1. bookAHEAD OF PRINT
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Otwarty dostęp

Financial Risk Prevention Model of Financial Institutions Based on Linear Partial Differential Equation

Data publikacji: 15 Jul 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 05 Mar 2022
Przyjęty: 04 May 2022
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Introduction

The capital markets of emerging market economies are constantly improving. There have been many “financial anomalies” in the financial market that classical financial theories cannot explain, such as the economic Sunday effect, capital market price surge and sell-off, contrarian investment strategy, seasonal effect, etc. The “financial anomaly” results from the joint efforts of the majority of social investors’ trading behavior. This synergy is again a function of investors’ heterogeneous beliefs [1]. Therefore, studying investors’ heterogeneous views can create a new explanatory path for uncovering “financial anomalies” in capital markets. Since the traditional financial theory is based on rational expectations, financial anomalies cannot be solved by mathematical models. The theoretical framework of rational expectations is even more unable to form a reasonable explanation for the price fluctuations of funds, futures, bonds, and other financial markets other than the stock market [2]. Thus, behavioral finance was born in the 1980s. It draws on psychological concepts such as “overconfidence” and “heterogeneous belief” in psychology. It tries to analyze the phenomenon of finance and finance with behavioral psychology theory. This article will explore the volatility of securities prices in the asset market. The research results of this paper inject new theoretical resources into the capital pricing theory.

Asset prices under the dimension of heterogeneous beliefs

There are many types of financial capital and its derivative products available to investors in the natural capital market. Chinese capital market started late and is immature. Financial investors have different preferences and beliefs [3]. These factors will make the solid and efficient capital market theory unable to explain phenomenal financial problems. It cannot effectively solve the pricing problem of financial products. Therefore, we need to adjust the research premise of traditional economic theory. We make the following assumptions: (1) The capital market consists of assets such as risk-free assets (such as treasury bills) and risky assets (such as stocks). (2) The number of investors with overconfidence beliefs is N. They are less likely to invest in risky assets. Its behavior has a particular impact on capital pricing. (3) Investors complete the investment transaction in three stages: t=1, 2, and 3 when the market transaction cost is 0. The cash flow produced is equal to V˜~N(V¯,σV˜2) \tilde V \sim N\left({\bar V,\,\sigma_{\tilde V}^2} \right) . (4) Based on Chinese unique capital system, we set the number of risky assets at t=1, 2, and 3 to be R¯S¯1 \bar R{\bar S_1} , R¯S¯2 \bar R{\bar S_2} , R¯S¯3 \bar R{\bar S_3} . (5) Capital market signals that investors can obtain: It=V˜+e˜t {I_t} = \tilde V + {\tilde e_t} , t = 1, 2, 3 obeys a normal distribution. e˜t {\tilde e_t} is the noise signal, and the noise signals are independent of each other. So the market information set can be expressed as Φ1i={I˜1} {\Phi_{1i}} = \left\{{{{\tilde I}_1}} \right\} , Φ2i={I˜2} {\Phi_{2i}} = \left\{{{{\tilde I}_2}} \right\} , Φ3i={I˜2I˜3}T {\Phi_{3i}} = {\left\{{{{\tilde I}_2}\,{{\tilde I}_3}} \right\}^T} . (6) Conservative capital market investors are more rational [4]. Their correct valuation of the market signal is σe2 \sigma_e^2 . The overestimation rate is k. So the overestimated market signal is kσe2 k\sigma_e^2 , k ∈ [0, 1]. (7) When the market investor is a risk-averse market type, its function distribution law is U (Wn) = eaWn. Here a is the absolute risk aversion coefficient, and W is the wealth level. Our investor decision problem at time t = 1, 2,3 is given by: maxRStiE0[eaWi+1,i|Φti]s.t.PtSti+FSti=PtRSt1+FSt1,i \matrix{{\mathop {\max}\limits_{R{S_{ti}}} \,{E_0}\left[{- {e^{- a{W_{i + 1,i}}}}\left| {{\Phi_{ti}}} \right.} \right]} \hfill \cr {s.t.\,{P_t}{S_{ti}} + F{S_{ti}} = {P_t}R{S_{t - 1}} + F{S_{t - 1,i}}} \hfill \cr}

Where RSt−1 is the number of investors’ risky assets. The investor's amount of risk-free assets is given by FSti. We make the investor's investment limit a constraint of the formula. In the above equation, the right side of the procedure is the investor's investment amount at time t. We let the risky asset price at time t be Pt, so the scenario at the time t = 2, 3 is as follows:

Scenario 1: The investor decision at time t=3 can be transformed into: maxRS3iE0[eaW4,i|Φ3i]s.t.P3S3i+FS3i=P3RS2+FS2,i \matrix{{\mathop {\max}\limits_{R{S_{3i}}} \,{E_0}\left[{- {e^{- a{W_{4,i}}}}\left| {{\Phi_{3i}}} \right.} \right]} \hfill \cr {s.t.\,{P_3}{S_{3i}} + F{S_{3i}} = {P_3}R{S_2} + F{S_{2,i}}} \hfill \cr}

We are based on the standard normal distribution principle and obtained from the above formula: maxRS3iE0[eaW4,i|Φ3i]=E0[ea(FS3i+V˜RS3i)|Φ3i]=eaFS3iE0[eaV˜RS3i|Φ3i]=eaFS3ieaRS3iE0[(V˜|Φ3i)+12a2RS3i2Var0(V˜|Φ3i)] \matrix{{\mathop {\max}\limits_{R{S_{3i}}} {E_0}\left[{- {e^{- a{W_{4,i}}}}\left| {{\Phi_{3i}}} \right.} \right] = {E_0}\left[{- {e^{- a\left({F{S_{3i}} + \tilde VR{S_{3i}}} \right)}}\left| {{\Phi_{3i}}} \right.} \right]} \hfill \cr {= - {e^{- aF{S_{3i}}}}{E_0}\left[{- {e^{- a\tilde VR{S_{3i}}}}\left| {{\Phi_{3i}}} \right.} \right]} \hfill \cr {= - {e^{- aF{S_{3i}}}}{e^{- aR{S_{3i}}{E_0}\left[{\left({\tilde V\left| {{\Phi_{3i}}} \right.} \right) + {1 \over 2}{a^2}RS_{3i}^2{Var}_0\left({\tilde V\left| {{\Phi_{3i}}} \right.} \right)} \right]}}} \hfill \cr}

