Differential equation model of financial market stability based on big data

In a specific economic environment, legal environment, credit environment, market environment and institutional environment, financial market entities provide financial services based on their interests and behaviours [3]. There are shared interests and survival foundations among all market entities. They participate in the market game while carrying out mutually beneficial cooperation to form a symbiosis and coexistence relationship. For example, guarantee companies provide banks with intermediary services such as verification, notarisation, evaluation, coordination and arbitration in market competition, transaction order and dispute resolution as a third party, but also require banks to provide customer credit information for their services. They and commercial banks are mutually beneficial, and they cannot survive after separation. As a result, a symbiotic coexistence relationship is formed.

The foundation of the symbiotic relationship between the financial market entities lies in a symbiotic matrix between each one and the others. They are mainly complementary resources among financial market entities. To promote the smooth development of symbiosis activities, financial market entities need to develop symbiosis communication methods and mechanisms, such as establishing regular meetings, mutual visits and other communication mechanisms and methods [4]. We can call these communication mechanisms and methods a symbiotic interface. The efficiency and function of the symbiosis interface are the key factors that determine the efficiency of financial symbiosis and the stability of the symbiosis relationship.

Symbiosis energy is a concrete manifestation of the viability and value-added ability of financial symbiosis. In the financial symbiosis relationship, the symbiosis energy mainly refers to the funds provided by the bank to the financial market entities [5]. The business development, the improvement of management ability, the enhancement of the ability to resist risks, and net profit increase. The characteristic value of the symbiosis interface and the degree of symbiosis are two key variables that affect the energy of the symbiosis. The characteristic value of the symbiosis interface is an important parameter to measure the communication resistance on the symbiosis interface. The more symbiotic the interfaces, the larger the contact surface and the better the contact medium, the smaller the communication resistance, and the closer the corresponding characteristic value is to zero. In addition, the greater the degree of symbiosis, the greater the energy of symbiosis. The level of symbiotic energy is the prerequisite for the continuous development of symbiotic relationships.

According to the differences in the energy distribution of financial market entities, we divide the financial symbiosis relationship into two types: favour symbiosis and mutual symbiosis. The symbiosis of partial benefit has the following characteristics: (1) New energy is produced, but one party obtains most of the new energy. (2) The symbiotic relationship is beneficial to one party but not harmful to the other party. (3) There are bilateral two-way exchanges between financial market entities. The reciprocal symbiosis relationship has the following characteristics: (1) New energy is generated, and there is a broad-spectrum distribution of new energy. (2) There is a bilateral exchange mechanism and a multilateral exchange mechanism between the financial market entities. (3) The growth and evolution of financial market entities are synchronised. Figure 1 shows the financial symbiosis relationship.

Fig. 1

(1) The profits of financial market entities can reflect the endogenous and exogenous changes experienced by various financial market entities in the evolution of financial symbiosis. We may explain the evolution process of the symbiosis relationship by describing the profit changes of the main body of the financial market. (2) The profit x of financial market entities is a function of time t. (3) The growth of financial market entities is constrained by various factors of production. In a given time and geographical space, the total production factors such as technology, raw materials, labour and capital are certain. We assume that the maximum amount is N. As the main body of the financial market grows, the economic benefits brought by its scale will rise to a certain level. It will inevitably be restricted by factor endowments and enter a critical stage. Therefore, the main body of the financial market has the most significant quantity of production factors N. When x = N, the main body of the financial market stops growing. (4) The growth rate r of financial market entities decreases with the increase of their production factors. With the growth of the main body of the financial market, the factors of production that they can take are getting less and less, and the growth is slowing down. While the main body of the financial market grows, other market central bodies’ technological innovation, key resource acquisition, differentiated marketing and other business behaviours will inhibit the growth of the main body of the financial market. (5) Each additional unit of production factors of the main body of the financial market will have a 1/N inhibitory effect on the growth of itself and other market-main bodies.

