American Institute of Mathematical Sciences

Advanced
October  2022, 17(5): 803-825. doi: 10.3934/nhm.2022027

Effect of time delay on flocking dynamics

Hyeong-Ohk Bae 1, , Seung Yeon Cho 2, , Jane Yoo 3,, and  Seok-Bae Yun 4,

1. 

Department of Financial Engineering, Ajou University, Republic of Korea
2. 

Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Republic of Korea
3. 

Department of Financial Engineering, Ajou University, Republic of Korea
4. 

Department of Mathematics, Sungkyunkwan University, Republic of Korea

*Corresponding author: Jane Yoo

Received  February 2022 Published  October 2022 Early access  June 2022

Fund Project: Bae was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2021R1A2C1093383), Cho by PRIN Project 2017 (No. 2017KKJP4X entitled "Innovative numerical methods for evolutionary partial differential equations and applications") funded by Italian Ministry of Instruction, University and Research (MIUR), Yoo by Ajou University Research Fund, and Yun by Samsung Science and Technology Foundation under Project Number SSTF-BA1801

We propose a time-delayed Cucker-Smale type model(CS model), which can be applied to modeling (1) collective dynamics of self-propelling agents and (2) the dynamical system of stock return volatility in a financial market. For both models, we assume that it takes a certain amount of time to collect/process information about the current position/return configuration until velocity/volatility adjustment is made. We provide a sufficient condition under which flocking phenomena occur. We also identify the initial configuration for a two-agent case, in which collective behaviors are accelerated by changes in the delay parameter. Numerical illustrations and financial simulations are carried out to verify the validity of the model.

Keywords: Cucker–Smale model, flocking, volatility, time delay.
Mathematics Subject Classification: Primary: 82C22, 91B70, 91G80.
Citation: Hyeong-Ohk Bae, Seung Yeon Cho, Jane Yoo, Seok-Bae Yun. Effect of time delay on flocking dynamics. Networks and Heterogeneous Media, 2022, 17 (5) : 803-825. doi: 10.3934/nhm.2022027
References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137. 

[2]

S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.

[3]

G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Mod. Meth. Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[4]

G. Albi and L. Pareschi, Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics, Appl. Math. Lett., 26 (2013), 397-401.  doi: 10.1016/j.aml.2012.10.011.

[5]

T. G. Andersen, Stochastic autoregressive volatility: A framework for volatility modeling, Math. Fin., 42 (1994), 75-102. 

[6]

H.-O. Bae, S.-Y. Cho, J.-H. Kim and S.-B. Yun, A kinetic description for the herding behavior in financial market, J. Stat Phys., 176 (2019), 398-424.  doi: 10.1007/s10955-019-02305-4.

[7]

H.-O. Bae, S.-Y. Cho, S.-H. Lee, J. Yoo and S.-B. Yun, A particle model for the herding phenomena induced by dynamic market signals, J. Stat. Phys., 177 (2019), 365-398.  doi: 10.1007/s10955-019-02371-8.

[8]

H.-O. Bae, S.-Y. Ha, M. Kang, Y. Kim, H. Lim and J. Yoo, Time-delayed stochastic volatility model, Phys. D: Nonlinear Phen., 430 (2022), 133088, 14 pp. doi: 10.1016/j.physd.2021.133088.

[9]

H.-O. Bae, S.-Y. Ha, D. Kim, Y. Kim, H. Lim and J. Yoo, Emergent dynamics of the first-order stochastic Cucker-Smale model and application to finance, Math. Methods Appl. Sci., 42 (2019), 6029-6048.  doi: 10.1002/mma.5697.

[10]

H.-O. Bae, S.-Y. Ha, Y. Kim, S.-Y. Lee, H. Lim and J. Yoo, A mathematical model for volatility flocking with a regime switching mechanism in a stock market, Math. Models Methods Appl. Sci., 25 (2015), 1299-1335.  doi: 10.1142/S0218202515500335.

[11]

H.-O. Bae, S.-Y. Ha, Y. Kim, H. Lim and J. Yoo, Volatility flocking by cucker-smale mechanism in financial markets, Asia-Pacific Fin. Mkts., 27 (2020), 387-414. 

[12]

R. T. Baillie, T. Bollerslev and H. O. Mikkelson, Fractionally integrated generalized autoregressive conditional heteroskedasticity, J. Econom., 74 (1996), 3-30.  doi: 10.1016/S0304-4076(95)01749-6.

[13]

N. Bellomo, H. Berestycki, F. Brezzi and J. Nadal, Mathematics and complexity in life and human sciences, Math. Mod. Meth. Appl. Sci., 20 (2010), 1391-1395.  doi: 10.1142/S0218202510004702.

[14]

T. Bollerslev, On the correlation structure of the generalize autoregressive conditional heteroscedastic process, J. Time. Ser. Anal., 9 (1988), 121-131.  doi: 10.1111/j.1467-9892.1988.tb00459.x.