Optimizing the above equation shows that E0[eaW4,i|Φ3i]RS3i=0 {{\partial {E_0}\left[{- {e^{- a{W_{4,i}}}}\left| {{\Phi_{3i}}} \right.} \right]} \over {\partial R{S_{3i}}}} = 0 . Therefore, the optimal decision-making solution for investors at this time is: RS3i=E0[V˜|Φ3i]P3aVar0[V˜|Φ3i]=0 R{S_{3i}} = {{{E_0}\left[{\tilde V\left| {{\Phi_{3i}}} \right.} \right] - {P_3}} \over {a{Var}_0\left[{\tilde V\left| {{\Phi_{3i}}} \right.} \right]}} = 0

The above E0[ ] and Var0 [ ] respectively represent the investor's investment expectation and its variance value under the belief of overconfidence [5]. According to the projection theorem, the expected value and its variance can be obtained as: E0[V˜|Φ3i]=V¯+COV(V˜,Φ3i)TCOV0(Φ3i,Φ3i)1{Φ3iE0(Φ3i)}Var0[V˜|Φ3i]=σV2+COV(V˜,Φ3i)1COV0(Φ3i,Φ3i)1(V˜,Φ3i) \matrix{{{E_0}\left[{\tilde V\left| {{\Phi_{3i}}} \right.} \right] = \bar V + COV{{\left({\tilde V,{\Phi_{3i}}} \right)}^T}\,{COV}_0{{\left({{\Phi_{3i}},{\Phi_{3i}}} \right)}^{- 1}}\left\{{{\Phi_{3i}} - {E_0}\left({{\Phi_{3i}}} \right)} \right\}} \hfill \cr {{Var}_0\left[{\tilde V\left| {{\Phi_{3i}}} \right.} \right] = \sigma_V^2 + COV{{\left({\tilde V,{\Phi_{3i}}} \right)}^{- 1}}{COV}_0{{\left({{\Phi_{3i}},{\Phi_{3i}}} \right)}^{- 1}}\left({\tilde V,{\Phi_{3i}}} \right)} \hfill \cr}

Therefore, the expected function value and covariance value of investor investment are expressed as follows: E0[Φ3i]=E0[(I˜2,I˜3)T]=E0[E0(I˜2),E0(I˜3)]T=(V¯,V¯)TCOV0(Φ3i,Φ3i)=(COV0(I˜2,I˜2)COV0(I˜2,I˜3)COV0(I˜3,I˜2)COV0(I˜3,I˜3))=(σV¯2+kσe2,σV¯2σV¯2,σV¯2+kσe2)COV0(V˜,Φ3i)=COV0(V˜,(I˜2,I˜3)T)=[COV0(V˜,I˜2),COV0(V˜,I˜3)]T=(σV¯2,σV¯2)2 \matrix{{{E_0}\left[{{\Phi_{3i}}} \right] = {E_0}\left[{{{\left({{{\tilde I}_2},{{\tilde I}_3}} \right)}^T}} \right] = {E_0}{{\left[{{E_0}\left({{{\tilde I}_2}} \right),\,{E_0}\left({{{\tilde I}_3}} \right)} \right]}^T} = {{\left({\bar V,\,\bar V} \right)}^T}} \hfill \cr {{COV}_0\left({{\Phi_{3i}},{\Phi_{3i}}} \right) = \left({\matrix{{{COV}_0\left({{{\tilde I}_2},\,{{\tilde I}_2}} \right)} \hfill & {{COV}_0\left({{{\tilde I}_2},\,{{\tilde I}_3}} \right)} \hfill \cr {{COV}_0\left({{{\tilde I}_3},\,{{\tilde I}_2}} \right)} \hfill & {{COV}_0\left({{{\tilde I}_3},\,{{\tilde I}_3}} \right)} \hfill \cr}} \right) = \left({\matrix{{\sigma_{\bar V}^2 + k\sigma_e^2,\,\sigma_{\bar V}^2} \hfill \cr {\sigma_{\bar V}^2,\,\sigma_{\bar V}^2 + k\sigma_e^2} \hfill \cr}} \right)} \hfill \cr {{COV}_0\left({\tilde V,{\Phi_{3i}}} \right) = {COV}_0\left({\tilde V,\,{{\left({{{\tilde I}_2},\,{{\tilde I}_3}} \right)}^T}} \right) = {{\left[{{COV}_0\left({\tilde V,\,\,{{\tilde I}_2}} \right),\,\,{COV}_0\left({\tilde V,\,\,{{\tilde I}_3}} \right)} \right]}^T} = {{\left({\sigma_{\bar V}^2,\,\sigma_{\bar V}^2} \right)}^2}} \hfill \cr}

