﻿ 马尔可夫调制的随机时滞IS-LM模型的随机稳定性仿真 Simulation of Stochastic Stability of Delayed IS-LM Model with Markovian Switching

Statistics and Application
Vol.07 No.01(2018), Article ID:23225,6 pages
10.12677/SA.2018.71001

Simulation of Stochastic Stability of Delayed IS-LM Model with Markovian Switching

Lujun Zhou

College of Economics, Hunan Agricultural University, Changsha Hunan

Received: Dec. 7th, 2017; accepted: Dec. 21st, 2017; published: Dec. 28th, 2017

ABSTRACT

This paper studies that the economic system is in the random environment which has delayed time in taxation and Markov chain with two discrete values. By using the method of stochastic stability and the given parameters, this paper analyzes the stability and simulation of the solution of the standard random IS-LM model with Markovian Switching and delayed time.

Keywords:IS-LM Model, Stochastic Delayed Differential Equation, Markov Chain, Stability

Copyright © 2018 by author and Hans Publishers Inc.

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

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

1. 引言

2. 预备知识及模型构建

Mao (2001)研究了如下的马尔可夫调制的随机微分方程 [1] ：

$\text{d}x\left(t\right)=f\left(x\left(t\right),x\left(t-\tau \right),t,r\left(t\right)\right)\text{d}t+g\left(x\left(t\right),x\left(t-\tau \right),t,r\left(t\right)\right)\text{d}w\left( t \right)$

$P\left\{r\left(t+\Delta \right)=j|r\left(t\right)=i\right\}=\left\{\begin{array}{l}{\gamma }_{ij}\Delta +o\left(\Delta \right),\\ 1+{\gamma }_{ij}\Delta +o\left(\Delta \right),\end{array}$ $\begin{array}{l}i\ne j\\ i=j\end{array}$

V. Torre (1977)用分岔理论分析了非均衡市场的完全凯恩斯系统 [2]

$\left\{\begin{array}{l}\stackrel{˙}{Y}\left(t\right)=\alpha \left(I\left(Y\left(t\right),R\left(t\right)\right)-S\left(Y\left(t\right),R\left(t\right)\right)\right),\\ \stackrel{˙}{R}\left(t\right)={\beta }_{0}\left(L\left(Y\left(t\right),R\left(t\right)\right)-{L}_{s}\right),\end{array}$

$\left\{\begin{array}{l}\stackrel{˙}{Y}\left(t\right)=\alpha \left(\left(\sigma +\gamma -1\right)Y\left(t\right)-\sigma \epsilon Y\left(t-\tau \right)-{\beta }_{1}R\left(t\right)+G\right),\\ \stackrel{˙}{R}\left(t\right)=\beta {}_{0}\left(uY\left(t\right)-{\beta }_{2}R\left(t\right)-{L}_{s}\right)\end{array}$ (1)

$\left({Y}^{*},{R}^{*}\right)=\left(\frac{{\beta }_{2}G+{\beta }_{1}{L}_{s}}{u{\beta }_{1}-{\beta }_{2}\left(\sigma \left(1-\epsilon \right)-1+\gamma \right)},\frac{uG+\left(\sigma \left(1-\epsilon \right)-1+\gamma \right){L}_{s}}{u{\beta }_{1}-{\beta }_{2}\left(\sigma \left(1-\epsilon \right)-1+\gamma \right)}\right)$

$x\left(t\right)=Y\left(t\right)-{Y}^{*},x\left(t-\tau \right)=Y\left(t-\tau \right)-{Y}^{*}$$y\left(t\right)=R\left(t\right)-{R}^{*}$ ，则模型(1)变成

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\alpha \left(\left(\sigma +\gamma -1\right)x\left(t\right)-\sigma \epsilon x\left(t-\tau \right)-{\beta }_{1}y\left(t\right)\right),\\ \stackrel{˙}{y}\left(t\right)={\beta }_{0}\left(ux\left(t\right)-{\beta }_{2}y\left(t\right)\right)\end{array}$ (2)