The main body of the financial market uses the 1/N factor of production for every additional unit of production factor and increases x units of production factor of x/N. Then the ‘surplus production factor’ that financial market entities can obtain is only (1−xN) \left( {1 - {x \over N}} \right) . Taking into account the continuity of the growth of financial market entities, we can establish the profit change equation of financial symbiosis through the differential method (1) dxdt=rx(1−xN) {{dx} \over {dt}} = rx\left( {1 - {x \over N}} \right)

In the symbiosis of partial benefits, we call the main body of the financial market that benefits more as the leader and the other party as the dependent. We denote the profit scale of the leader as x₁(t) and the profit scale of the dependents as x₂(t). The dominant person can develop on his own before the dependent person appears. At this time, its profit satisfies the Logistic model: (2) dx1dt=r1x1(1−x1N1) {{d{x_1}} \over {dt}} = {r_1}{x_1}\left( {1 - {{{x_1}} \over {{N_1}}}} \right) N₁ is the amount of production factors taken by the leader. According to the hypothesis of this article, the profit level of the leader can be described as (3) dx1dt=r1x1(1−x1N1+ax1N1), a>0 {{d{x_1}} \over {dt}} = {r_1}{x_1}\left( {1 - {{{x_1}} \over {{N_1}}} + {{a{x_1}} \over {{N_1}}}} \right),\quad a > 0 α is the contribution of each unit leader's natural market size saturation to the leader's profit level. We count the natural negative growth rate of the dependant's profit level as r₂. In a market structure where only the dependants exist alone, we can describe the profit level of the dependants as dx2dt=−r2x2 {{d{x_2}} \over {dt}} = - {r_2}{x_2} . After the leader is introduced into the market, the profit level of the dependents will increase [7]. The profit level of the dependant can be further described as (4) dx2dt=r2x2(−1−βx1N1), β>0 {{d{x_2}} \over {dt}} = {r_2}{x_2}\left( { - 1 - {{\beta {x_1}} \over {{N_1}}}} \right),\quad \beta > 0 β is the contribution of the natural market size saturation of the dominant per unit to the profit level of the dependents. The saturation of the ideal market size of the dependents has a hindering effect on their profit levels [8]; therefore, the behaviour of the dependant can ultimately be described as (5) dx2dt=r2x2(−1−βx1N1−x2N2) {{d{x_2}} \over {dt}} = {r_2}{x_2}\left( { - 1 - {{\beta {x_1}} \over {{N_1}}} - {{{x_2}} \over {{N_2}}}} \right) From this, we can infer the partial benefit symbiosis model of financial symbiosis to be the following: (6) dx1dt=r1x1(1−x1N1+ax2N1)dx2dt=r2x2(−1+βx1N1−x2N2)} \left. {\matrix{ \hfill{{{d{x_1}} \over {dt}} = {r_1}{x_1}\left( {1 - {{{x_1}} \over {{N_1}}} + {{a{x_2}} \over {{N_1}}}} \right)} \cr \hfill{{{d{x_2}} \over {dt}} = {r_2}{x_2}\left( { - 1 + {{\beta {x_1}} \over {{N_1}}} - {{{x_2}} \over {{N_2}}}} \right)} \cr } } \right\}

Before the two financial market entities formed a symbiosis, their profit level satisfies the Logistic model: (7) dx1dt=r1x1(1−x1N1)dx2dt=r2x2(1−x2N2)} \left. {\matrix{\hfill {{{d{x_1}} \over {dt}} = {r_1}{x_1}\left( {1 - {{{x_1}} \over {{N_1}}}} \right)} \cr \hfill {{{d{x_2}} \over {dt}} = {r_2}{x_2}\left( {1 - {{{x_2}} \over {{N_2}}}} \right)} \cr } } \right\} x₁(t) and x₂(t) are the profit levels of financial market entities 1 and 2, respectively. When the main body of the financial market 1 and that of the financial market 2 coexist, the formation of the symbiotic relationship can promote the profit level of both parties [9]. Therefore, the description of the profit level of financial market entities 1 and 2 should be changed to the mutually beneficial symbiosis model in financial symbiosis, as follows: (8) dx1(t)dt=r1x1(1−x1N1+δ1x2N2)dx1(t)dt=r1x1(1−x1N1+δ2x2N2)} \left. {\matrix{\hfill {{{d{x_1}(t)} \over {dt}} = {r_1}{x_1}\left( {1 - {{{x_1}} \over {{N_1}}} + {{{\delta _1}{x_2}} \over {{N_2}}}} \right)} \cr \hfill {{{d{x_1}(t)} \over {dt}} = {r_1}{x_1}\left( {1 - {{{x_1}} \over {{N_1}}} + {{{\delta _2}{x_2}} \over {{N_2}}}} \right)} \cr } } \right\} δ₁ > 0, δ₂ > 0, δ₁ is the contribution of the profit level of the financial market entity 1 unit to the profit level of the financial market entity 2. δ₂ represents the contribution of the profit level of the financial market entity 2 to the profit level of the financial market entity 1.