[15]

T. Bollerslev and R. F. Engle, Common persistence in conditional variances, Econometrica, 61 (1993), 167-186.  doi: 10.2307/2951782.

[16]

T. Bollerslev, R. F. Engle and J. M. Wooldridge, A capital asset pricing model with time-varying covariances, J. Pol. Econ., 96 (1988), 116-131. 

[17]

J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[18]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.

[19]

Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789-804.  doi: 10.3934/nhm.2019032.

[20]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[21]

L. C. Davis, Modifications of the optimal velocity traffic model to include delay due to driver reaction time, Phys. A: Stat. Mech. and its Appl., 319 (2003), 557-567. 

[22]

J.-G. Dong, S.-Y. Ha and D. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596.  doi: 10.3934/dcdsb.2019072.

[23]

J.-G. Dong, S.-Y. Ha, D. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.

[24]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302. 

[25]

R. F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50 (1982), 987-1007.  doi: 10.2307/1912773.

[26]

S. Galam and J. Zucker, From individual choice to group decision-making, Phys A: Stat. Mech. and its Appl., 287 (2000), 644-659.  doi: 10.1016/S0378-4371(00)00399-X.

[27]

S. A. Gourley, J. H. So and J. H. Wu, Nonlocality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153.  doi: 10.1023/B:JOTH.0000047249.39572.6d.

[28]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. 

[29]

S.-Y. Ha, S. E. Noh and J. Park, Synchronization of Kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194.  doi: 10.1137/15M101484X.

[30]

A. C. Harvey, E. Ruiz and E. Santana, Unobserved component time series models with ARCH disturbances, J. Econom., 52 (1992), 129-157. 

[31]

G. A. Karolyi, A multivariate GARCH model of international transmission of stock returns and volatility: The case of United States and Canada, J. Bus. Econ. Stat., 13 (1995), 11-25. 

[32]

Y. Kazmerchuk, A. Swishchuk and J. Wu, A continuous-time Garch model for stochastic volatility with delay, Can. Appl. Math. Q., 13 (2005), 123-149. 

[33]

Y. N. Kyrychko and S. J. Hogan, On the use of delay equations in engineering applications, J. Vib. Cont., 16 (2010), 943-960.  doi: 10.1177/1077546309341100.

[34]

S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Soc., 96 (1986), 75-78.  doi: 10.2307/2045656.

[35]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.

[36]

J. Lu, D. W. Ho and J. Kurths, Consensus over directed static networks with arbitrary finite communication delays, Phys. Rev. E., 80 (2009), 066121. 

[37]

X. Lu, H. Zhang, W. Wang and K. L. Teo, Kalman filtering for multiple time-delay systems, Automatica, 41 (2005), 1455-1461.  doi: 10.1016/j.automatica.2005.03.018.

[38]

A. Martin and S. Ruan, Predator-prey models with delay and prey harvesting, J. Math. Biol., 43 (2001), 247-267.  doi: 10.1007/s002850100095.

[39]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[40]

J. Park, H. J. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Cont., 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.

[41]

M. R. Roussel, The use of delay differential equations in chemical kinetics, J. Phys. Chem., 100 (1996), 8323-8330. 

[42]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.

[43]

R. Sipahi, F. M. Atay and S. I. Niculescu, Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers, SIAM J. Appl. Math., 68 (2008), 738-759.  doi: 10.1137/060673813.

[44]

J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A., 366 (2007), 397-402.  doi: 10.1016/j.physleta.2007.01.083.

[45]

G. Stoica, A stochastic delay financial model, Proc. Amer. Math. Soc., 133 (2005), 1837-1841.  doi: 10.1090/S0002-9939-04-07765-2.

[46]

T. Vicsek, A. Cziròk, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[47]

J. Zhang, J. Zhu and C. Qian, On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyn. Syst., 17 (2018), 1716-1741.  doi: 10.1137/17M112765X.

show all references

References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137. 

[2]

S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.

[3]

G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Mod. Meth. Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[4]

G. Albi and L. Pareschi, Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics, Appl. Math. Lett., 26 (2013), 397-401.  doi: 10.1016/j.aml.2012.10.011.

[5]

T. G. Andersen, Stochastic autoregressive volatility: A framework for volatility modeling, Math. Fin., 42 (1994), 75-102. 

[6]

H.-O. Bae, S.-Y. Cho, J.-H. Kim and S.-B. Yun, A kinetic description for the herding behavior in financial market, J. Stat Phys., 176 (2019), 398-424.  doi: 10.1007/s10955-019-02305-4.

[7]

H.-O. Bae, S.-Y. Cho, S.-H. Lee, J. Yoo and S.-B. Yun, A particle model for the herding phenomena induced by dynamic market signals, J. Stat. Phys., 177 (2019), 365-398.  doi: 10.1007/s10955-019-02371-8.

