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Differential equation model of financial market stability based on Internet big data

Publié en ligne: 22 Nov 2021
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 17 Jun 2021
Accepté: 24 Sep 2021
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License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Abstract

In the context of Internet big data, the market characteristics of the financial market can be used to feed back its stability with the help of differential equation models. China's financial market is roughly divided into three main markets: stocks, currency and foreign exchange. The interaction of the three has promoted the development of the financial market. With this as a background, the paper aims at these three financial markets and selects relevant indicators that can reflect the indications of the financial market to construct differential equations to analyse the relationship between the three. The paper uses the nonlinear characteristics of ordinary differential equations and related algorithms to solve the three types of market models. It uses an example to demonstrate that the differential equation model proposed in this paper can feed back the evolutionary characteristics of the three, and this model can help investors produce more correct investment decisions.

Keywords

MSC 2010

Introduction

The close connections and interactions among financial sub-markets have affected the development, changes and operation laws of each market, forming an overall linkage development trend. Existing research results fully demonstrate that the three markets of the stock market, foreign exchange market and currency market are significant. They mutually influence the relationship. Changes in one market directly or indirectly affect the changes in the other two markets. This article attempts to use the ideas and analysis methods of evolutionary finance and system dynamics to dig out the evolutionary characteristics and interaction relationships of financial markets [1]. Then we will reveal the operating laws of the financial market, the direction of market fluctuations and the nonlinear evolution mechanism. It is well known that the devaluation of the RMB in the foreign exchange market will cause capital outflows and cause the stock market to be depressed. When the interest rate is lowered, a large amount of money flows into the stock market from the money market, which enhances the attractiveness of the stock market. Then, how will the stock market change when the currency depreciates and interest rates are lowered at the same time? There are numerous such problems among the three markets. The study of such issues has important theoretical and practical significance. For the government, when an emergency occurs in one market, it can indirectly restore the abnormally volatile market to normal by adjusting the other two markets. For investors, when a market fluctuates, investors can use the nonlinear evolution structure to anticipate changes in the other two markets to avoid risk. This shows that it is particularly important to study the nonlinear evolution of the three markets.

In response to the above problems, this article selects 12 important decision-making indicators from 19 important indicators in the three markets through causality testing. On this basis, we have established a comprehensive indicator that can fully reflect the changes in the three markets [2]. Then the article is based on three overall markets as the research object established a three-market nonlinear evolution model based on ordinary differential equations. Ordinary differential equations can fully reflect the time-varying characteristics of variables, and it can more comprehensively reflect the evolutionary relationship between the three markets. Finally, the paper uses China's monthly data from January 2005 to May 2019 to simulate the specific evolution between the three markets. Form, the established nonlinear evolution model with constraints better describes the nonlinear evolution structure of the three markets and the mutual influence between the three markets.

Market indicator selection and causality test

This article starts from two aspects: theoretical analysis and references. This choice not only considers the economic significance of the indicators in the real market, but also considers the market indicators that are often used in existing studies. We select the market indicators of this article based on three factors: (1) important indicators of a single market. The stock turnover (ST) of the stock market can reflect the trading status of the stock market, and it is one of the important indicators of the stock market; (2) important indicators between markets. Money market money supply M0 and broad money (M2) can explain the stock market price changes to a certain extent, so we choose M0 and M2 as money market indicators to reflect part of the stock market information at the same time; (3) We select available and complete indicators. Based on the consideration of the above three factors, we selected 19 market indicators that can accurately reflect the characteristics of the market from the stock, currency and foreign exchange markets, as shown in Table 1.

Market indicators selected in the article.

Market Market indicators

Stock market SHCI, SZCI, ST, SCC, STC, TR
Currency market M0, M2, IBOR, FTD, M1, RRR, RR
Foreign exchange market The weighted average exchange rate of U.S. dollar to RMB (U/C), the weighted average exchange rate of Japanese yen to RMB (J/C), the weighted average exchange rate of Hong Kong dollar to RMB (H/C), IEV, FOFE, FER

FOFE, foreign exchange account; FTD, benchmark deposit and loan interest rate; FER, foreign exchange reserves; IEV, import and export trade volume; IBOR, interbank lending rate; M0, cash in circulation; M1, narrow currency; M2, broad money; RRR, deposit reserve ratio; RR, rediscount rate; SHCI, Shanghai Composite Index; SZCI, Shenzhen Stock Exchange Component Index; SCC, stock market value; ST, stock turnover; STC, total stock market value; TR, turnover rate.

Whether we choose market indicators to reflect the interaction between markets is very important. Since the existing research has not conducted a comprehensive test on the interaction of 19 market indicators, this paper conducts a comprehensive causality test on the indicators [3]. We screened out decision-making indicators that not only reflect the market conditions in which they are located, but are also closely related to the other two markets. This article uses the causality test process shown in Figure 1. This causality test method speeds up the calculation on one hand, and on the other, sufficient causality judgements have been made on the indicators.