Further results: E0[V˜|Φ3i]=V¯+(σV¯2,σV¯2)(σV¯2+kσe2,σV¯2σV¯2,σV¯2+kσe2)(I˜2V¯I˜3V¯)=V¯+σV¯2kσe2(2σV¯2+kσe2)(I˜2+I˜3V¯)=kσe2V¯+σV¯2(I˜2+I˜3)2σV¯2+kσe2 \matrix{{{E_0}\left[{\tilde V\left| {{\Phi_{3i}}} \right.} \right] = \bar V + \left({\sigma_{\bar V}^2,\sigma_{\bar V}^2} \right)\left({\matrix{{\sigma_{\bar V}^2 + k\sigma_e^2,\,\sigma_{\bar V}^2} \hfill \cr {\sigma_{\bar V}^2,\,\sigma_{\bar V}^2 + k\sigma_e^2} \hfill \cr}} \right)\left({\matrix{{{{\tilde I}_2} - \bar V} \hfill \cr {{{\tilde I}_3} - \bar V} \hfill \cr}} \right) = \bar V +} \hfill \cr {{{\sigma_{\bar V}^2} \over {k\sigma_e^2\left({2\sigma_{\bar V}^2 + k\sigma_e^2} \right)}}\left({{{\tilde I}_2} + {{\tilde I}_3} - \bar V} \right) = {{k\sigma_e^2\bar V + \sigma_{\bar V}^2\left({{{\tilde I}_2} + {{\tilde I}_3}} \right)} \over {2\sigma_{\bar V}^2 + k\sigma_e^2}}} \hfill \cr} Var0[V˜|Φ3i]=σV˜2(σV˜2,σV˜2)(σV˜2+kσe2,σV˜2σV˜2,σV˜2+kσe2)1(σV˜2σV˜2)=σV˜2σV˜22kσe2kσe2(2σV˜2+kσe2)=kσV˜2+σe22σV˜2+kσe2 \matrix{{{Var}_0\left[{\tilde V\left| {{\Phi_{3i}}} \right.} \right] = \sigma_{\tilde V}^2 - \left({\sigma_{\tilde V}^2,\sigma_{\tilde V}^2} \right){{\left({\matrix{{\sigma_{\tilde V}^2 + k\sigma_e^2,\,\sigma_{\tilde V}^2} \hfill \cr {\sigma_{\tilde V}^2,\,\sigma_{\tilde V}^2 + k\sigma_e^2} \hfill \cr}} \right)}^{- 1}}\left({\matrix{{\sigma_{\tilde V}^2} \hfill \cr {\sigma_{\tilde V}^2} \hfill \cr}} \right)} \hfill \cr {= \sigma_{\tilde V}^2 - {{\sigma_{\tilde V}^22k\sigma_e^2} \over {k\sigma_e^2\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)}} = {{k\sigma_{\tilde V}^2 + \sigma_e^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr}

From the above function values of E0[ ] and Var0[ ], the first-order optimal condition can be obtained as: RS3i=σV˜2(I˜2+I˜3)+kσe2V˜2σV˜2+kσe2P3aσV˜2kσe2kσe2+2σV˜2 R{S_{3i}} = {{{{\sigma_{\tilde V}^2\left({{{\tilde I}_2} + {{\tilde I}_3}} \right) + k\sigma_e^2\tilde V} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} - {P_3}} \over {a{{\sigma_{\tilde V}^2k\sigma_e^2} \over {k\sigma_e^2 + 2\sigma_{\tilde V}^2}}}}

According to the principle of equilibrium of supply and demand in the capital market: RS3i=R¯S¯3 R{S_{3i}} = \bar R{\bar S_3} , i=1NRS3i=N=R¯S¯3 \sum\limits_{i = 1}^N {R{S_{3i}} = N = \bar R{{\bar S}_3}} . So the asset pricing at t=3 is: P3=σV˜2(I˜2+I˜3)+kσe2V˜2σV˜2+kσe2akσV˜2σe2R¯S¯32σV˜2+kσe2=kσe2V˜akσV˜2R¯S¯32σV˜2+kσe2+σV˜22σV˜2+kσe2I˜2+σV˜22σV˜2+kσe2I˜3 \eqalign{& {P_3} = {{\sigma_{\tilde V}^2\left({{{\tilde I}_2} + {{\tilde I}_3}} \right) + k\sigma_e^2\tilde V} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} - {{ak\sigma_{\tilde V}^2\sigma_e^2\bar R{{\bar S}_3}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} \cr & = {{k\sigma_e^2\tilde V - ak\sigma_{\tilde V}^2\bar R{{\bar S}_3}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} + {{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}{{\tilde I}_2} + {{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}{{\tilde I}_3} \cr}

Scenario 2: The investor's decision at time t=2 can be transformed into: maxRS2iE0[eaW3,i|Φ2i]s.t.P2S2i+FS2i=P2RS1i+FS1i \matrix{{\mathop {\max}\limits_{R{S_{2i}}} {E_0}\left[{- {e^{- a{W_{3,i}}}}\left| {{\Phi_{2i}}} \right.} \right]} \hfill \cr {s.t.\,{P_2}{S_{2i}} + F{S_{2i}} = {P_2}R{S_{1i}} + F{S_{1i}}} \hfill \cr}

We set the expected value of the investor's investment at this time to be P3. So the investor's optimal investment decision function at time t=2 is as follows: RS2i=E0[P3|Φ2i]P2aVar0[P3|Φ2i]=0 R{S_{2i}} = {{{E_0}\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right] - {P_2}} \over {a{Var}_0\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right]}} = 0

According to the principle of investment equilibrium, the following functions can be obtained: RS2i=E0[P3|Φ2i]P2aVar0[P3|Φ2i]=R¯S¯2 R{S_{2i}} = {{{E_0}\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right] - {P_2}} \over {a{Var}_0\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right]}} = \bar R{\bar S_2} . According to the capital pricing market, equilibrium conditions can be obtained: P2=E0[P3|Φ2i]aVar0[P3|Φ2i]R¯S¯2 {P_2} = {E_0}\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right] - a{Var}_0\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right]\bar R{\bar S_2} .

