﻿ 含记忆系数的资本资产定价模型 An Asset Pricing Model with Memory Coefficient

Vol.06 No.07(2017), Article ID:22553,9 pages
10.12677/AAM.2017.67108

An Asset Pricing Model with Memory Coefficient

Xuxia Li

College of Mathematics and Systems Science, Xinjiang University, Urumqi Xinjiang

Received: Oct. 9th, 2017; accepted: Oct. 23rd, 2017; published: Oct. 31st, 2017

ABSTRACT

This paper develops heterogeneous beliefs of the asset pricing model with the evolution fitness in the case of the two types of agents. Introducing memory coefficient, the memory parameter in deviation between the fundamentalists market fraction and chartists market fraction, we develop an asset pricing model with heterogeneous beliefs. By using the theory of difference equation, we discuss about the system of the model for local stability bifurcation analysis, and obtain the effect of main parameters on the stability of the model.

Keywords:Heterogeneous Belief, Memory Parameter, Stability

1. 引言

2. 含记忆系数的动态模型

BP文献中建立的动态模型如下：

$\left\{\begin{array}{l}{x}_{t}=\frac{k}{R}\frac{1-{m}_{t-1}}{\frac{{\alpha }_{C,t}}{{\alpha }_{F}}+1+{m}_{t-1}\left(\frac{{\alpha }_{C,t}}{{\alpha }_{F}}-1\right)}{x}_{t-1},\\ {m}_{t}=\text{tanh}\left[\frac{\beta }{2{\alpha }_{F}{\sigma }^{2}}\left(R{x}_{t-1}-{x}_{t}\right)\left(R{x}_{t-1}+\frac{k{x}_{t-2}-R{x}_{t-1}}{\frac{{\alpha }_{C,t}}{{\alpha }_{F}}}\right)-\frac{\beta c}{2}\right],\\ {\alpha }_{C,t}=\left(1-\varpi \right){\overline{\alpha }}_{C}\mathrm{exp}\left[\tau \left({x}_{t-1}-R{x}_{t-2}\right)\right]+\varpi {\alpha }_{C,t-1}.\end{array}$ (1.1)

2.1. 记忆系数的引入

BP主要考虑 ${U}_{F,t}={\pi }_{F,t}$${U}_{C,t}={\pi }_{C,t}$ ，这里 ${\pi }_{h,t}$ 表示t时h类投资者的净收益， ${\pi }_{h,t}={R}_{t-1}{z}_{h,t-1}=\left({x}_{t}-R{x}_{t-1}+{\delta }_{t}\right){z}_{h,t-1}$${\delta }_{t}$ 是鞅差序列，令 ${\delta }_{t}=0$${z}_{h,t-1}$ 为t − 1时h类投资者对风险资产的需求， ${U}_{h,t}$ 为进化适应度函数。本文在此基础上引入记忆参数 $\gamma$$\eta$ ，则基本面分析者和图表分析者的进化适应度函数为：

$\left\{\begin{array}{l}{U}_{F,t}=\gamma {\pi }_{F,t}+\eta {U}_{F,t-1},\\ {U}_{C,t}=\gamma {\pi }_{C,t}+\eta {U}_{C,t-1}.\end{array}$ (1.2)

$\gamma$$\eta \in \left[0,1\right)$ 为记忆参数， $\gamma$ 衡量t − 1时h类投资者的净收益对t时的适应度函数的影响程度， $\eta$ 衡量t − 1时h类投资者的进化适应度函数对t时的适应度函数的影响程度。当 $\gamma =1,\eta =0$ 时即为BP的情形。

${m}_{t}=\mathrm{tan}h\left[\frac{\beta }{2}\left({U}_{F,t-1}-{U}_{C,t-1}\right)\right]+\delta {m}_{t-1}$(1.3)