Fig. 1

Causality test process.

Linear causality test

The linear causality test with threshold is a method that compares the variance of the error term between the AR model and the binary AR model to determine whether there is a causal relationship between the two time series [4]. This method is aimed at two time series, namely time series X and Time series Y. We have established the following AR model and binary AR model, respectively: {x(n)=i=1qrα1,i(n)x(ni)+θ1(n),σ11=Var(θ1)y(n)=i=1qeβ1,i(n)y(ni)+υ1(n),σ21=Var(υ1) \left\{{\matrix{{x(n) = \sum\limits_{i = 1}^{qr}{\alpha_{1,i}}(n)x(n - i) + {\theta_1}(n),{\sigma_{11}} = Var({\theta_1})}\cr {y(n) = \sum\limits_{i = 1}^{qe}{\beta_{1,i}}(n)y(n - i) + {\upsilon_1}(n),{\sigma_{21}} = Var({\upsilon_1})}\cr}} \right. {x(n)=i=1qrα2,i(n)x(ni)+i=1qeβ2,i(n)y(ni)+θ2(n),σ12=Var(θ2)y(n)=i=1qeβ3,i(n)y(ni)+i=1qrα3,i(n)x(ni)+υ2(n),σ22=Var(υ2) \left\{{\matrix{{x(n) = \sum\limits_{i = 1}^{qr}{\alpha_{2,i}}(n)x(n - i) + \sum\limits_{i = 1}^{qe}{\beta_{2,i}}(n)y(n - i) + {\theta_2}(n),{\sigma_{12}} = Var({\theta_2})}\cr {y(n) = \sum\limits_{i = 1}^{qe}{\beta_{3,i}}(n)y(n - i) + \sum\limits_{i = 1}^{qr}{\alpha_{3,i}}(n)x(n - i) + {\upsilon_2}(n),{\sigma_{22}} = Var({\upsilon_2})}\cr}} \right. where x and y represent the observed values of time series X and time series Y, respectively. qe, qr is the lag order. Estimating the parameters of Eqs. (1) and (2) can obtain the corresponding variances σ11, σ21, σ12 and σ22. The strength of linear causality between X versus Y and Y versus X can be calculated from the following formula: FXY=ln(σ21σ22),FYX=ln(σ11σ12) {F_{X \to Y}} = \ln \left({{{{\sigma_{21}}} \over {{\sigma_{22}}}}} \right),\quad {F_{Y \to X}} = \ln \left({{{{\sigma_{11}}} \over {{\sigma_{12}}}}} \right)

Through the linear causality test of X versus Y, the strength of linear causality FXY can be obtained. Then comparing it with the threshold, if FXY is larger than the threshold, it is considered that X has a linear causality relationship with Y. Otherwise, it is considered that X has a wireless causality relationship with Y. It can be seen that the selection of the threshold is directly related to the inspection effect [5]. The specific steps are as follows:

First, we randomly sort the sequence Y to get YK. We use the time series X and YK to calculate the new statistic FXY new(k) and repeat it N times. Next, we sort FXY new(k) to FXY new(N) in descending order and set the 5% value as the threshold. Then we can calculate the linear causality test of Y versus X result.