Combined with the projection theorem, we get: E0[P3|Φ2i]=E0[P3]+COV(P3,Φ2i)COV0(Φ2i,Φ2i)1{Φ2iE0(Φ2i)}Var0[P3|Φ2i]=Var0[P3]COV(P3,Φ2i)COV0(Φ2i,Φ2i)1(P3,Φ2i)T \matrix{{{E_0}\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right] = {E_0}\left[{{P_3}} \right] + COV\left({{P_3},{\Phi_{2i}}} \right)\,{COV}_0{{\left({{\Phi_{2i}},{\Phi_{2i}}} \right)}^{- 1}}\left\{{{\Phi_{2i}} - {E_0}\left({{\Phi_{2i}}} \right)} \right\}} \hfill \cr {{Var}_0\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right] = {Var}_0\left[{{P_3}} \right] - COV\left({{P_3},{\Phi_{2i}}} \right){COV}_0{{\left({{\Phi_{2i}},{\Phi_{2i}}} \right)}^{- 1}}{{\left({{P_3},{\Phi_{2i}}} \right)}^T}} \hfill \cr}

The investment expectations of an investor with overconfidence heterogeneous beliefs at a price P3 are as follows: E0(P3)=kσe2V˜akσV˜2σe2R¯S¯32σV˜2+kσe2+σV˜22σV˜2+kσe2E0(I˜2)+σV˜22σV˜2+kσe2E0(I˜3)=(2σV˜2+kσe2)V˜akσV˜2σe2R¯S¯32σV˜2+kσe2Var0(P3)=Var0[kσe2V˜akσV˜2σe2R¯S¯32σV˜2+kσe2+σV˜22σV˜2+kσe2I˜2+σV˜22σV˜2+kσe2I˜3]=2[σV˜22σV˜2+kσe2](σV˜2+kσe2)+2[σV˜22σV˜2+kσe2]σV˜2=σV˜22σV˜2+kσe2 \matrix{{{E_0}\left({{P_3}} \right) = {{k\sigma_e^2\tilde V - ak\sigma_{\tilde V}^2\sigma_e^2\bar R{{\bar S}_3}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} + {{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}{E_0}\left({{{\tilde I}_2}} \right) +} \hfill \cr {{{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}{E_0}\left({{{\tilde I}_3}} \right) = {{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)\tilde V - ak\sigma_{\tilde V}^2\sigma_e^2\bar R{{\bar S}_3}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr {{Var}_0\left({{P_3}} \right) = {Var}_0\left[{{{k\sigma_e^2\tilde V - ak\sigma_{\tilde V}^2\sigma_e^2\bar R{{\bar S}_3}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} + {{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}{{\tilde I}_2} +} \right.} \hfill \cr {\left. {{{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}{{\tilde I}_3}} \right] = 2\left[{{{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \right]\left({\sigma_{\tilde V}^2 + k\sigma_e^2} \right) + 2\left[{{{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \right]\sigma_{\tilde V}^2 = {{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr}

The above formula can be converted to: E0[P3|Φ2i]=(2σV˜2+kσe2)V¯akσV˜2σe2R¯S¯32σV˜2+kσe2+σV˜22σV˜2+kσe2σV˜2+kσe2+σV˜22σV˜2+kσe2(I˜2iV¯)=(2σV˜2+kσe2)V¯akσV˜2σe2R¯S¯32σV˜2+kσe2σV˜22σV˜2+kσe2V¯+σV˜2σV˜2+kσe2I˜2i \matrix{{{E_0}\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right] = {{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)\bar V - ak\sigma_{\tilde V}^2\sigma_e^2\bar R{{\bar S}_3}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} + {{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr {{{\sigma_{\tilde V}^2 + k\sigma_e^2 + \sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}\left({{{\tilde I}_{2i}} - \bar V} \right) = {{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)\bar V - ak\sigma_{\tilde V}^2\sigma_e^2\bar R{{\bar S}_3}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} - {{\sigma_{\tilde V}^2} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr {\bar V + {{\sigma_{\tilde V}^2} \over {\sigma_{\tilde V}^2 + k\sigma_e^2}}{{\tilde I}_{2i}}} \hfill \cr} Var0[P3|Φ2i]=2σV˜42σV˜2+kσe2[σV˜2(σV˜2+kσe2+σV˜2)2σV˜2+kσe2]21σV˜2+kσe22σV˜42σV˜2+kσe2σV˜4σV˜2+kσe2 \matrix{{{Var}_0\left[{{P_3}\left| {{\Phi_{2i}}} \right.} \right] = {{2\sigma_{\tilde V}^4} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} - {{\left[{{{\sigma_{\tilde V}^2\left({\sigma_{\tilde V}^2 + k\sigma_e^2 + \sigma_{\tilde V}^2} \right)} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \right]}^2}{1 \over {\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr {{{2\sigma_{\tilde V}^4} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}} - {{\sigma_{\tilde V}^4} \over {\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr}

According to formula (10) and the functional equations (6)~(9), the asset pricing model at time t=2 is: P2=(2σV˜2+kσe2)V¯akσV˜2σe2R¯S¯32σV˜4aR¯S¯22σV˜2+kσe2σV˜2(V¯aσV˜2R¯S¯3)σV˜2+kσe2+σV˜2σV˜2+kσe2I˜2 \matrix{{{P_2} = {{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)\bar V - ak\sigma_{\tilde V}^2\sigma_e^2\bar R{{\bar S}_3} - 2\sigma_{\tilde V}^4a\bar R{{\bar S}_2}} \over {2\sigma_{\tilde V}^2 + k\sigma_e^2}}} \hfill \cr {{{\sigma_{\tilde V}^2\left({\bar V - a\sigma_{\tilde V}^2\bar R{{\bar S}_3}} \right)} \over {\sigma_{\tilde V}^2 + k\sigma_e^2}} + {{\sigma_{\tilde V}^2} \over {\sigma_{\tilde V}^2 + k\sigma_e^2}}{{\tilde I}_2}} \hfill \cr}

Differential characteristics and price volatility of capital market prices under heterogeneous beliefs
Differential Characteristics of Equilibrium Prices in Capital Markets under Heterogeneous Beliefs