2.2. 于是我们得到如下的非线性动态系统

$\left\{\begin{array}{l}{x}_{t}=\frac{k}{R}\frac{1-{m}_{t-1}}{\frac{{\alpha }_{C,t}}{{\alpha }_{F}}+1+{m}_{t-1}\left(\frac{{\alpha }_{C,t}}{{\alpha }_{F}}-1\right)}{x}_{t-1},\\ {m}_{t}=\text{tanh}\left[\frac{\beta }{2}\left({U}_{F,t-1}-{U}_{C,t-1}\right)\right]+\delta {m}_{t-1},\\ {\alpha }_{C,t}=\left(1-\varpi \right){\overline{\alpha }}_{C}\mathrm{exp}\left[\tau \left({x}_{t-1}-R{x}_{t-2}\right)\right]+\varpi {\alpha }_{C,t-1},\\ {U}_{F,t}=\frac{\gamma }{{\alpha }_{F}{\sigma }^{2}}R{x}_{t-1}\left(R{x}_{t-1}-{x}_{t}\right)-c+\eta {U}_{F,t-1},\\ {U}_{C,t}=\frac{\gamma }{{\alpha }_{C,t}{\sigma }^{2}}\left(k{x}_{t-2}-R{x}_{t-1}\right)\left({x}_{t}-R{x}_{t-1}\right)+\eta {U}_{C,t-1}.\end{array}$ (1.4)

$\left\{\begin{array}{l}{x}_{t}=\frac{k}{R}\frac{1-{m}_{t-1}}{\frac{{\alpha }_{C,t}}{{\alpha }_{F}}+1+{m}_{t-1}\left(\frac{{\alpha }_{C,t}}{{\alpha }_{F}}-1\right)}{x}_{t-1},\\ {y}_{t}={x}_{t-1},\\ {z}_{t}={y}_{t-1},\\ {u}_{F,t}={U}_{F,t-1},\\ {u}_{C,t}={U}_{C,t-1},\\ {m}_{t}=\mathrm{tanh}\left[\frac{\beta \gamma }{2{\alpha }_{F}{\sigma }^{2}}\left(R{y}_{t-1}-{x}_{t-1}\right)\left(R{y}_{t-1}+\frac{k{z}_{t-1}-R{y}_{t-1}}{\frac{{\alpha }_{C,t}}{{\alpha }_{F}}}\right)-\frac{\beta c}{2}+\frac{\beta \eta }{2}\left({u}_{F,t-1}-{u}_{C,t-1}\right)\right]+\delta {m}_{t-1},\\ {\alpha }_{C,t}=\left(1-\varpi \right){\overline{\alpha }}_{C}\mathrm{exp}\left[\tau \left({x}_{t-1}-R{y}_{t-1}\right)\right]+\varpi {\alpha }_{C,t-1},\\ {U}_{F,t}=\frac{\gamma }{{\alpha }_{F}{\sigma }^{2}}R{x}_{t-1}\left(R{x}_{t-1}-{x}_{t}\right)-c+\eta {U}_{F,t-1},\\ {U}_{C,t}=\frac{\gamma }{{\alpha }_{C,t}{\sigma }^{2}}\left(k{y}_{t-1}-R{x}_{t-1}\right)\left({x}_{t}-R{x}_{t-1}\right)+\eta {U}_{C,t-1}.\end{array}$ (1.5)

3. 确定性系统分析

${X}_{t}={\left({x}_{t},{y}_{t},{z}_{t},{u}_{F,t},{u}_{C,t},{m}_{t},{\alpha }_{C,t},{U}_{F,t},{U}_{C,t}\right)}^{\prime }$ ，模型的确定性系统为： $F:{R}^{9}\to {R}^{9}$ ，即 ${X}_{t}=F\left({X}_{t-1}\right)$

${F}_{1}\left({X}_{t-1}\right)=\frac{k}{R}\frac{1-{m}_{t-1}}{\frac{{F}_{7}\left({X}_{t-1}\right)}{{\alpha }_{F}}+1+{m}_{t-1}\left(\frac{{F}_{7}\left({X}_{t-1}\right)}{{\alpha }_{F}}-1\right)}{x}_{t-1}$${F}_{2}\left({X}_{t-1}\right)={x}_{t-1}$