Nonlinear causality test

Nonlinear causality test means that when predicting the time series Y, if the past information of the time series X is added to the prediction of Y, it is considered that X has a nonlinear causality relationship with Y. From this we can determine whether the time series X has a causal relationship with the time series Y. First, consider the following two time series X and Y as follows: {XLr(nLr)=(XnLr,XnLr+1,,Xn1),Lr=1,2,;n=Lr+1,Lr+2,YLe(nLe)=(YnLe,YnLe+1,,Yn1),Le=1,2,;n=Le+1,Le+2,Ym(n)=(Yn,Yn+1,,Yn+m1),m=1,2,;n=1,2,, \left\{{\matrix{{{X^{Lr}}(n - Lr) = ({X_{n - Lr}},{X_{n - Lr + 1}}, \cdots,{X_{n - 1}}),} \hfill & {Lr = 1,2, \cdots ;n = Lr + 1,Lr + 2, \cdots} \hfill\cr {{Y^{Le}}(n - Le) = ({Y_{n - Le}},{Y_{n - Le + 1}}, \cdots,{Y_{n - 1}}),} \hfill & {Le = 1,2, \cdots ;n = Le + 1,Le + 2, \cdots} \hfill\cr {{Y^m}(n) = ({Y_n},{Y_{n + 1}}, \cdots,{Y_{n + m - 1}}),} \hfill & {m = 1,2, \cdots ;n = 1,2, \cdots,} \hfill\cr}} \right. where Ym(n) is the value of m order lag of Y at time n. XLr(nLr) and YLe(nLe) are the lag values of the Lr order of X and the Le order of Y at the corresponding time. When m ≥ 1, Lr ≥ 1, Le ≥ 1 and θ ≥ 0, if the following conditional probability Eq. (5) holds, it means that X does not have a nonlinear causal relationship with Y, that is, Pr(ym(n)ym(s)<θ|yLe(nLe)yLe(sLe)<θ,xLr(nLr)xLr(sLr)<θ)=Pr(ym(n)ym(s)<θ|yLe(nLe)yLe(sLe)<θ) \matrix{{\Pr \left({\left\| {{y^m}(n) - {y^m}(s)} \right\| < \theta \left| {\left\| {{y^{Le}}(n - Le) - {y^{Le}}(s - Le)} \right\|} \right. < \theta,\left\| {{x^{Lr}}(n - Lr) - {x^{Lr}}(s - Lr)} \right\| < \theta} \right)} \hfill\cr {= \Pr \left({\left\| {{y^m}(n) - {y^m}(s)} \right\| < \theta \left| {\left\| {{y^{Le}}(n - Le) - {y^{Le}}(s - Le)} \right\|} \right. < \theta} \right)} \hfill\cr} where Pr(·) stands for probability; ||·|| represents the maximum norm; x and y represent the observed values of X and Y, respectively and θ is the threshold: C1(m+Le,Lr,θ)C2(Le,Lr,θ)=C3(m+Le,θ)C4(Le,θ){C1(m+Le,Lr,θ)=Pr(ym+Le(nLe)ym+Le(sLe)<θ,xLr(nLr)xLr(sLr)<θ)C2(Le,Lr,θ)=Pr(yLe(nLe)yLe(sLe)<θ,xLr(nLr)xLr(sLr)<θ)C3(m+Le,θ)=Pr(ym+Le(nLe)ym+Le(sLe)<θ)C4(Le,θ)=Pr(yLe(nLe)yLe(sLe)<θ) \matrix{{{{{C_1}(m + Le,Lr,\theta)} \over {{C_2}(Le,Lr,\theta)}} = {{{C_3}(m + Le,\theta)} \over {{C_4}(Le,\theta)}}} \hfill\cr {\left\{{\matrix{{{C_1}(m + Le,Lr,\theta) = \Pr (\left\| {{y^{m + Le}}(n - Le) - {y^{m + Le}}(s - Le)} \right\| < \theta,\left\| {{x^{Lr}}(n - Lr) - {x^{Lr}}(s - Lr)} \right\| < \theta)} \hfill\cr {{C_2}(Le,Lr,\theta) = \Pr (\left\| {{y^{Le}}(n - Le) - {y^{Le}}(s - Le)} \right\| < \theta,\left\| {{x^{Lr}}(n - Lr) - {x^{Lr}}(s - Lr)} \right\| < \theta)} \hfill\cr {{C_3}(m + Le,\theta) = \Pr (\left\| {{y^{m + Le}}(n - Le) - {y^{m + Le}}(s - Le)} \right\| < \theta)} \hfill\cr {{C_4}(Le,\theta) = \Pr (\left\| {{y^{Le}}(n - Le) - {y^{Le}}(s - Le)} \right\| < \theta)} \hfill\cr}} \right.} \hfill\cr}

The statistics in Eq. (6) are estimated by Eq. (7) {Cl1(Le,Lr,θ)=2nc(nc1)s=max(Le,Lr)+1Nm+1n<sNm+1I(yLe(nLe)yLe(sLe)<θ)(xLr(nLr)xLr(sLr)<θ)Cl2(Le,θ)=2nc(nc1)s=max(Le,Lr)+1Nm+1n<sNm+1I(yLe(nLe)yLe(sLe)<θ) \left\{{\matrix{{{C_{{l_1}}}(Le,Lr,\theta) = {2 \over {{n_c}({n_c} - 1)}}\sum\limits_{s = \max (Le,Lr) + 1}^{N - m + 1}\sum\limits_{n < s}^{N - m + 1}I(\left\| {{y^{Le}}(n - Le) - {y^{Le}}(s - Le)} \right\| < \theta)} \hfill\cr {(\left\| {{x^{Lr}}(n - Lr) - {x^{Lr}}(s - Lr)} \right\| < \theta)} \hfill\cr {{C_{{l_2}}}(Le,\theta) = {2 \over {{n_c}({n_c} - 1)}}\sum\limits_{s = \max (Le,Lr) + 1}^{N - m + 1}\sum\limits_{n < s}^{N - m + 1}I(\left\| {{y^{Le}}(n - Le) - {y^{Le}}(s - Le)} \right\| < \theta)} \hfill\cr}} \right.