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left(\alpha \left(\sigma +\gamma -1\right)+{\sigma }_{1}\omega \left(t\right)\right)x\left(t\right)+\left(-\alpha \sigma \epsilon +{\sigma }_{2}\omega \left(t\right)\right)x\left(t-\tau \right)+\left(-\alpha {\beta }_{1}+{\sigma }_{3}w\left(t\right)\right)y\left(t\right),\\ \stackrel{˙}{y}\left(t\right)=\left(\left({\beta }_{0}u+{\sigma }_{4}\omega \left(t\right)\right)x\left(t\right)+\left(-{\beta }_{0}{\beta }_{2}+{\sigma }_{5}w\left(t\right)\right)y\left(t\right)\right)\end{array}$ (3)

$\text{d}X\left(t\right)=\left(AX\left(t\right)+BX\left(t-\tau \right)\right)dt+\left(CX\left(t\right)+DX\left(t-\tau \right)\right)\text{d}w\left(t\right)$ (4)

$A=\left(\begin{array}{cc}\alpha \left(\sigma +\gamma -1\right)& -\alpha {\beta }_{1}\\ {\beta }_{0}u& -{\beta }_{0}{\beta }_{2}\end{array}\right),B=\left(\begin{array}{cc}-\alpha \sigma \epsilon & 0\\ 0& 0\end{array}\right)$$C=\left(\begin{array}{cc}{\sigma }_{1}& {\sigma }_{3}\\ {\sigma }_{4}& {\sigma }_{5}\end{array}\right),D=\left(\begin{array}{cc}{\sigma }_{2}& 0\\ 0& 0\end{array}\right)$

$dX\left(t\right)=\left(A\left(r\left(t\right)\right)X\left(t\right)+B\left(r\left(t\right)\right)X\left(t-\tau \right)\right)\text{d}t+\left(C\left(r\left(t\right)\right)X\left(t\right)+D\left(r\left(t\right)\right)X\left(t-\tau \right)\right)\text{d}w\left(t\right)$ (5)

3. 案例仿真

${\alpha }_{1}=0.011,{\sigma }_{1}=0.018,{\gamma }_{1}=0.049,{\beta }_{01}=0.024,{\beta }_{11}=0.056,{\beta }_{21}=0.035,{u}_{1}=0.046$

${\epsilon }_{1}=0.021,{\sigma }_{11}=0.048,{\sigma }_{21}=0.024,{\sigma }_{31}=0.015,{\sigma }_{41}=0.071,{\sigma }_{51}=0.171$

${\alpha }_{2}=0.011,{\sigma }_{2}=0.016,{\gamma }_{2}=0.029,{\beta }_{02}=0.025,{\beta }_{12}=0.065,{\beta }_{22}=0.065,{u}_{2}=0.026$

${\epsilon }_{2}=0.031,{\sigma }_{12}=0.058,{\sigma }_{22}=0.016,{\sigma }_{32}=0.025,{\sigma }_{42}=0.061,{\sigma }_{52}=0.271$

Figure 1. Solution x(t), y(t) of sde and r(t) (time lag = 0.2)

Figure 2. Solution x(t), y(t) of sde and r(t) (time lag = 0.2)

Figure 3. Solution x(t), y(t) of sde and r(t) (time lag = 0.8)

Figure 4. Solution x(t), y(t) of sde and r(t) (time lag = 0.8)

Simulation of Stochastic Stability of Delayed IS-LM Model with Markovian Switching[J]. 统计学与应用, 2018, 07(01): 1-6. http://dx.doi.org/10.12677/SA.2018.71001

1. 1. Mao, X., Matasov, A. and Piunovskiy, A.B. (2000) Stochastic Differential Delay Equations with Markovian Switching. Bernoulli, 1, 73-90. https://doi.org/10.2307/3318634

2. 2. Torre, V. (1977) The Existence of Limit Cycles and Control in Complete Keynesian Systems by Theory of Bifurcations. Econometrica, 45, 1457-1466. https://doi.org/10.2307/1912311

3. 3. Turnovsky, S.J. (1996) Methods of Macroeconomic Dynamics. MIT Press, Boston.

4. 4. Bing, L., Dingshi, L. and Daoyi, X. (2013) Stability Analysis for Impulsive Stochastic Delay Differential Equations with Markovian Switching. Journal of the Franklin Institute, 350, 1848-1864.