Now we use differential equations to solve the stability of the partial benefit symbiosis model. (9) f(x1,x2)≡dx1dt=r1, r1(1−x1N1+δ1x2N1)=0g(x1,x2)≡dx2dt=r2, r2(−1+δ1x1N1−x2N2)=0} \left. {\matrix{ \hfill {f({x_1},{x_2}) \equiv {{d{x_1}} \over {dt}} = {r_1},\quad {r_1}\left( {1 - {{{x_1}} \over {{N_1}}} + {{{\delta _1}{x_2}} \over {{N_1}}}} \right) = 0}\cr \hfill {g({x_1},{x_2}) \equiv {{d{x_2}} \over {dt}} = {r_2},\quad {r_2}\left( { - 1 + {{{\delta _1}{x_1}} \over {{N_1}}} - {{{x_2}} \over {{N_2}}}} \right) = 0} \cr } } \right\} The equilibrium point P₁(N,0), P2(N1(1−δ1)1−δ1δ2,N3(δ2−1)1−δ1δ2) {P_2}\left( {{{{N_1}(1 - {\delta _1})} \over {1 - {\delta _1}{\delta _2}}},{{{N_3}({\delta _2} - 1)} \over {1 - {\delta _1}{\delta _2}}}} \right) , P₃(0,0) can be obtained by solving the differential equation. We analyse the stability of the above three equilibrium points and find that point P₃ is an unstable point under any conditions. For the balance point P₂ to have practical meaning, it needs to be located in the I quadrant (x₁,x₂ ≥ 0) of the phase plane, and one of the following two conditions must be met: (10) δ1<1, δ2>1 δ1δ2<1 {\delta _1} < 1,\quad {\delta _2} > 1\quad {\delta _1}{\delta _2} < 1 (11) δ1>1, δ2<1 δ1δ2>1 {\delta _1} > 1,\quad {\delta _2} < 1\quad {\delta _1}{\delta _2} > 1 We can judge the stability of the equilibrium point according to the direct method. The first-order Taylor expansion of the differential equation can be performed at the equilibrium point x₁^*, x₂^* to obtain (12) dx1dt=r1(1+δ1x2N2−2x1N1)(x1−x1*)+r1x1δ1(x2−x2*)N2dx2dt=r2x2δ2(x1−x1*)N1+r2(δ2x1N1−1−2x2N2)(x2−x2*)} \left. {\matrix{ \hfill {{{d{x_1}} \over {dt}} = {r_1}\left( {1 + {{{\delta _1}{x_2}} \over {{N_2}}} - {{2{x_1}} \over {{N_1}}}} \right)\left( {{x_1} - {x_1}^*} \right) + {r_1}{x_1}{{{\delta _1}({x_2} - {x_2}^*)} \over {{N_2}}}} \cr \hfill {{{d{x_2}} \over {dt}} = {{{r_2}{x_2}{\delta _2}({x_1} - {x_1}^*)} \over {{N_1}}} + {r_2}\left( {{{{\delta _2}{x_1}} \over {{N_1}}} - 1 - {{2{x_2}} \over {{N_2}}}} \right)\left( {{x_2} - {x_2}^*} \right)}\cr } } \right\} Then the coefficient matrix A can be denoted as A=(r1(1+δ1x2N2−2x1N1)r1x1δ1r1x2δ2N1r2(δ2x2N1−1−2x2N2)) A = \left( {\matrix{ {{r_1}\left( {1 + {{{\delta _1}{x_2}} \over {{N_2}}} - {{2{x_1}} \over {{N_1}}}} \right)} & {{r_1}{x_1}{\delta _1}} \cr {{{{r_1}{x_2}{\delta _2}} \over {{N_1}}}} & {{r_2}\left( {{{{\delta _2}{x_2}} \over {{N_1}}} - 1 - {{2{x_2}} \over {{N_2}}}} \right)} \cr } } \right) . We substitute the equilibrium point P2(N1(1−δ1)1−δ1δ2,N2(δ1−1)1−δ1δ2) {P_2}\left( {{{{N_1}(1 - {\delta _1})} \over {1 - {\delta _1}{\delta _2}}},{{{N_2}({\delta _1} - 1)} \over {1 - {\delta _1}{\delta _2}}}} \right) into the coefficient matrix A. According to the method of determining the stable point of the differential equation, the conditions for obtaining the stable node of the model are: (13) δ1<1 δ2>1 δ1δ2<1 {\delta _1} < 1\quad {\delta _2} > 1\quad {\delta _1}{\delta _2} < 1 Three conditions must be met to achieve a stable symbiosis of partial interest among the financial market entities: (1) δ₁ < 1 means that the dependent person's contribution to the leader's profit is relatively small. (2) δ₂ > 1 means that the dominant player makes a relatively significant contribution to the profits of the dependents. (3) δ₁δ₂ < 1 means that when two financial market entities reach a stable state of symbiosis with partial benefits, δ₁ is required to be small and δ₂ is large [10]. This requires a larger scale of the leader. When the favoured symbiosis reaches a stable state, the symbiosis energy obtained by the dependent is N1(1−δ1)1−δ1δ2,N1(1−δ1)1−δ1δ2>N1 {{{N_1}(1 - {\delta _1})} \over {1 - {\delta _1}{\delta _2}}},{{{N_1}(1 - {\delta _1})} \over {1 - {\delta _1}{\delta _2}}} > {N_1} . The symbiotic energy obtained by the dominant player is N2(δ2−1)1−δ1δ2,N2(δ2−1)1−δ1δ2>N2 {{{N_2}({\delta _2} - 1)} \over {1 - {\delta _1}{\delta _2}}},{{{N_2}({\delta _2} - 1)} \over {1 - {\delta _1}{\delta _2}}} > {N_2} .