The above form is an ideal situation for nonlinear causality testing, but real data is difficult to meet its stringent requirements. Therefore, we have further converted it to make it more convenient to use. If X has no nonlinear causality with Y, there are nc(C1(m+Le,Lr,θ,nc)C2(Le,Lr,θ,nc)C3(m+Le,θ,nc)C4(Leθ,nc))N(0,σ2(m,Le,Lr,θ)) \sqrt {{n_c}} ({{{C_1}(m + Le,Lr,\theta,{n_c})} \over {{C_2}(Le,Lr,\theta,{n_c})}} - {{{C_3}(m + Le,\theta,{n_c})} \over {{C_4}(Le\theta,{n_c})}}) \sim N(0,{\sigma^2}(m,Le,Lr,\theta))

When the value of the statistic in formula (8) falls at the two ends of the normal distribution, it means that X has a nonlinear causality relationship with Y, that is, adding the information of the time series X when predicting the time series Y helps to improve the prediction accuracy. The nonlinearity of Y versus X causality is judged in the same way.

Empirical test results of causality

We use the causality matrix to select the 12 market indicators that have the greatest degree of mutual influence, and each market has 4 indicators. In this way, these 12 market indicators can not only reflect the characteristics of the market in which they are located, but also reflect their relationship with each other [6]. The connection between the other two markets is the 12 decision-making indicators selected through the causality test, as shown in Table 2. The causal relationship between the 12 decision-making indicators is shown in Table 3. In Table 3, ‘1’ represents the left indicator There is a causal relationship with the upper indicator ‘0’, which means that the left indicator has no causal relationship with the upper indicator. It is not difficult to see that the causal relationship is asymmetric.

12 decision-making indicators selected.

Stock market Currency market Foreign exchange market

SHCI Cash in circulation (M0) Weighted average exchange rate of US dollar to RMB (U/C)
SZCI M1 Hong Kong dollar to renminbi weighted average exchange rate (H/C)
SCC M2 FER
ST Interbank Offered Rate (IBOR) IEV

FER, foreign exchange reserves; IEV, import and export trade volume; M1, narrow currency; M2, broad money; SHCI, Shanghai Composite Index; SZCI, Shenzhen Stock Exchange Component Index; SCC, Stock market capitalisation; ST, stock turnover.

Causal relationship matrix between 12 decision indicators.

SCC SHCI SZCI ST M0 M1 M2 IBOR U/C H/C FER IEV

SCC 1 1 0 1 1 1 1 1 0 1 1 1
SHCI 1 1 1 1 0 1 1 1 1 1 1 0
SZCI 1 1 1 1 1 1 1 0 1 1 1 1
ST 1 1 1 1 1 1 1 1 1 0 1 1
M0 1 1 0 1 1 1 1 1 0 0 1 1
M1 1 1 0 1 1 1 1 1 0 1 1 1
M2 1 1 0 1 1 1 1 1 0 0 1 1
IBOR 1 1 1 1 1 1 1 1 0 1 1 1
U/C 1 1 0 0 1 1 1 1 1 1 1 1
H/C 1 0 0 0 1 1 1 1 1 1 1 1
FER 1 1 1 1 1 1 1 1 0 0 1 1
IEV 0 1 1 1 1 1 1 1 0 0 1 1

FER, foreign exchange reserves; IEV, import and export trade volume; M1, narrow currency; M2, broad money; SHCI, Shanghai Composite Index; SZCI, Shenzhen Stock Exchange Component Index; SCC, stock market value; ST, stock turnover.

Establish a comprehensive index of the three markets

This study takes three overall markets as the research objectives. We need to combine multiple decision-making indicators in the stock, currency, and foreign exchange markets into corresponding market comprehensive indexes to represent each market. The construction method is similar to the foreign exchange market stress index. In the stock market, the market value of stocks and the turnover of stocks are extremely large indicators [7]. The larger the value, the more capital investors invest in the stock market, and the more active stock transactions are. Similarly, the Shanghai Composite Index (SHCI) and the Shenzhen Stock Exchange Component Index (SZCI) are also very large indicators. The larger the value the higher the investor's expectations of the stock market are. Based on this, the following stock market comprehensive index is established in this article: SMI=SCCμSCCσSCC+SCCμSHCIσSHCI+SZCIμSZCIσSZCI+STμSTσST SMI = {{SCC - \mu SCC} \over {\sigma SCC}} + {{SCC - \mu SHCI} \over {\sigma SHCI}} + {{SZCI - \mu SZCI} \over {\sigma SZCI}} + {{ST - \mu ST} \over {\sigma ST}}

The larger the value of M0, M1 and M2, the greater the money supply. The interbank lending rate (IBOR) is a very small indicator. The smaller the value, the bank can obtain funds at a lower cost. Based on this, we build the following currency market composite index: MMI=M0μM0σM0+M1μM1σM1+M2μM2σM2IBORμIBORσIBOR MMI = {{M0 - \mu M0} \over {\sigma M0}} + {{M1 - \mu M1} \over {\sigma M1}} + {{M2 - \mu M2} \over {\sigma M2}} - {{IBOR - \mu IBOR} \over {\sigma IBOR}}

