## Appendix A  
### Existence and Uniqueness Proof  
Prove the existence and uniqueness of solutions of Equations (8)-(11) according to the following theorem.  
**Theorem A1**.: _For \(d(x)t=f(t,x(t))dt+g(t,x(t))dw(t),t\in J\), Let the functions \(f(t,x)\) and \(g(t,x)\) satisfy the following conditions in \(J\times R^{d}\). (i) Consistent Lipschitz conditions. (ii) Linear growth condition [68]._  
For Equations (8)-(11), since \(x\in[0,1],y\in[0,1],z\in[0,1],u\in[0,1],t>0\), only the local Lipschitz conditions and nonlinear conditions need to be satisfied. The proof is as follows:  
Write Equations (8)-(11) as Equations (A1)-(A4), respectively:  
\[dx(t)=f_{1}(x,t)dt+g_{1}(x,t)dw(t)\] (A1)  
\[dy(t)=f_{2}(y,t)dt+g_{2}(y,t)dw(t)\] (A2)  
\[dz(t)=f_{3}(z,t)dt+g_{3}(z,t)dw(t)\] (A3)  
\[du(t)=f_{4}(u,t)dt+g_{4}(u,t)dw(t)\] (A4)  
The individual Equations are expanded as follows:  
\[f_{1}(x,t)=x[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N _{Y}+(1-z)N_{Z}+(1-u)N_{U}+G_{X}]g_{1}(x,t)=\sigma x\] (A5)  
\[f_{2}(y,t)=y[(1+x\beta_{YX}+z\beta_{YZ}+u\beta_{YU}-xa_{YX})E_{Y}-C_{Y}+\Pi_{ Y}+G_{Y}+xN_{Y}]g_{2}(y,t)=\sigma y\] (A6)  
\[f_{3}(z,t)=z[(1+x\beta_{ZX}+y\beta_{ZY}+u\beta_{ZU}-xa_{ZX}-y\alpha_{ZY})E_{ Z}-C_{Z}+\Pi_{Z}+G_{Z}+xN_{Z}]g_{3}(z,t)=\sigma z\] (A7)  
\[f_{4}(u,t)=u[(1+x\beta_{UX}+y\beta_{UY}+z\beta_{UZ}-xa_{UX}-y\alpha_{UY}-za_ {UZ})E_{U}-C_{U}+\Pi_{U}+G_{U}+xN_{U}]g_{4}(u,t)=\sigma u\] (A8)  
Clearly, Equations (A5)-(A8) are continuous on \(x\in[0,1],y\in[0,1],z\in[0,1],u\in[0,1]\).  
First, prove for Equation (8) that it has a local Lipschitz condition:  
Equation (A9) holds for \(\forall x,x^{*}\in[0,1]\).  
\[\begin{array}{l}|f_{1}(x,t)-f_{1}(x^{*},t)|=  
\[\begin{array}{l}|f_{1}(x,t)|^{2}=|[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}- C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x|^{2}\\ =|[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N_ {Z}+(1-u)N_{U}+G_{X}]|^{2}|x|^{2}\\ \leq|[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1- z)N_{Z}+(1-u)N_{U}+G_{X}]|^{2}(1+|x|^{2})=K^{2}(1+|x|^{2})\\ |g_{1}(x)|^{2}=|\sigma x|^{2}=|\sigma|^{2}|x|^{2}\leq|\sigma|^{2}(1+|x|^{2}) \end{array} \tag{12}\]  
From Equation (12), Equation (13) can be obtained.  
\[\begin{array}{l}|f_{1}(x,t)|^{2}+|g_{1}(x)|^{2}=|[(1+y\beta_{XY}+z\beta_{XZ }+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x|^{2 }+|\sigma x|^{2}\\ =|[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N _{Z}+(1-u)N_{U}+G_{X}]|^{2}|x|^{2}+|\sigma^{2}|x|^{2}\\ \leq|[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1- z)N_{Z}+(1-u)N_{U}+G_{X}]|^{2}(1+|x|^{2})+|\sigma|^{2}(1+|x|^{2})\\ =(K_{1}^{2}+|\sigma|^{2})(1+|x|^{2})\leq(K_{1}+|\sigma|^{2})^{2}(1+|x|^{2})=K^ {2}(1+|x|^{2})\end{array} \tag{13}\]  
Since \(K_{1}\geq 0\), \(|f_{1}(x,t)|^{2}+|g_{1}(x)|^{2}=(K_{1}^{2}+|\sigma|^{2})(1+|x|^{2})\leq(K_{1} +|\sigma|)^{2}(1+|x|^{2})=K^{2}(1+|x|^{2})\)  
Therefore, Equation (8) satisfies the linear growth condition.  
Since initially \(t=0\) and \(x(t)=0\), at this point, Equation (8) is independent of the Brownian motion.  
Therefore, there exists a unique solution to Equation (8).  
Similarly, it can be obtained that there exists a unique solution to Equations (9)-(11), which can be analyzed for stability.