-{\bar{\sigma }}_1^2\left(2c_1+{\sigma }_1^2\right)\right)x^2\left(s\right)}+\left(2\left(b+1\right)-{\bar{\sigma }}_2^2\left(2c_2+{\sigma }_2^2\right)\right)y^2\left(s\right)+K_1{\left(z\left(s\right)+\frac{1}{K_1}\right)}^2\text{d}t\\ \le \hat{\mathbb{E}}V\left(x\left(0\right),y\left(0\right),z\left(0\right)\right)+\left(a+b+2\right)t+\left(\frac{{\sigma }_1^2}{2}-c_1\right){\bar{\sigma }}_1^{2}t+\left(\frac{{\sigma }_2^2}{2}-c_2\right){\bar{\sigma }}_2^{2}t+\frac{t}{K_1},\end{array}$

其中， $K_1=4\left(1-c\right)-{\bar{\sigma }}_3^2\left(4c_3+2{\sigma }_3^2\right)$。因此：

$\begin{array}{l}\underset{t\to \infty }{\lim \sup }\frac{1}{t}\hat{\mathbb{E}}{\displaystyle {\int }_0^t\left(2\left(a+1\right)-{\bar{\sigma }}_1^2\left(2c_1+{\sigma }_1^2\right)\right)x^2\left(s\right)}+\left(2\left(b+1\right)-{\bar{\sigma }}_2^2\left(2c_2+{\sigma }_2^2\right)\right)y^2\left(s\right)+K_1{\left(z\left(s\right)+\frac{1}{K_1}\right)}^2\text{d}t\\ \le a+b+2+\left(\frac{{\sigma }_1^2}{2}-c_1\right){\bar{\sigma }}_1^2+\left(\frac{{\sigma }_2^2}{2}-c_2\right){\bar{\sigma }}_2^2+\frac{1}{K_1}.\end{array}$

令

$K_2=\min \left\{2\left(a+1\right)-{\bar{\sigma }}_1^2\left(2c_1+{\sigma }_1^2\right),2\left(b+1\right)-{\bar{\sigma }}_2^2\left(2c_2+{\sigma }_2^2\right),K_1\right\},$

那么有，

$\begin{array}{l}\underset{t\to \infty }{\lim \sup }\frac{1}{t}\hat{\mathbb{E}}{\displaystyle {\int }_0^t{x}^2}\left(s\right)+y^2\left(s\right)+{\left(z\left(s\right)+\frac{1}{K_1}\right)}^2\text{d}t\\ \le \frac{1}{K_2}\left(a+b+2+\left(\frac{{\sigma }_1^2}{2}-c_1\right){\bar{\sigma }}_1^2+\left(\frac{{\sigma }_2^2}{2}-c_2\right){\bar{\sigma }}_2^2+\frac{1}{K_1}\right).\end{array}$

定理4.2假设 $e_1,e_2,e_3>0$ 成立。则对给定初解 $E_1=\left(x^{*},y^{*},z^{*}\right)\in {ℝ}^3$，G-布朗运动驱动的随机金融风险系统模型(1.2)的解在有风险均衡点有以下性质：

$\begin{array}{l}\underset{t\to \infty }{\lim \sup }\frac{1}{t}\hat{\mathbb{E}}{\displaystyle {\int }_0^t{\left(x\left(s\right)-\left(1+g_1\right)x^{*}\right)}^2}+{\left(y\left(s\right)-\left(1+g_2\right)y^{*}\right)}^2+{\left(z\left(s\right)-\left(1+g_3\right)z^{*}\right)}^2\text{d}t\\ \le \frac{1}{K_3}\left(\frac{1}{2}{\sigma }_1^2{\bar{\sigma }}_1^2{x}^{*}{}^2+\frac{1}{2}{\sigma }_2^2{\bar{\sigma }}_2^2{y}^{*}{}^2+{\sigma }_3^2{\bar{\sigma }}_3^2{z}^{*}+\frac{f_1^2}{4e_1}+\frac{f_2^2}{4e_2}+\frac{f_3^2}{4e_3}\right),\end{array}$

其中，

$e_1=a-z^{*}{}^2-c_1{\bar{\sigma }}_1^2-\frac{1}{2}{\sigma }_1^2{\bar{\sigma }}_1^2,e_2=b-1-c_2{\bar{\sigma }}_2^2-\frac{1}{2}{\sigma }_2^2{\bar{\sigma }}_2^2,$

$e_3=2-2c-\frac{x^{*}{}^2}{4}-\frac{y^{*}{}^2}{4}-2c_3{\bar{\sigma }}_3^2-{\sigma }_3^2{\bar{\sigma }}_3^2,f_1=y^{*}{z}^{*}-\left(a+1\right)x^{*}+c_1{\bar{\sigma }}_1^2{x}^{*}+{\sigma }_1^2{\bar{\sigma }}_1^2{x}^{*},$

$f_2=x^{*}{z}^{*}-\left(b+1\right)y^{*}+c_2{\bar{\sigma }}_2^2{y}^{*}+{\sigma }_2^2{\bar{\sigma }}_2^2{y}^{*},f_3=2\left(c-1\right)z^{*}-2x^{*}{y}^{*}+2c_3{\bar{\sigma }}_3^2{z}^{*}+2{\sigma }_3^2{\bar{\sigma }}_3^2{z}^{*},$

$\left(x^{*},y^{*},z^{*}\right)=\left(\sqrt{\left(b+1\right)\left(c-1\right)},\sqrt{\left(a+1\right)\left(c-1\right)},\sqrt{\left(a+1\right)\left(b+1\right)}\right),$