According to the second part of the investor's investment decision and its equilibrium price model, it can be known that the investor's decision at time t=3 can be converted into formula (2). The asset pricing model is formula (6). And the investor's decision at time t=2 can be reversed into formula (7). The asset pricing model is formula (11). This article aims to obtain the investment characteristics of investors with different beliefs about the equilibrium price of assets. We consider the following differential equation: max{κ1(θD),κ2(θD)}ϕ+12σ2ϕλϕ+D=0 \max \left\{{{\kappa_1}\left({\theta - D} \right),\,{\kappa_2}\left({\theta - D} \right)} \right\}{\phi^{'}} + {1 \over 2}{\sigma^2}{\phi^{''}} - \lambda \phi + D = 0

Since this differential equation under Dθ has a unique solution, the above function can be transformed into: κ1(θD)ϕ+12σ2ϕλϕ+D=0 {\kappa_1}\left({\theta - D} \right){\phi^{'}} + {1 \over 2}{\sigma^2}{\phi^{''}} - \lambda \phi + D = 0

Suppose ψ(D)=ϕ(D)Dλ+κ1+θκ1λ(λ+κ1) \psi \left(D \right) = \phi \left(D \right) - {D \over {\lambda + {\kappa_1}}} + {{\theta {\kappa_1}} \over {\lambda \left({\lambda + {\kappa_1}} \right)}} , then ψ (D) satisfies the following conditions: κ1(θD)ψ+12σ2ψλψ+D=0 {\kappa_1}\left({\theta - D} \right){\psi^{'}} + {1 \over 2}{\sigma^2}{\psi^{''}} - \lambda \psi + D = 0

Let σ^=σ/2κ1 \hat \sigma = \sigma /\sqrt {2{\kappa_1}} , ψ (D) = ew24ψ(ω), ω=(Dθ)/σ^ \omega = \left({D - \theta} \right)/\hat \sigma , v1 = λ / κ1 again. The above function can then be converted into a Weber differential equation: (ω2+12v1ω24)ψ(ω)=0 \left({\partial_\omega^2 + {1 \over 2} - {v_1} - {{{\omega^2}} \over 4}} \right)\psi \left(\omega \right) = 0

According to the formal definition of the general solution of Weber's differential equation, we can get: ψ(D)=A1Fv1(θDσ/2κ1)+B1Fv1(θDσ/2κ1) \psi \left(D \right) = {A_1}{F_{- {v_1}}}\left({{{\theta - D} \over {\sigma /\sqrt {2{\kappa_1}}}}} \right) + {B_1}{F_{- {v_1}}}\left({{{\theta - D} \over {\sigma /\sqrt {2{\kappa_1}}}}} \right)

Here, F=(1κ1Γ(v1+22)Γ(v2+12)+1κ2Γ(v1+12)Γ(v2+22))1 F = {\left({{1 \over {\sqrt {{\kappa_1}} \Gamma \left({{{{v_1} + 2} \over 2}} \right)\Gamma \left({{{{v_2} + 1} \over 2}} \right)}} + {1 \over {\sqrt {{\kappa_2}\Gamma} \left({{{{v_1} + 1} \over 2}} \right)\Gamma \left({{{{v_2} + 2} \over 2}} \right)}}} \right)^{- 1}}

The positive solution of the function of formula (7) in this case Dθ is as follows: ϕ(D)=C1Fv1(θDσ/2κ1)+C1Fv1(Dθσ/2κ1)+Dλ+κ1+θκ1λ(λ+κ1) \phi \left(D \right) = {C_1}{F_{- {v_1}}}\left({{{\theta - D} \over {\sigma /\sqrt {2{\kappa_1}}}}} \right) + {C_1}{F_{- {v_1}}}\left({{{D - \theta} \over {\sigma /\sqrt {2{\kappa_1}}}}} \right) + {D \over {\lambda + {\kappa_1}}} + {{\theta {\kappa_1}} \over {\lambda \left({\lambda + {\kappa_1}} \right)}}

Correspondingly, the positive solution of the function of formula (7) in the case of D > θ is: ϕ(D)=C2Fv2(θDσ/2κ1)+C2Fv2(Dθσ/2κ2)+Dλ+κ2+θκ2λ(λ+κ2)C1=2v22σ(κ1κ2)πΓ(v1+2)λ(λ+κ1)(λ+κ2)FC2=2v12σ(κ1κ2)πΓ(v1+12)λ(λ+κ1)(λ+κ2)F \matrix{{\phi \left(D \right) = {C_2}{F_{- {v_2}}}\left({{{\theta - D} \over {\sigma /\sqrt {2{\kappa_1}}}}} \right) + {C_2}{F_{- {v_2}}}\left({{{D - \theta} \over {\sigma /\sqrt {2{\kappa_2}}}}} \right) + {D \over {\lambda + {\kappa_2}}} + {{\theta {\kappa_2}} \over {\lambda \left({\lambda + {\kappa_2}} \right)}}} \hfill \cr {{C_1} = {{{2^{{{{v_2}} \over 2}\sigma \left({{\kappa_1} - {\kappa_2}} \right)}}} \over {\sqrt \pi \Gamma \left({{{{v_1} +} \over 2}} \right)\lambda \left({\lambda + {\kappa_1}} \right)\left({\lambda + {\kappa_2}} \right)}}F} \hfill \cr {{C_2} = {{{2^{{{{v_1}} \over 2}\sigma \left({{\kappa_1} - {\kappa_2}} \right)}}} \over {\sqrt \pi \Gamma \left({{{{v_1} + 1} \over 2}} \right)\lambda \left({\lambda + {\kappa_1}} \right)\left({\lambda + {\kappa_2}} \right)}}F} \hfill \cr}

Combining the above derivation process, the solution of equation (12) is obtained: ϕ(D)={C1Fv1(θDσ/2κ1)+Dλ+κ1+θκ1λ(λ+κ1),DθC2Fv2(θDσ/2κ1)+Dλ+κ2+θκ2λ(λ+κ2),Dθ \phi \left(D \right) = \left\{{\matrix{{{C_1}{F_{- {v_1}}}\left({{{\theta - D} \over {\sigma /\sqrt {2{\kappa_1}}}}} \right) + {D \over {\lambda + {\kappa_1}}} + {{\theta {\kappa_1}} \over {\lambda \left({\lambda + {\kappa_1}} \right)}},\,D \le \theta} \hfill \cr {{C_2}{F_{- {v_2}}}\left({{{\theta - D} \over {\sigma /\sqrt {2{\kappa_1}}}}} \right) + {D \over {\lambda + {\kappa_2}}} + {{\theta {\kappa_2}} \over {\lambda \left({\lambda + {\kappa_2}} \right)}},\,D \ge \theta} \hfill \cr}} \right.