In the foreign exchange market, we used the foreign exchange market stress index to establish the following foreign exchange market comprehensive index: FMI=FERμFERσFER+IEVμIEVσIEV+U/CμU/CσU/CH/CμH/CσH/C FMI = {{FER - \mu FER} \over {\sigma FER}} + {{IEV - \mu IEV} \over {\sigma IEV}} + {{U/C - \mu U/C} \over {\sigma U/C}} - {{H/C - \mu H/C} \over {\sigma H/C}}

In order to illustrate the effectiveness of the three market composite indexes, this article gives a historical trend chart of the three market composite indexes from January 2005 to May 2019 (as shown in Figure 2).

Fig. 2

Historical trends of the three market composite indexes.

Nonlinear evolution model
Unconstrained nonlinear evolution model

Considering that there may not only be linear relationships but also nonlinear relationships between financial markets, this paper constructs three market data-driven models based on differential equations. The specific form is shown in Eq. (12). {dX1(t)dt=A1X+XTB1X+C1dX2(t)dt=A2X+XTB2X+C2dX3(t)dt=A3X+XTB3X+C3 \left\{{\matrix{{{{d{X_1}(t)} \over {dt}} = {A_1}X + {X^T}{B_1}X + {C_1}}\cr {{{d{X_2}(t)} \over {dt}} = {A_2}X + {X^T}{B_2}X + {C_2}}\cr {{{d{X_3}(t)} \over {dt}} = {A_3}X + {X^T}{B_3}X + {C_3}}\cr}} \right. where Xi(t), i = 1, 2, 3 is the composite index of the three markets at time t. x = (X1(t − 1), X2(t − 1), X3(t − 1), X1(t − 2),⋯, X3(tm))T is the lagging term of the three-market composite index; m is the lag order; Ai is a 1 × 3 m matrix and Bi is a 3 m × 3 m upper triangular matrix and Ci is a constant term. i = 1, 2, 3. It is not difficult to find that the unconstrained nonlinear evolution model (12) contains both the linear structure between variables and the nonlinear structure between variables. Compared with the regression model, it is more in line with the complex and changeable real market and is easy to analyse the market [8] and the nonlinear evolution structure between.

Constrained nonlinear evolution model

In the real market, due to the excessive parameters of the basic model (12), the complex structure, and the inability to clearly determine the influence and relationship between the three markets, it brings a lot of difficulties to actual research and analysis. In fact, the existing research results show that the truth is that the impact relationship between some indicators in the market is not significant [9]. There may also be an insignificant relationship between the composite indexes and their lags in the study of this article. This means that the weight parameter between these indicators should be zero. For this problem, we propose a constrained nonlinear evolution model. This model adds cardinality constraints on the basis of the unconstrained model. It is assumed that only the few variables with the most significant influence are non-zero. In view of the above analysis, we construct the following nonlinear evolution model with constraints: {dX1(t)dt=A1X+XTB1X+C1dX2(t)dt=A2X+XTB2X+C2dX3(t)dt=A3X+XTB3X+C3 \left\{{\matrix{{{{d{X_1}(t)} \over {dt}} = {A_1}X + {X^T}{B_1}X + {C_1}}\cr {{{d{X_2}(t)} \over {dt}} = {A_2}X + {X^T}{B_2}X + {C_2}}\cr {{{d{X_3}(t)} \over {dt}} = {A_3}X + {X^T}{B_3}X + {C_3}}\cr}} \right.

Of them, K1K2 is a non-negative integer, and the rest of the symbols are the same as above. It is easy to see from Eq. (13) that the non-zero values in matrix Ai and matrix Bi are greatly reduced after we add constraints [10]. The weight is concentrated on a few parameters and the structure is clearer.