$\frac{f_1}{2e_1}=g_1{x}^{*},\frac{f_2}{2e_2}=g_2{y}^{*},\frac{f_3}{2e_3}=g_3{z}^{*},K_3=\min \left\{e_1,e_2,e_3\right\}.$

证明：首先我们作一个变换 $u=x-x^{*}$，$v=y-y^{*}$，$w=z-z^{*}$，则模型随机金融风险系统模型(1.2)可以写成

$\{\begin{array}{l}\text{d}u\left(t\right)=\left(\left(v\left(t\right)+y^{*}\right)\left(w\left(t\right)+z^{*}\right)-\left(a+1\right)\left(u\left(t\right)+x^{*}\right)\right)\text{d}t+c_1\left(u\left(t\right)+x^{*}\right)\text{d}{\langle B_1,B_1\rangle }_t+{\sigma }_1\left(u\left(t\right)+x^{*}\right)\text{d}{B}_1\left(t\right),\\ \text{d}v\left(t\right)=\left(\left(u\left(t\right)+x^{*}\right)\left(w\left(t\right)+z^{*}\right)-\left(b+1\right)\left(v\left(t\right)+y^{*}\right)\right)\text{d}t+c_2\left(v\left(t\right)+y^{*}\right)\text{d}{\langle B_2,B_2\rangle }_t+{\sigma }_2\left(v\left(t\right)+y^{*}\right)\text{d}{B}_2\left(t\right),\\ \text{d}w\left(t\right)=\left(\left(c-1\right)\left(w\left(t\right)+z^{*}\right)-\left(u\left(t\right)+x^{*}\right)\left(v\left(t\right)+y^{*}\right)\right)\text{d}t+c_3\left(w\left(t\right)+z^{*}\right)\text{d}{\langle B_3,B_3\rangle }_t+{\sigma }_3\left(w\left(t\right)+z^{*}\right)\text{d}{B}_3\left(t\right).\end{array}$

构造Lyapunov函数：

$V\left(u,v,w\right)=\frac{1}{2}{u}^2+\frac{1}{2}{v}^2+w^2,$

且由G-伊藤公式可得：

$\begin{array}{c}V\left(u,v,w\right)=V\left(u\left(0\right),v\left(0\right),w\left(0\right)\right)+{\displaystyle {\int }_0^{t}L}V\text{d}s+{\displaystyle {\int }_0^t{\sigma }_1}u\left(s\right)\left(u\left(s\right)+x^{*}\right)\\ +{\sigma }_{2}v\left(s\right)\left(v\left(s\right)+y^{*}\right)+2{\sigma }_{3}w\left(s\right)\left(w\left(s\right)+z^{*}\right)\text{d}B\left(s\right)\\ +{\displaystyle {\int }_0^t{c}_1}u\left(s\right)\left(u\left(s\right)+x^{*}\right)+\frac{1}{2}{\sigma }_1^2{\left(u\left(s\right)+x^{*}\right)}^2\text{d}{\langle B_1,B_1\rangle }_s\\ +{\displaystyle {\int }_0^t{c}_2}v\left(s\right)\left(v\left(s\right)+y^{*}\right)+\frac{1}{2}{\sigma }_2^2{\left(v\left(s\right)+y^{*}\right)}^2\text{d}{\langle B_2,B_2\rangle }_s\\ +{\displaystyle {\int }_0^{t}2}{c}_{3}w\left(s\right)\left(w\left(s\right)+z^{*}\right)+{\sigma }_3^2{\left(w\left(s\right)+z^{*}\right)}^2\text{d}{\langle B_3,B_3\rangle }_s,\end{array}$ (4.5)

其中，

$\begin{array}{c}LV=u\left(t\right)\left(\left(v\left(t\right)+y^{*}\right)\left(w\left(t\right)+z^{*}\right)-\left(a+1\right)\left(u\left(t\right)+x^{*}\right)\right)\\ +v\left(t\right)\left(\left(u\left(t\right)+x^{*}\right)\left(w\left(t\right)+z^{*}\right)-\left(b+1\right)\left(v\left(t\right)+y^{*}\right)\right)\\ +w\left(t\right)\left(\left(c-1\right)\left(w\left(t\right)+z^{*}\right)-\left(u\left(t\right)+x^{*}\right)\left(v\left(t\right)+y^{*}\right)\right).\end{array}$

通过基本不等式，可以得到：

$\begin{array}{c}LV=-\left(a+1\right)u^2\left(s\right)-\left(b+1\right)v^2\left(s\right)-2\left(1-c\right)w^2\left(s\right)-y^{*}u\left(t\right)w\left(t\right)-x^{*}v\left(t\right)w\left(t\right)+2z^{*}u\left(t\right)v\left(t\right)\\ +u\left(t\right)\left(y^{*}{z}^{*}-\left(a+1\right)x^{*}\right)+v\left(t\right)\left(x^{*}{z}^{*}-\left(b+1\right)y^{*}\right)+2w\left(t\right)\left(\left(c-1\right)z^{*}-x^{*}{y}^{*}\right)\\ \le -\left(a-z^{*}{}^2\right)u^2\left(s\right)-\left(b-1\right)v^2\left(s\right)-\left(2-2c-\frac{x^{*}{}^2}{4}-\frac{y^{*}{}^2}{4}\right)w^2\left(s\right)\\ +u\left(t\right)\left(y^{*}{z}^{*}-\left(a+1\right)x^{*}\right)+v\left(t\right)\left(x^{*}{z}^{*}-\left(b+1\right)y^{*}\right)+2w\left(t\right)\left(\left(c-1\right)z^{*}-x^{*}{y}^{*}\right).\end{array}$