According to the above function solution, it can be known that ϕ is an increasing function of D. When D → ∞, then there is ϕ1=Dλ+κ1+θκ1λ(λ+κ1)+0+ {\phi_1} = {D \over {\lambda + {\kappa_1}}} + {{\theta {\kappa_1}} \over {\lambda \left({\lambda + {\kappa_1}} \right)}} + {0^ +} . Correspondingly, D → ∞ has ϕ2=Dλ+κ2+θκ2λ(λ+κ2)+0+ {\phi_2} = {D \over {\lambda + {\kappa_2}}} + {{\theta {\kappa_2}} \over {\lambda \left({\lambda + {\kappa_2}} \right)}} + {0^ +} . According to the inequality principle, 1λ+κ1<ϕ(D)<1λ+κ2 {1 \over {\lambda + {\kappa_1}}} < \phi {\,^{'}}\left(D \right) < {1 \over {\lambda + {\kappa_2}}} . Therefore, ϕ can be obtained as the equilibrium market price of the asset. The equilibrium price is strictly increasing and is a convex function.

Neutral belief investors who neither overestimate nor underestimate their decision-making ability can pay capital prices that are not frothy [6]. They are the equilibrium prices of the asset market. Then given any D0θ, we have: τθ = inf {t > 0: D(t) = θ}, and τθ > 0, so in D0 < θ, we have: 0<t<τθmax{κ1(θD(t)),κ2(θD(t)))}=κ1(θD(t)) \forall 0 < t < {\tau_\theta}\exists \max \left\{{\left. {{\kappa_1}\left({\theta - D\left(t \right)} \right),\,{\kappa_2}\left({\theta - D\left(t \right)} \right)} \right)} \right\} = {\kappa_1}\left({\theta - D\left(t \right)} \right)

Arranged: EQ1{eλτθϕ(D(τθ))}=ϕ(D0)+EQ10τθeλt[κ1(θD)ϕ+12σ2ϕλϕ]dt=ϕ(D0)+EQ10τθeλt[D(t)]dt \matrix{{{E^{{Q_1}}}\left\{{{e^{- \lambda {\tau_\theta}}}\phi \left({D\left({{\tau_\theta}} \right)} \right)} \right\} = \phi \left({{D_0}} \right) + {E^{{Q_1}}}\int_0^{{\tau_\theta}} {{e^{- \lambda t}}\left[{{\kappa_1}\left({\theta - D} \right){\phi^{'}} +} \right.}} \hfill \cr {\left. {{1 \over 2}{\sigma^2}{\phi^{''}} - \lambda \phi} \right]dt = \phi \left({{D_0}} \right) + {E^{{Q_1}}}\int_0^{{\tau_\theta}} {{e^{- \lambda t}}} \left[{- D\left(t \right)} \right]dt} \hfill \cr}

Simultaneous formulas (12) and (14), we have: ϕ(D)=EQ1{0τθeλsD(s)ds+eλtθΦ(D(τθ))|D(0)=D} \phi \left(D \right) = {E^{{Q_1}}}\left\{{\int_0^{{\tau_\theta}} {{e^{- \lambda s}}D\left(s \right)ds + {e^{- \lambda {t_\theta}}}\Phi \left({D\left({{\tau_\theta}} \right)} \right)\left| {D\left(0 \right) = D} \right.}} \right\}

Because of 0 < t < τθ, we get: max{{κ1(θD(t)),κ2(θD(t))}κ2(θD(t)) \max \left\{{\left\{{{\kappa_1}\left({\theta - D\left(t \right)} \right),\,{\kappa_2}\left({\theta - D\left(t \right)} \right)} \right\} \ge {\kappa_2}\left({\theta - D\left(t \right)} \right)} \right.

So for any τ > 0 the following inequality holds: EQ2{eλτϕ(D(τ))}=ϕ(D0)+EQ20τeλt[κ2(θD)ϕ+12σ2ϕλϕ]dt<ϕ(D0)+EQ20τeλt[D(t)]dt \matrix{{{E^{{Q_2}}}\left\{{{e^{- \lambda \tau}}\phi \left({D\left(\tau \right)} \right)} \right\} = \phi \left({{D_0}} \right) + {E^{{Q_2}}}\int_0^\tau {{e^{- \lambda t}}\left[{{\kappa_2}\left({\theta - D} \right){\phi^{'}} +} \right.}} \hfill \cr {\left. {{1 \over 2}{\sigma^2}{\phi^{''}} - \lambda \phi} \right]dt < \phi \left({{D_0}} \right) + {E^{{Q_2}}}\int_0^\tau {{e^{- \lambda t}}} \left[{- D\left(t \right)} \right]dt} \hfill \cr}

Which in turn converts to: ϕ(D)=EQ2{0τeλsD(s)ds+eλtΦ(D(τθ))|D(0)=D} \phi \left(D \right) = {E^{{Q_2}}}\left\{{\int_0^\tau {{e^{- \lambda s}}D\left(s \right)ds + {e^{- \lambda t}}\Phi \left({D\left({{\tau_\theta}} \right)} \right)\left| {D\left(0 \right) = D} \right.}} \right\}

From model (16) and model (17), it can be obtained that the assets in the case Dθ are held by optimistic investors. Investors with overconfident beliefs are more likely to impact asset pricing than overly pessimistic investors. It is easier to have a say in the capital market.