Parameter estimation

Because models (12) and (13) have too many parameters, the efficiency of using traditional parameter estimation methods is extremely low. In addition, there are constraints in model (13), and we cannot effectively solve the problem using the traditional parameter estimation methods. This paper uses the improved differential evolution (COMDE) algorithm to estimate the parameters of models (12) and (13). The differential evolution algorithm is a relatively new group-based random optimization method. It is simple, fast and robust, along with other characteristics. Different from other evolutionary algorithms, its mutation operator is obtained from the difference in multiple pairs of vectors selected arbitrarily in the population [11]. This algorithm is mainly used for real parameter optimization problems, especially for nonlinear and non-differentiable continuous space problems Solving has obvious advantages over other evolutionary algorithms. In order to understand how to use the COMDE algorithm to solve the parameters of models (12) and (13), we take model (12) as an example, if the information before t′ is known, That is, given that X(t) = (X1(t − 1), X2(t − 1), X3(t − 1), X1(t − 2),⋯, X3(tm))T, t = m+1, m+2,⋯, t′ defines ΔXi(t) = Xi(t)−Xi(t −1) the discretisation form of dXi,tdt {{d{X_{i,t}}} \over {dt}} (real market data is discrete data), then model (12) can be rewritten as: {ΔX1(t)=A1X(t)+X(t)TB1X(t)+C1ΔX2(t)=A2X(t)+X(t)TB2X(t)+C2ΔX3(t)=A3X(t)+X(t)TB3X(t)+C3,t=m+1,m+2,,t' \left\{{\matrix{{\Delta {X_1}(t) = {A_1}X(t) + X{{(t)}^T}{B_1}X(t) + {C_1}} \hfill\cr {\Delta {X_2}(t) = {A_2}X(t) + X{{(t)}^T}{B_2}X(t) + {C_2}} \hfill\cr {\Delta {X_3}(t) = {A_3}X(t) + X{{(t)}^T}{B_3}X(t) + {C_3},t = m + 1,m + 2, \cdots,{t^{'}}} \hfill\cr}} \right.

At time t′, it is easy to see that the data before time t′ in each equation in Eq. (14) are known. We can regard each equation of Eq. (14) as a sub-problem. Based on the principle of minimising errors, we establish the following model for the i comprehensive index (i.e. the i equation) of model (14) as: Minεi=t=m+1t'(Xi(t1)+AiX(t)+X(t)TBiX(t)+Ci)2,i=1,2,3 Min\;{\varepsilon_i} = \sum\limits_{t = m + 1}^{{t^{'}}}{\left({{X_i}(t - 1) + {A_i}X(t) + X{{(t)}^T}{B_i}X(t) + {C_i}} \right)^2},\quad i = 1,2,3

Similarly, we discretise model (13) to get: {ΔX1(t)=A1X(t)+X(t)TB1X(t)+C1ΔX2(t)=A2X(t)+X(t)TB2X(t)+C2ΔX3(t)=A3X(t)+X(t)TB3X(t)+C3 \left\{{\matrix{{\Delta {X_1}(t) = {A_1}X(t) + X{{(t)}^T}{B_1}X(t) + {C_1}}\cr {\Delta {X_2}(t) = {A_2}X(t) + X{{(t)}^T}{B_2}X(t) + {C_2}}\cr {\Delta {X_3}(t) = {A_3}X(t) + X{{(t)}^T}{B_3}X(t) + {C_3}}\cr}} \right.

The following model is established for the i comprehensive index of model (16) as: {Minεi=t=m+1t'(Xi(t)Xi(t1)+(AiX(t)+X(t)TBiX(t)+Ci))2,i=1,2,3 \left\{{\matrix{{Min\;{\varepsilon_i} = \sum\limits_{t = m + 1}^{{t^{'}}}{{\left({{X_i}(t) - {X_i}(t - 1) + \left({{A_i}X(t) + X{{(t)}^T}{B_i}X(t) + {C_i}} \right)} \right)}^2},\quad i = 1,2,3}\cr}} \right.

Of them, Xi(t) is the observation value of the i comprehensive index at time t, and the remaining symbols are the same as before [12]. The COMDE algorithm can be used to calculate the unknown parameters in models (12) and (13) through models (14) and (15).

Empirical test and result analysis

In order to discuss the effects of the above two non-constrained and constrained nonlinear evolution models, this paper uses the BP neural network model and the multiple linear regression model to model under the same data. We compare and analyse the effects of the unconstrained nonlinear evolution model and the constrained nonlinear evolution model. The data are a comprehensive index obtained through the causality test and the index construction process. We randomly select 80% of the full sample as the learning sample, and the remaining 20% as the test sample. In order to improve the accuracy of the model, the number of iterations of the COMDE algorithm is 10,000 generations, the number of populations is 200 and the remaining parameters are default values [13]. The number of iterations of the BP neural network is set to 10,000 times, and the remaining parameters are default values. The values of K1 and K2 are more flexible; when the values of K1 and K2 are large, the model has a higher learning accuracy, but the structure is relatively unclear. When the values of K1 and K2 are small, the model has a clear structure, but the learning accuracy is poor. This article considers two factors: K1 = 5 and K2 = 10. Generally, the information in the past period of time has a guiding effect on the future market.

From the perspective of model accuracy, the constrained nonlinear evolution model has the best overall performance under different values of m. It has strong learning ability for learning samples and evolutionary ability for testing samples. From the definition of model form, the constrained nonlinear evolution model and the multiple linear regression model have the simplest structure. The constrained nonlinear evolution model has only a few non-zero parameters that can clearly analyse the influence relationship between each comprehensive index. Compared with the unconstrained nonlinear evolution model the structure of is relatively unclear [14]. The BP neural network model is a black box operation. It is not difficult to find that the effect and structure of the constrained nonlinear evolution model are optimal, and it can analyse the nonlinear evolution relationship of the three markets well.