Differential equations solve the stability of the reciprocal symbiosis model. (14) f(x1,x2)≡dx1dt=r1x1(1−x1N1+δ1x2N1)=0g(x1,x2)≡dx2dt=r2x2(1−x2N2+δ2x2N1)=0} \left. {\matrix{ {f({x_1},{x_2}) \equiv {{d{x_1}} \over {dt}} = {r_1}{x_1}\left( {1 - {{{x_1}} \over {{N_1}}} + {{{\delta _1}{x_2}} \over {{N_1}}}} \right) = 0} \hfill \cr {g({x_1},{x_2}) \equiv {{d{x_2}} \over {dt}} = {r_2}{x_2}\left( {1 - {{{x_2}} \over {{N_2}}} + {{{\delta _2}{x_2}} \over {{N_1}}}} \right) = 0} \hfill \cr } } \right\} The equilibrium point P(x1,x2)=(N1(1+δ1)1−δ1δ2,N2(1+δ2)1−δ1δ2) P({x_1},{x_2}) = \left( {{N_1}{{(1 + {\delta _1})} \over {1 - {\delta _1}{\delta _2}}},{N_2}{{(1 + {\delta _2})} \over {1 - {\delta _1}{\delta _2}}}} \right) , can be obtained by solving the differential equation. When x₁ > 0, x₂ > 0, N1(1+δ1)1−δ1δ2>0 {N_1}{{(1 + {\delta _1})} \over {1 - {\delta _1}{\delta _2}}} > 0 , N2(1+δ2)1−δ1δ2>0 {N_2}{{(1 + {\delta _2})} \over {1 - {\delta _1}{\delta _2}}} > 0 it means that the two financial market entities reciprocally coexist. Therefore, a condition for the mutual symbiosis of the two financial market entities is δ₁δ₂ < 1.