Capital market price volatility under heterogeneous beliefs

Overconfident investors easily sway capital market pricing. They have a more significant say in capital market pricing. As a result, asset prices will be bid up when most investors in the market have overconfident beliefs [7]. This exceeds its actual equilibrium price and creates a market bubble. It causes volatility in capital market prices.

Continuing the asset price equation above, we build on Model (6) and Model (11). We use the variance value to describe the volatility of capital market prices: Var(P3)=σV˜4(2σV˜2+kσe2)2Var(I˜2+I˜3)=2σV˜4(2σV˜2+σe2)(2σV˜2+kσe2)2Var(P2)=σV˜4(2σV˜2+kσe2)2Var(I˜2)=σV˜4(2σV˜2+σe2)(2σV˜2+kσe2)2 \matrix{{Var\left({{P_3}} \right) = {{\sigma_{\tilde V}^4} \over {{{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)}^2}}}Var\left({{{\tilde I}_2} + {{\tilde I}_3}} \right) = {{2\sigma_{\tilde V}^4\left({2\sigma_{\tilde V}^2 + \sigma_e^2} \right)} \over {{{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)}^2}}}} \hfill \cr {Var\left({{P_2}} \right) = {{\sigma_{\tilde V}^4} \over {{{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)}^2}}}Var\left({{{\tilde I}_2}} \right) = {{\sigma_{\tilde V}^4\left({2\sigma_{\tilde V}^2 + \sigma_e^2} \right)} \over {{{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)}^2}}}} \hfill \cr}

Taking the partial derivative of the above function, we get: Var(P3)k=2σV˜4(2σV˜2+σe2)(2σV˜2+kσe2)3σe2<0 {{\partial Var\left({{P_3}} \right)} \over {\partial k}} = {{2\sigma_{\tilde V}^4\left({2\sigma_{\tilde V}^2 + \sigma_e^2} \right)} \over {{{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)}^3}}}\sigma_e^2 < 0 Var(P2)k=σV˜4(σV˜2+σe2)(2σV˜2+kσe2)3σe2<0 {{\partial Var\left({{P_2}} \right)} \over {\partial k}} = {{\sigma_{\tilde V}^4\left({\sigma_{\tilde V}^2 + \sigma_e^2} \right)} \over {{{\left({2\sigma_{\tilde V}^2 + k\sigma_e^2} \right)}^3}}}\sigma_e^2 < 0

From formula (18) and formula (19), it can be judged that the capital market price volatility is a decreasing function of investors’ overconfidence beliefs [8]. This means that the more optimistic investors are about capital markets, the more control they may have over capital pricing. The greater the volatility of asset prices, the more likely it will bring the risk of price fluctuations in the capital market.

Discussion

Financial anomalies such as capital market price surges and sell-offs can be explained by investors’ heterogeneous beliefs and partial differential characteristics. Generally speaking, investors’ confidence in the capital market will bring a signal to the market that returns are expected [9]. This has contributed to the asset price being in a rising channel. As a result, it promotes the rise of the stock index and brings about the scene of individual stocks rising together. At this time, the investor is in a profitable state as a whole, and the prosperous state of the account will enhance the investor's confidence in investment decision-making. They are more willing to believe that they are “very insightful” and have more accurate and effective market information. Based on this belief, investors will further increase their investment in assets, eventually driving the capital market to rise sharply [10].

On the other hand, too pessimistic beliefs when the market is in a general downtrend channel will cause the market to sell. The “cut the meat out” strategy will become the norm. To this end, it is necessary to make reasonable use of and control investors’ heterogeneous beliefs and prevent the volatility risks brought by heterogeneous beliefs to the capital market to the greatest extent. For this, the following work needs to be done:

Standardize the capital market information disclosure system to reduce the mistransmission of investors’ heterogeneous beliefs to the capital market. At present, many listed companies are more inclined to disclose positive information rather than damaging information when disclosing information [11]. This is a detrimental approach to investors. This would be a big mistake for investors with overconfident beliefs. So we need to govern this behavior of listed companies. On the one hand, the securities regulatory authorities should start with the three primary regulatory goals of ensuring market fairness and transparency, protecting investors’ rights and interests, and reducing capital market risks. It strengthens the crackdown on the misreporting and omission of important information of listed companies to form a deterrent to securities crimes. On the other hand, maintain the collection and arrangement of listed companies’ market transactions and financial data. Strive to build a capital market information database and a dynamic supervision platform for the financial data of listed companies. The regulatory authorities implement classified management of listed companies’ information disclosure and focus on supervision of problem enterprises.

Strengthen the supervision of information disclosure and increase the “cost” of violations. Build a long-term mechanism to investigate and deal with false positives and omissions of important information. At the same time, the regulatory authorities intensified the punishment and gradually formed a regulatory deterrent and effective restraint on the information disclosure violations of listed companies [12]. At the same time, they need to improve the protection of the rights and interests of investors, continuously expand the scope of securities civil compensation litigation, and strengthen the free will of investors to exercise their right to sue.

Strengthen the intervention and supervision of inefficient markets. It can be seen from the above research that heterogeneous beliefs bring price fluctuations in the capital market. Therefore, as the maintainer of the market order, the government should take measures such as reducing speculation, limiting the daily price of the capital market, and setting up market transaction circuit breakers. In this way, the extreme, irrational behavior of the public can be reduced to reduce the risk of financial panic. Or the government builds the investor's investment utility framework by establishing a market economic climate index and an investor overconfidence index. This guides investors to recognize the cognitive biases and investment risks caused by heterogeneous beliefs. Ultimately, this leads investors to make sound investments.