From the different values of m, the error of the constrained nonlinear evolution model is the smallest when m = 6. As m increases, the error begins to increase. It shows that the stock market, the currency market and the foreign exchange market have the most significant interactions in the short term. When the government regulates the market, it should pay attention to short-term market information. The effectiveness of market information gradually decreases over time as: {ΔSMIt=0.11560.3096FMIt20.4919FMIt3+0.9231FMIt40.1646SMIt6+0.0506FMIt6+0.1185FMIt1FMIt20.1479FMIt1FMIt4+0.5909SMIt2FMIt30.8077SMIt2FMIt5+1.0156SMIt3FMIt50.7162SMIt3FMIt60.3323FMIt3FMIt6+0.3052FMIt4FMIt51.0760SMIt4FMIt5+0.9167SMIt4FMIt6ΔMMIt=0.12050.5259FMIt10.0366MMIt20.7861FMIt2+0.1343FMIt3+1.2355FMIt40.2591MMIt1FMIt60.1529FMIt1MMIt50.1968FMIt2SMIt4+0.7831FMIt2FMIt52.4403FMIt3MMIt5+2.3672FMIt3FMIt5+4.2843FMIt4MMIt52.4791FMIt4FMIt50.4506FMIt4FMIt61.6089MMIt5FMIt5ΔFMIt=0.00410.1914FMIt1+0.2331FMIt2+0.1648FMIt40.1495FMIt50.0559FMIt6+0.0399FMIt1FMIt40.2672FMIt1FMIt60.5993FMIt2MMIt3+0.4584FMIt2FMIt6+0.5579MMIt3FMIt3+0.0740MMIt3FMIt40.3212FMIt320.0252FMIt3SMIt4+0.0735SMIt4FMIt4+0.0155FMIt4FMIt6 \left\{{\matrix{{\Delta SM{I_t} = 0.1156 - 0.3096FM{I_{t - 2}} - 0.4919FM{I_{t - 3}} + 0.9231FM{I_{t - 4}} - 0.1646SM{I_{t - 6}}} \hfill\cr {\quad \quad \quad+ 0.0506FM{I_{t - 6}} + 0.1185FM{I_{t - 1}}FM{I_{t - 2}} - 0.1479FM{I_{t - 1}}FM{I_{t - 4}}} \hfill\cr {\quad \quad \quad+ 0.5909SM{I_{t - 2}}FM{I_{t - 3}} - 0.8077SM{I_{t - 2}}FM{I_{t - 5}} + 1.0156SM{I_{t - 3}}FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.7162SM{I_{t - 3}}FM{I_{t - 6}} - 0.3323FM{I_{t - 3}}FM{I_{t - 6}} + 0.3052FM{I_{t - 4}}FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 1.0760SM{I_{t - 4}}FM{I_{t - 5}} + 0.9167SM{I_{t - 4}}FM{I_{t - 6}}} \hfill\cr {\Delta MM{I_t} = 0.1205 - 0.5259FM{I_{t - 1}} - 0.0366MM{I_{t - 2}} - 0.7861FM{I_{t - 2}} + 0.1343FM{I_{t - 3}}} \hfill\cr {\quad \quad \quad+ 1.2355FM{I_{t - 4}} - 0.2591MM{I_{t - 1}}FM{I_{t - 6}} - 0.1529FM{I_{t - 1}}MM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.1968FM{I_{t - 2}}SM{I_{t - 4}} + 0.7831FM{I_{t - 2}}FM{I_{t - 5}} - 2.4403FM{I_{t - 3}}MM{I_{t - 5}}} \hfill\cr {\quad \quad \quad+ 2.3672FM{I_{t - 3}}FM{I_{t - 5}} + 4.2843FM{I_{t - 4}}MM{I_{t - 5}} - 2.4791FM{I_{t - 4}}FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.4506FM{I_{t - 4}}FM{I_{t - 6}} - 1.6089MM{I_{t - 5}}FM{I_{t - 5}}} \hfill\cr {\Delta FM{I_t} = 0.0041 - 0.1914FM{I_{t - 1}} + 0.2331FM{I_{t - 2}} + 0.1648FM{I_{t - 4}} - 0.1495FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.0559FM{I_{t - 6}} + 0.0399FM{I_{t - 1}}FM{I_{t - 4}} - 0.2672FM{I_{t - 1}}FM{I_{t - 6}}} \hfill\cr {\quad \quad \quad- 0.5993FM{I_{t - 2}}MM{I_{t - 3}} + 0.4584FM{I_{t - 2}}FM{I_{t - 6}} + 0.5579MM{I_{t - 3}}FM{I_{t - 3}}} \hfill\cr {\quad \quad \quad+ 0.0740MM{I_{t - 3}}FM{I_{t - 4}} - 0.3212FMI_{t - 3}^2 - 0.0252FM{I_{t - 3}}SM{I_{t - 4}}} \hfill\cr {\quad \quad \quad+ 0.0735SM{I_{t - 4}}FM{I_{t - 4}} + 0.0155FM{I_{t - 4}}FM{I_{t - 6}}} \hfill\cr}} \right.

