The paper establishes a related differential equation model about changes in financial interest rates. It uses information related to liquidity to feedback the law and stability of differential equations in interest rate changes. The article applies stochastic processes and partial differential equations to complex financial networks to confirm node yields in financial market networks. It confirms the existence of interest rate stickiness in Chinese financial markets. The advantage of this interest rate model is that when the external economic environment changes, the state of interest rates will also change accordingly.

#### Keywords

- rate of return
- liquidity
- differential equation
- the principle of equilibrium
- financial market
- interest rate changes

#### MSC 2010

- 91G30

In modern economics, the rate of return often refers to the rate of return to maturity of funds. Therefore, it is an important research content of finance. Investing funds in vital areas to obtain the most significant yield is the most concerning issue for all investors.

Some scholars have established basic models for calculating movements and fluctuations in the stock market, including the interest rate and liquidity equations, stock buying and selling equations, stock price change rate and so on. In addition, they put forward the basic principles of calculating the trends on the stock market from the perspective of market operation laws and the combination of financial instrument trading principles and practices [1] . As examples, we may mention the principle of potentiality and the principle of supply and demand difference.

Assume that the entire financial market of the national economy is regarded as a network. It has m nodes, which are represented by Arabic numerals 1, 2,..., respectively. Each node represents various economic sectors or industries, such as the stock and securities industry, banking and finance industry, real estate industry etc. Since the rate of return between nodes changes over time, and since investors are motivated towards maximising their profit, there will be capital flows between nodes. The flow of funds between different nodes leads to changes in the instantaneous rate of return of each node, thereby obtaining the set of equations that the instantaneous rate of return of each node should satisfy [2] . Hence, it is called the stock market rate of return-liquidity equation.

In the above financial network, node i represents each financial industry with different functions. The internal capital flow of the financial industry with the same function is not considered. The financial industry with the same functions contains several different entities. They also form a network by themselves. Taking the stock and securities industry as an example, we observe that investors have multiple choices when choosing to buy stocks, and thus the flow of funds between different stocks will occur frequently. Take the banking industry as an example to analyse the financial network. There will naturally be capital flows between them. Investors will also choose according to the characteristics of each bank. Therefore, each node in the above equation is a subnetwork composed of different smaller nodes. Such a network is called a complex network [3] . In a complex financial network, there are capital flows between nodes in two different sub-networks. There will also be capital flows between two different nodes within the same sub-network. Figure 1 shows the capital flow process of the bank’s financial network. The literature mentioned above only considers the case of simple networks and does not study complex financial networks.

To summarise, the primary purpose of this paper is to regard the entire financial network as a complex network composed of several sub-networks with different functions. Each sub-network represents a financial industry with particular functions. It is composed of several nodes, and each node represents a specific financial entity [4] . This article will establish and study a mathematical model similar to the stock market return-liquidity equation. We reveal the changing laws of each node’s instantaneous rate of return in the complex financial network.

For the sake of simplicity, we only consider a complex financial network consisting of two sub-networks. Suppose a complex network N contains two sub-networks _{i}_{j}_{1}, _{2},··· ,_{m}_{1}, _{2},··· ,_{n}

To consider the change law of the node interest rate and the number of funds in circulation with time t, we denote R as the totality of real numbers ^{+}
= {_{i}_{j}_{i}_{j}_{i}_{j}_{i}_{j}_{i}_{j}

There is a flow of funds between the nodes of the subnet X. This capital flow has existence for any _{ij}_{i}_{j}_{ij}_{i}_{j}_{ij}_{j}_{i}_{ij}_{i}_{j}

Time _{ij}_{i}_{j}_{ij}_{j}_{i}_{ij}_{i}_{j}_{ij}_{i}_{j}_{ij}_{j}_{i}

The following relationship exists for any ^{+}
,

We mark _{i}_{j}_{i}_{j}^{*}, ^{*} is an average number.

According to economic market experience, we find that the cause of capital flow is the difference in the rate of return. However, the difference between the introductory rate of return is acceptable. Therefore, the difference in the introductory rate of return between the nodes will not cause the flow of funds. Similarly, if the difference between the immediate rate of return is equal to the difference between the introductory rate of return, there will be no capital flow between nodes [8] . Thus, the real reason for the flow of funds is the difference in the amount of change.

Let _{i}_{j}_{ij}_{j}_{i}_{ij}

The same can be expressed as:

On the other hand, for a certain _{i}_{j}

We assume that node _{i}_{j}

Where _{i}_{j}

Substituting Eqs (5)–(7) into Eqs (10)–(11), and then substituting Eqs (8) and (9), we obtain the differential equations of the complex financial network’s instant return rate change, which is called complex financial network The rate of return equation is the following:

We use differential equation theory to discuss Eq. (12). Eq. (12) is a system of equations composed of _{ik}_{ki}_{jk}_{kj}

Among them _{1}, _{2},··· ,_{m}_{1}, _{2},···_{n}_{ik}_{m}_{×}_{m}_{jk}_{n}_{×}_{n}

C and D are both invertible matrices. So the equation system in Eq. (13) can be rewritten as

We assume ^{(}^{m}^{)}
= (1, 1,··· ,1) ∈ ^{m}^{(}^{n}^{)}
= (1, 1,··· ,1) ∈ ^{n}_{ik}_{ki}

The same can be proved as:

Combining the above three equations and substituting them into Eq. (16), we have

Therefore, there are the following conclusions:

_{1}(_{2}(_{m}_{1}(_{2}(_{n}^{+}
,

We assume that if

Thus there is a constant r such that for any

The certificate is complete.

To find the constant r, we take the instant interest rate vector at a particular initial moment

And substituting into Eq. (17), we can get

From Eq. (17), we have

Eq. (17) shows that the weighted sum of the returns of each node in the complex financial network remains constant, and it does not change with time t. Eq. (19) shows that the weighted sum of the changes in the network is always zero. This conclusion is called the principle of interest rate equilibrium of the financial network in the literature [11]
. This shows that an increased node in the network will inevitably lead to a decrease in the rate of return of other nodes. Thus, in Eqs (17) or (19), the weighting coefficients _{i}_{j}

Theorem 1 shows that the principle of interest rate equilibrium is still valid in a complex financial network.

Integrating both sides of Eqs. (8) and (9) from _{0} to t, we can get

It can be seen that _{i}_{j}

We assume that the equation set in Eq. (12) describes the nodes of the return rate change rule of a complex financial network, including two sub-networks of Bank (_{i}_{j}_{i}_{j}

Then _{i}_{j}_{i}_{j}^{+}
, let us recall that

The thesis first established the differential equation model of the first-rate flux of interest rate in the closed system. It proved that the weighted sum of interest rates at each node is a constant – i.e. the principle of interest rate equilibrium in the financial market. The interest rate limit of each node is the average interest rate of the entire network. Second, when the introductory interest rate of each node is not the same, the article establishes the inhomogeneous interest rate first-rate flux differential equation model. It proves that the weighted sum of interest rates at each node of the financial network is still a constant. The real-time interest rate difference between each node will eventually steadily tend to its primary interest rate difference.

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