${F}_{3}\left({X}_{t-1}\right)={y}_{t-1}$${F}_{4}\left({X}_{t-1}\right)={U}_{F,t-1}$${F}_{5}\left({X}_{t-1}\right){u}_{C,t}={U}_{C,t-1},$

${F}_{6}\left({X}_{t-1}\right)=\text{tanh}\left[\frac{\beta \gamma }{2{\alpha }_{F}{\sigma }^{2}}\left(R{y}_{t-1}-{x}_{t-1}\right)\left(R{y}_{t-1}+\frac{k{z}_{t-1}-R{y}_{t-1}}{\frac{{F}_{7}\left({X}_{t-1}\right)}{{\alpha }_{F}}}\right)-\frac{\beta c}{2}+\frac{\beta \eta }{2}\left({u}_{F,t-1}-{u}_{C,t-1}\right)\right]+\delta {m}_{t-1},$

${F}_{7}\left({X}_{t-1}\right)=\left(1-\varpi \right){\overline{\alpha }}_{C}\mathrm{exp}\left[\tau \left({x}_{t-1}-R{y}_{t-1}\right)\right]+\varpi {\alpha }_{C,t-1},$

${F}_{8}\left({X}_{t-1}\right)=\frac{\gamma }{{\alpha }_{F}{\sigma }^{2}}R{x}_{t-1}\left(R{x}_{t-1}-{F}_{1}\left({X}_{t-1}\right)\right)-c+\eta {U}_{F,t-1},$

${F}_{9}\left({X}_{t-1}\right)=\frac{\gamma }{{F}_{7}\left({X}_{t-1}\right){\sigma }^{2}}\left(k{y}_{t-1}-R{x}_{t-1}\right)\left({F}_{1}\left({X}_{t-1}\right)-R{x}_{t-1}\right)+\eta {U}_{C,t-1}$ .

${X}^{*}={\left({x}^{*},{y}^{*},{z}^{*},{u}_{F}^{*},{u}_{C}^{*},{m}^{*},{\alpha }_{C}^{*},{U}_{F}^{*},{U}_{C}^{*}\right)}^{\text{T}}$ 为其平衡解，满足 ${X}^{*}=F\left({X}^{*}\right)$ ，即为以下等式：

$\left\{\begin{array}{l}{x}^{\ast }=\frac{k}{R}\frac{1-{m}^{\ast }}{\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}+1+{m}^{\ast }\left(\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}-1\right)}{x}^{\ast },\\ {y}^{\ast }={x}^{\ast },\\ {z}^{\ast }={y}^{\ast },\\ {u}_{F}^{\ast }={U}_{F}^{\ast },\\ {u}_{C}^{\ast }={U}_{C}^{\ast },\\ {m}^{\ast }=\frac{1}{1-\delta }\text{tanh}\left[\frac{\beta \gamma }{2{\alpha }_{F}{\sigma }^{2}}\left(R-1\right)\left(R+\frac{k-R}{\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}}\right){\left({x}^{\ast }\right)}^{2}-\frac{\beta c}{2}+\frac{\beta \eta }{2}\left({u}_{F}^{\ast }-{u}_{C}^{\ast }\right)\right],\\ {\alpha }_{C}^{\ast }={\overline{\alpha }}_{C}\mathrm{exp}\left[\tau \left(1-R\right){x}^{\ast }\right],\\ {U}_{F}^{\ast }=\frac{1}{1-\eta }\left[\frac{\gamma }{{\alpha }_{F}{\sigma }^{2}}R\left(R-1\right){\left({x}^{\ast }\right)}^{2}-c\right],\\ {U}_{C}^{\ast }=\frac{1}{1-\eta }\frac{\gamma }{{\alpha }_{C}^{\ast }{\sigma }^{2}}\left(k-R\right)\left(1-R\right){\left({x}^{\ast }\right)}^{2}.\end{array}$ (2.1)

${m}^{eq}=\text{tanh}\left(\frac{\beta c}{2\left(\eta -1\right)}\right)$${\alpha }_{C}^{eq}={\overline{\alpha }}_{C}$${u}_{F}^{eq}={U}_{F}^{eq}=\frac{c}{\eta -1}$${u}_{C}^{eq}={U}_{C}^{eq}=0$$±{x}^{\ast }$ 为下式的解(若存在)