In order to study the nonlinear evolution relationship between the three markets, the results of the constrained nonlinear evolution model when m is 6 are given here as shown in Eq. (18). At the same time, this is also a nonlinear evolution structure between the three markets. Among them, SMI, MMI and FMI represent the stock, currency and foreign exchange market composite indexes, respectively [15]. The relationship between the three market composite indexes is shown in Table 4.

Interrelationships among the three markets.

Stock market Currency market Foreign exchange market

Stock market 7 items, linear and nonlinear 1 term, nonlinear 2 terms, nonlinear
Currency market 0 items 6 items, linear and nonlinear 3 terms, nonlinear
Foreign exchange market 14 items, linear and nonlinear 14 items, linear and nonlinear 15 items, linear and nonlinear

The items in Table 4 represent the way and degree of influence of the left market on the upper market. From Table 4, the following conclusions can be drawn: (1) each market has the most significant impact on itself, and all participate in the evolution of its own market in linear and nonlinear forms [16]. It shows that the operation mechanism of China's financial sub-markets is relatively independent. Although the various sub-markets are closely connected, the independent development of each sub-market is still the main. (2) The foreign exchange market has an important influence on the stock market and the money market. The foreign exchange market participates in the evolution of the stock and money market in a linear and nonlinear form, and the degree of influence is huge. When the foreign exchange market fluctuates, the other two markets will also fluctuate in the short term. (3) The stock market has less effect on the evolution of the other two markets. This is due to the strong government regulation in the currency and foreign exchange markets, and fluctuations in the stock market are difficult to transmit to the currency and foreign exchange markets.

Conclusion

This paper constructs three market-comprehensive indexes of stocks, currency and foreign exchange through index selection and causality test, and establishes three nonlinear evolution models of markets driven by data. We use the COMDE algorithm to obtain the nonlinear evolution structure and evolution relationship of the stock, currency and foreign exchange markets, and break through the nonlinear evolution research based on theoretical research. The constrained nonlinear evolution model is for the nonlinear evolution problem between the three markets. They have good learning and evolution ability and they have the clearest structure. The three markets not only have independent operating mechanisms, but also have a significant influence relationship with the other two markets. In particular, the impact of the foreign exchange market on the stock and currency markets is the most obvious of the three markets.

Fig. 1

Causality test process.
Causality test process.

Fig. 2

Historical trends of the three market composite indexes.
Historical trends of the three market composite indexes.

Market indicators selected in the article.

Market Market indicators

Stock market SHCI, SZCI, ST, SCC, STC, TR
Currency market M0, M2, IBOR, FTD, M1, RRR, RR
Foreign exchange market The weighted average exchange rate of U.S. dollar to RMB (U/C), the weighted average exchange rate of Japanese yen to RMB (J/C), the weighted average exchange rate of Hong Kong dollar to RMB (H/C), IEV, FOFE, FER

Causal relationship matrix between 12 decision indicators.

SCC SHCI SZCI ST M0 M1 M2 IBOR U/C H/C FER IEV

SCC 1 1 0 1 1 1 1 1 0 1 1 1
SHCI 1 1 1 1 0 1 1 1 1 1 1 0
SZCI 1 1 1 1 1 1 1 0 1 1 1 1
ST 1 1 1 1 1 1 1 1 1 0 1 1
M0 1 1 0 1 1 1 1 1 0 0 1 1
M1 1 1 0 1 1 1 1 1 0 1 1 1
M2 1 1 0 1 1 1 1 1 0 0 1 1
IBOR 1 1 1 1 1 1 1 1 0 1 1 1
U/C 1 1 0 0 1 1 1 1 1 1 1 1
H/C 1 0 0 0 1 1 1 1 1 1 1 1
FER 1 1 1 1 1 1 1 1 0 0 1 1
IEV 0 1 1 1 1 1 1 1 0 0 1 1

Interrelationships among the three markets.

Stock market Currency market Foreign exchange market

Stock market 7 items, linear and nonlinear 1 term, nonlinear 2 terms, nonlinear
Currency market 0 items 6 items, linear and nonlinear 3 terms, nonlinear
Foreign exchange market 14 items, linear and nonlinear 14 items, linear and nonlinear 15 items, linear and nonlinear

12 decision-making indicators selected.

Stock market Currency market Foreign exchange market

SHCI Cash in circulation (M0) Weighted average exchange rate of US dollar to RMB (U/C)
SZCI M1 Hong Kong dollar to renminbi weighted average exchange rate (H/C)
SCC M2 FER
ST Interbank Offered Rate (IBOR) IEV

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