Conclusion

In behavioral capital investment, heterogeneous belief refers to the psychological orientation that parties, including operators and investors, have different expectations for future capital returns, such as investor pessimism, overconfidence, and so on. We find that heterogeneous beliefs are an essential factor affecting capital market prices through the above model construction. Especially overconfidence beliefs. It has a substantial urging effect on the formation of capital market bubbles. Analysis of capital market price volatility from the capital pricing model under heterogeneous beliefs is a decreasing function of investors’ overconfidence beliefs. This shows that the more optimistic investors are about the capital market, the stronger their control over capital pricing and the greater the volatility of asset prices. Investor behavior is more likely to interfere with the price trend of the capital market and bring about the risk of price fluctuations in the capital market.

Wang, J., Xia, B., & Qiao, H. Time-varying impact of housing price fluctuations on banking financial risk. Managerial and Decision Economics., 2022; 43(2): 457–467 WangJ. XiaB. QiaoH. Time-varying impact of housing price fluctuations on banking financial risk Managerial and Decision Economics 2022 43 2 457 467 10.1002/mde.3393 Search in Google Scholar

Binder, A., Jadhav, O., & Mehrmann, V. Model order reduction for the simulation of parametric interest rate models in financial risk analysis. Journal of Mathematics in Industry., 2021; 11(1): 1–34 BinderA. JadhavO. MehrmannV. Model order reduction for the simulation of parametric interest rate models in financial risk analysis Journal of Mathematics in Industry 2021 11 1 1 34 10.1186/s13362-021-00105-8 Search in Google Scholar

Wang, C., & Wei, Y. Simulation of financial risk spillover effect based on ARMA-GARCH and fuzzy calculation model. Journal of Intelligent & Fuzzy Systems., 2021; 40(4): 6555–6566 WangC. WeiY. Simulation of financial risk spillover effect based on ARMA-GARCH and fuzzy calculation model Journal of Intelligent & Fuzzy Systems 2021 40 4 6555 6566 10.3233/JIFS-189493 Search in Google Scholar

Vieira, M., Paulo, H., Pinto-Varela, T., & Barbosa-Póvoa, A. P. Assessment of financial risk in the design and scheduling of multipurpose plants under demand uncertainty. International Journal of Production Research., 2021; 59(20): 6125–6145 VieiraM. PauloH. Pinto-VarelaT. Barbosa-PóvoaA. P. Assessment of financial risk in the design and scheduling of multipurpose plants under demand uncertainty International Journal of Production Research 2021 59 20 6125 6145 10.1080/00207543.2020.1804638 Search in Google Scholar

Payzan-LeNestour, E., & Woodford, M. Outlier blindness: A neurobiological foundation for neglect of financial risk. Journal of Financial Economics., 2022; 143(3): 1316–1343 Payzan-LeNestourE. WoodfordM. Outlier blindness: A neurobiological foundation for neglect of financial risk Journal of Financial Economics 2022 143 3 1316 1343 10.1016/j.jfineco.2021.06.019 Search in Google Scholar

Xie, T., Liu, R., & Wei, Z. Improvement of the Fast Clustering Algorithm Improved by-Means in the Big Data. Applied Mathematics and Nonlinear Sciences., 2020; 5(1): 1–10 XieT. LiuR. WeiZ. Improvement of the Fast Clustering Algorithm Improved by-Means in the Big Data Applied Mathematics and Nonlinear Sciences 2020 5 1 1 10 10.2478/amns.2020.1.00001 Search in Google Scholar

Gençoğlu, M. T., & Agarwal, P. Use of quantum differential equations in sonic processes. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 21–28 GençoğluM. T. AgarwalP. Use of quantum differential equations in sonic processes Applied Mathematics and Nonlinear Sciences 2021 6 1 21 28 10.2478/amns.2020.2.00003 Search in Google Scholar

Zhang, H., Shi, Y., & Tong, J. Online supply chain financial risk assessment based on improved random forest. Journal of Data, Information and Management., 2021; 3(1): 41–48 ZhangH. ShiY. TongJ. Online supply chain financial risk assessment based on improved random forest Journal of Data, Information and Management 2021 3 1 41 48 10.1007/s42488-021-00042-6 Search in Google Scholar

Ma, J., Lu, S., & Wang, S. Research on GSEC's Restructuring Financial Risk. ratio., 2021; 2(4): 6–8 MaJ. LuS. WangS. Research on GSEC's Restructuring Financial Risk. ratio 2021 2 4 6 8 Search in Google Scholar

Mironova, M. D., & Ibragimov, L. G. Financial risk management of companies operating in the oil sector in the context of globalization based on the COVID-19 economic impact. International Journal of Engineering Research and Technology., 2021; 13(12): 4500–4504 MironovaM. D. IbragimovL. G. Financial risk management of companies operating in the oil sector in the context of globalization based on the COVID-19 economic impact International Journal of Engineering Research and Technology 2021 13 12 4500 4504 Search in Google Scholar

Mohit, A., & Amit, U. A Modified Iterative Method for Solving Nonlinear Functional Equation. Applied Mathematics and Nonlinear Sciences., 2021; 6(2): 347–360 MohitA. AmitU. A Modified Iterative Method for Solving Nonlinear Functional Equation Applied Mathematics and Nonlinear Sciences 2021 6 2 347 360 10.2478/amns.2020.2.00055 Search in Google Scholar

Huang, X., & Guo, F. A kernel fuzzy twin SVM model for early warning systems of extreme financial risks. International Journal of Finance & Economics., 2021; 26(1): 1459–1468 HuangX. GuoF. A kernel fuzzy twin SVM model for early warning systems of extreme financial risks International Journal of Finance & Economics 2021 26 1 1459 1468 10.1002/ijfe.1858 Search in Google Scholar

Polecane artykuły z Trend MD

Zaplanuj zdalną konferencję ze Sciendo