${m}^{\ast }=\frac{1}{1-\delta }\text{tanh}\left[\frac{\beta \gamma }{2{\alpha }_{F}{\sigma }^{2}}\left(R-1\right)\left(R+\frac{k-R}{\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}}\right){\left({x}^{\ast }\right)}^{2}-\frac{\beta c}{2}+\frac{\beta \eta }{2}\left({u}_{F}^{\ast }-{u}_{C}^{\ast }\right)\right]$ (2.2)

1、当 $0 时， ${X}_{0}$ 为唯一的平衡解， ${X}_{0}=\left(0,0,0,{u}_{F}^{eq},0,{m}^{eq},{\alpha }_{C}^{eq},{U}_{F}^{eq},0\right)$

2、当 $k>R\left(\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}+1\right),R>1$ 时，

(1) 若 $\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}<1$ 时，存在平衡解 ${X}_{0},{X}_{1}$${X}_{1}=\left({x}^{\ast },{x}^{\ast },{x}^{\ast },{u}_{F}^{\ast },{u}_{C}^{\ast },{m}^{\ast },{\alpha }_{C}^{\ast },{U}_{F}^{\ast },{U}_{C}^{\ast }\right)$

(2) 若 $\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}>1$ 时，存在平衡解 ${X}_{0},{X}_{2}$${X}_{2}=\left(-{x}^{\ast },-{x}^{\ast },-{x}^{\ast },{u}_{F}^{\ast },{u}_{C}^{\ast },{m}^{\ast },{\alpha }_{C}^{\ast },{U}_{F}^{\ast },{U}_{C}^{\ast }\right)$

3、当 $R 时，

(1) ${m}^{\ast }<{m}^{eq}$ 时， ${X}_{0}$ 是唯一的平衡解；

(2) ${m}^{\ast }>{m}^{eq}$ 时，若 $\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}<1$ ，则存在平衡解 ${X}_{0},{X}_{1}$ ；若 $\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}>1$ ，则存在平衡解 ${X}_{0},{X}_{2}$

1、 $0 时， $\frac{R\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}}{k+R\left(\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}-1\right)}>\frac{R\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}}{R+R\left(\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}-1\right)}=1$ ，则 ${m}^{\ast }<-1$ ，又由(2.1)第7式可知 ${\alpha }_{C}^{\ast }>0$ ，但是 ${\alpha }_{C}^{\ast }=\frac{1-{m}^{\ast }}{1+{m}^{\ast }}\left(\frac{k}{R}-1\right){\alpha }_{F}<0$ ，两者相矛盾，故此时只有平衡解 ${X}_{0}$

2、 $k>R\left(\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}+1\right),\text{}R>1$ 时， $0<{m}^{\ast }<1$${\alpha }_{C}^{\ast }>0$ ，因此(2.2)式存在两个平衡解 $±{x}^{\ast }$ ，有(2.1)第7式可知 ${x}^{\ast }=\frac{\mathrm{ln}\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}}{\tau \left(1-R\right)}$

(1) 若 $\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}<1$ 时， $\mathrm{ln}\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}<0$ ，所以 ${x}^{\ast }>0$ ，则存在平衡解 ${X}_{0},{X}_{1}$

(2) 若 $\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}>1$ 时， $\mathrm{ln}\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}>0$ ，所以 ${x}^{\ast }<0$ ，则存在平衡解 ${X}_{0},{X}_{2}$

3、 $R 时， $-1<{m}^{\ast }<0$

(1) ${m}^{\ast }<{m}^{eq}$ 时，方程(2.2)无解，故只有平衡解 ${X}_{0}$

(2) ${m}^{\ast }>{m}^{eq}$ 时，方程(2.2)有解，又由(2.1)第7式得 ${x}^{\ast }=\frac{\mathrm{ln}\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}}{\tau \left(1-R\right)}$ ，若 $\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}<1$$\mathrm{ln}\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}<0$ ，则存在平衡解 ${X}_{0},{X}_{1}$ ；若 $\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}>1$$\mathrm{ln}\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}>0$ ，所以 ${x}^{\ast }<0$ ，则存在平衡解 ${X}_{0},{X}_{2}$