We carry out the first-order Taylor expansion of the differential equations to get (15) dx1dt=r1(1−2x1N1+δ2x2N2(x1−x1*))+r1x1δ1(x2−x2*)N2dx2dt=(r2x2δ2N1)(x1−x1*)+r2(1−2x2N2+δ2x1N1)(x2−x2*)} \left. {\matrix{ \hfill {{{d{x_1}} \over {dt}} = {r_1}\left( {1 - {{2{x_1}} \over {{N_1}}} + {{{\delta _2}{x_2}} \over {{N_2}}}({x_1} - {x_1}^*)} \right) + {r_1}{x_1}{{{\delta _1}({x_2} - {x_2}^*)} \over {{N_2}}}} \cr \hfill {{{d{x_2}} \over {dt}} = \left( {{{{r_2}{x_2}{\delta _2}} \over {{N_1}}}} \right)({x_1} - {x_1}^*) + {r_2}\left( {1 - {{2{x_2}} \over {{N_2}}} + {{{\delta _2}{x_1}} \over {{N_1}}}} \right)({x_2} - {x_2}^*)} \cr } } \right\} We substitute P(x1x2)=(N1(1−δ1)1−δ1δ2,N2(1+δ2)1−δ1δ2) P({x_1}{x_2}) = \left( {{N_1}{{(1 - {\delta _1})} \over {1 - {\delta _1}{\delta _2}}},{N_2}{{(1 + {\delta _2})} \over {1 - {\delta _1}{\delta _2}}}} \right) into Eq. (15), and the relative system matrix is (16) A=(r1(−1−δ1)1−δ1δ2r1x1δ1(1+δ1)N2(1−δ1δ2)N2r2δ2(1+δ2)N1(1−δ1δ2)r2(−1−δ2)1−δ1δ2) A = \left( {\matrix{ {{{{r_1}( - 1 - {\delta _1})} \over {1 - {\delta _1}{\delta _2}}}} & {{{{r_1}{x_1}{\delta _1}(1 + {\delta _1})} \over {{N_2}(1 - {\delta _1}{\delta _2})}}} \cr {{{{N_2}{r_2}{\delta _2}(1 + {\delta _2})} \over {{N_1}(1 - {\delta _1}{\delta _2})}}} & {{{{r_2}( - 1 - {\delta _2})} \over {1 - {\delta _1}{\delta _2}}}} \cr } } \right) According to the stability theory of differential equations, we find that the condition for a stable point to be a stable node is δ₁δ₂ < 1. Therefore, to achieve a stable symbiosis, the symbiosis relationship in the mutually beneficial symbiosis model is δ₁δ₂ < 1. Considering the symmetry of financial market entities, we can further determine the stable symbiosis condition δ₁ < 1,δ₂ < 1.

The two conditions that need to be met to achieve a stable reciprocal symbiosis relationship can be explained δ₁ < 1, δ₂ < 1. indicates that the two financial market entities in a mutually beneficial symbiosis relationship contribute relatively little to each other's output. When the reciprocal symbiosis reaches a stable state, the symbiosis energy obtained by the financial market entity 1 is N1(1+δ1)1−δ1δ2,N1(1+δ1)1−δ1δ2>N1 {N_1}{{(1 + {\delta _1})} \over {1 - {\delta _1}{\delta _2}}},{N_1}{{(1 + {\delta _1})} \over {1 - {\delta _1}{\delta _2}}} > {N_1} , whereas the symbiotic energy obtained by the financial market entity 2 is N2(1+δ2)1−δ1δ2,N2(1+δ2)1−δ1δ2>N2 {N_2}{{(1 + {\delta _2})} \over {1 - {\delta _1}{\delta _2}}},{N_2}{{(1 + {\delta _2})} \over {1 - {\delta _1}{\delta _2}}} > {N_2} .