1、 $0$\varpi <1$$\delta <1,\eta <1$ 时，基本平衡解 ${X}_{0}$ 是渐近稳定的；

2、 $k>R\left(\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}+1\right)$ ，或 $\varpi <1$$\delta >1$$\eta >1$ 时，基本平衡解 ${X}_{0}$ 不稳定；

3、 $R$\varpi <1$$\delta <1,\eta <1$ 时，

${m}^{\ast }<{m}^{eq}$$\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}<1$ 时，基本平衡解 ${X}_{0}$ 是渐近稳定的；

4、当 $\varpi <1,\delta <1,\eta <1$$\frac{k}{R}\frac{1-{m}^{eq}}{\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}+1+{m}^{eq}\left(\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}-1\right)}=1$ 时，出现音叉分支，当 $0<\beta <{\beta }^{\ast }$ 时，基本平衡解 ${X}_{0}$ 稳定，当 $\beta >{\beta }^{\ast }$ 时，基本平衡解 ${X}_{0}$ 不稳定， ${\beta }^{\ast }=\frac{\eta -1}{c}\mathrm{ln}\left[\frac{{\alpha }_{F}}{{\overline{\alpha }}_{C}}\left(1-\frac{k}{R}-\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}\right)\right]$

$\frac{\partial {F}_{1}}{\partial x}=\frac{k}{R}\frac{1-{m}^{eq}}{\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}+1+{m}^{eq}\left(\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}-1\right)}=A$ , $\frac{\partial {F}_{1}}{\partial X}=0$ , $\frac{\partial {F}_{2}}{\partial x}=1$ , $\frac{\partial {F}_{2}}{\partial X}=0$ , $\frac{\partial {F}_{3}}{\partial y}=1$ , $\frac{\partial {F}_{3}}{\partial X}=0$ , $\frac{\partial {F}_{4}}{\partial {U}_{F}}=1$ , $\frac{\partial {F}_{4}}{\partial X}=0$ , $\frac{\partial {F}_{5}}{\partial {U}_{C}}=1$ , $\frac{\partial {F}_{5}}{\partial X}=0$ , $\frac{\partial {F}_{6}}{\partial {u}_{F}}=\frac{\beta \eta }{2}\text{tan h ′}\left(\theta \right)=B$ , $\frac{\partial {F}_{6}}{\partial m}=\delta$ , $\frac{\partial {F}_{6}}{\partial {u}_{C}}=-\frac{\beta \eta }{2}\text{tan h ′}\left(\theta \right)=C$ , $\frac{\partial {F}_{6}}{\partial X}=0$ , $\frac{\partial {F}_{7}}{\partial x}=\tau \left(1-\varpi \right){\overline{\alpha }}_{C}=D$ , $\frac{\partial {F}_{7}}{\partial {\alpha }_{C}}=\varpi$ , $\frac{\partial {F}_{7}}{\partial y}=-\tau R\left(1-\varpi \right){\overline{\alpha }}_{C}=E$ , $\frac{\partial {F}_{7}}{\partial X}=0$ , $\frac{\partial {F}_{8}}{\partial {U}_{F}}=\eta$ , $\frac{\partial {F}_{8}}{\partial X}=0$ , $\frac{\partial {F}_{9}}{\partial {U}_{C}}=\eta$ , $\frac{\partial {F}_{9}}{\partial X}=0$ .

$J=\left[\begin{array}{ccccccccc}A& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& B& C& \delta & 0& 0& 0\\ D& E& 0& 0& 0& 0& \varpi & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& \eta & 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \eta \end{array}\right]$

$\begin{array}{c}|\lambda E-J|=|0\begin{array}{ccccccccc}\lambda -A& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& \lambda & 0& 0& 0& 0& 0& 0& 0\\ 0& -1& \lambda & 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \lambda & 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \lambda & 0& 0& 0& 0\\ 0& 0& 0& -B& -C& \lambda -\delta & 0& 0& 0\\ -D& -E& 0& 0& 0& 0& \lambda -\varpi & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& \lambda -\eta & 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \lambda -\eta \end{array}|\\ =\left(\lambda -A\right)\left(\lambda -\varpi \right)\left(\lambda -\delta \right){\left(}^{\lambda }{\lambda }^{4}=0,\end{array}$

${\lambda }_{1}=A=\frac{k}{R}\frac{1-{m}^{eq}}{\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}+1+{m}^{eq}\left(\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}-1\right)}$${\lambda }_{2}=\varpi$${\lambda }_{3}=\delta$${\lambda }_{4,5}=\eta$${\lambda }_{6,7,8,9}=0$

1、因为 $\frac{1-{m}^{eq}}{\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}+1+{m}^{eq}\left(\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}-1\right)}=\frac{1-{m}^{eq}}{\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}\left({m}^{eq}+1\right)+1-{m}^{eq}}<1$ ，所以当 $0 时， ${\lambda }_{1}<1$

$\varpi <1$ 时， ${\lambda }_{2}<1$$\delta <1$ 时， ${\lambda }_{3}<1$$\eta <1$ 时， ${\lambda }_{4,5}<1$ ，故基本平衡解 ${X}_{0}$ 是渐近稳定的；

2、 $k>R\left(\frac{{\alpha }_{C}^{\ast }}{{\alpha }_{F}}+1\right)$ 时， ${\lambda }_{1}>1$$\varpi >1$ 时， ${\lambda }_{2}>1$$\delta >1$ 时， ${\lambda }_{3}>1$$\eta >1$ 时， ${\lambda }_{4,5}>1$

3、 $R${m}^{\ast }<{m}^{eq}$ 时方程(2.2)无解，若 $\frac{{\alpha }_{C}^{\ast }}{{\overline{\alpha }}_{C}}<1$ ，则 ${\lambda }_{1}<1$ ，且 $\varpi <1$$\delta <1$$\eta <1$ 时， ${\lambda }_{2}<1$${\lambda }_{3}<1$${\lambda }_{4,5}<1$ ，所以基本平衡解 ${X}_{0}$ 是渐近稳定的；

4、 $\varpi <1,\delta <1,\eta <1$${\lambda }_{1}=1$ 时，此时特征根都为非负实数，有且仅有可能出现pitchfork分支，即 $\frac{k}{R}\frac{1-{m}^{eq}}{\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}+1+{m}^{eq}\left(\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}-1\right)}=1$ ，解得 ${\beta }^{\ast }=\frac{\eta -1}{c}\mathrm{ln}\left(\frac{\left(2-\delta \right)\left(\frac{k}{R}-1\right)+\delta \frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}}{\delta \left(\frac{k}{R}-1\right)+\left(2-\delta \right)\frac{{\overline{\alpha }}_{C}}{{\alpha }_{F}}}\right)$ ，故当 $0<\beta <{\beta }^{\ast }$ 时，基本平衡解 ${X}_{0}$ 稳定，当 $\beta >{\beta }^{\ast }$ 时，基本平衡解 ${X}_{0}$ 不稳定。

$R=1.1$ , ${\alpha }_{F}=1$ , $\varpi =0.1$ , $\tau =1$ , ${\overline{\alpha }}_{C}=0.9$ , ${\sigma }^{2}=1$ , $c=1$ , $k=0.5$ , $\eta =0.5$ , $\gamma =0.5$ , $\delta =0.2$ .

Figure 1. The sensitivity of β to deterministic systems

Figure 2. $\left({\alpha }_{C,t},{x}_{t}\right)$ plane by the change of γ and η

Figure 3. The effect of n and k on β

4. 总结

An Asset Pricing Model with Memory Coefficient[J]. 应用数学进展, 2017, 06(07): 896-904. http://dx.doi.org/10.12677/AAM.2017.67108

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