[8]

H.-O. Bae, S.-Y. Ha, M. Kang, Y. Kim, H. Lim and J. Yoo, Time-delayed stochastic volatility model, Phys. D: Nonlinear Phen., 430 (2022), 133088, 14 pp. doi: 10.1016/j.physd.2021.133088.

[9]

H.-O. Bae, S.-Y. Ha, D. Kim, Y. Kim, H. Lim and J. Yoo, Emergent dynamics of the first-order stochastic Cucker-Smale model and application to finance, Math. Methods Appl. Sci., 42 (2019), 6029-6048.  doi: 10.1002/mma.5697.

[10]

H.-O. Bae, S.-Y. Ha, Y. Kim, S.-Y. Lee, H. Lim and J. Yoo, A mathematical model for volatility flocking with a regime switching mechanism in a stock market, Math. Models Methods Appl. Sci., 25 (2015), 1299-1335.  doi: 10.1142/S0218202515500335.

[11]

H.-O. Bae, S.-Y. Ha, Y. Kim, H. Lim and J. Yoo, Volatility flocking by cucker-smale mechanism in financial markets, Asia-Pacific Fin. Mkts., 27 (2020), 387-414. 

[12]

R. T. Baillie, T. Bollerslev and H. O. Mikkelson, Fractionally integrated generalized autoregressive conditional heteroskedasticity, J. Econom., 74 (1996), 3-30.  doi: 10.1016/S0304-4076(95)01749-6.

[13]

N. Bellomo, H. Berestycki, F. Brezzi and J. Nadal, Mathematics and complexity in life and human sciences, Math. Mod. Meth. Appl. Sci., 20 (2010), 1391-1395.  doi: 10.1142/S0218202510004702.

[14]

T. Bollerslev, On the correlation structure of the generalize autoregressive conditional heteroscedastic process, J. Time. Ser. Anal., 9 (1988), 121-131.  doi: 10.1111/j.1467-9892.1988.tb00459.x.

[15]

T. Bollerslev and R. F. Engle, Common persistence in conditional variances, Econometrica, 61 (1993), 167-186.  doi: 10.2307/2951782.

[16]

T. Bollerslev, R. F. Engle and J. M. Wooldridge, A capital asset pricing model with time-varying covariances, J. Pol. Econ., 96 (1988), 116-131. 

[17]

J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[18]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.

[19]

Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789-804.  doi: 10.3934/nhm.2019032.

[20]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[21]

L. C. Davis, Modifications of the optimal velocity traffic model to include delay due to driver reaction time, Phys. A: Stat. Mech. and its Appl., 319 (2003), 557-567. 

[22]

J.-G. Dong, S.-Y. Ha and D. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596.  doi: 10.3934/dcdsb.2019072.

[23]

J.-G. Dong, S.-Y. Ha, D. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.

[24]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302. 

[25]

R. F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50 (1982), 987-1007.  doi: 10.2307/1912773.

[26]

S. Galam and J. Zucker, From individual choice to group decision-making, Phys A: Stat. Mech. and its Appl., 287 (2000), 644-659.  doi: 10.1016/S0378-4371(00)00399-X.

[27]

S. A. Gourley, J. H. So and J. H. Wu, Nonlocality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153.  doi: 10.1023/B:JOTH.0000047249.39572.6d.

[28]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. 

[29]

S.-Y. Ha, S. E. Noh and J. Park, Synchronization of Kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194.  doi: 10.1137/15M101484X.

[30]

A. C. Harvey, E. Ruiz and E. Santana, Unobserved component time series models with ARCH disturbances, J. Econom., 52 (1992), 129-157. 

[31]

G. A. Karolyi, A multivariate GARCH model of international transmission of stock returns and volatility: The case of United States and Canada, J. Bus. Econ. Stat., 13 (1995), 11-25. 

[32]

Y. Kazmerchuk, A. Swishchuk and J. Wu, A continuous-time Garch model for stochastic volatility with delay, Can. Appl. Math. Q., 13 (2005), 123-149. 

[33]

Y. N. Kyrychko and S. J. Hogan, On the use of delay equations in engineering applications, J. Vib. Cont., 16 (2010), 943-960.  doi: 10.1177/1077546309341100.

[34]

S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Soc., 96 (1986), 75-78.  doi: 10.2307/2045656.

[35]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.

[36]

J. Lu, D. W. Ho and J. Kurths, Consensus over directed static networks with arbitrary finite communication delays, Phys. Rev. E., 80 (2009), 066121. 

[37]

X. Lu, H. Zhang, W. Wang and K. L. Teo, Kalman filtering for multiple time-delay systems, Automatica, 41 (2005), 1455-1461.  doi: 10.1016/j.automatica.2005.03.018.

[38]

A. Martin and S. Ruan, Predator-prey models with delay and prey harvesting, J. Math. Biol., 43 (2001), 247-267.  doi: 10.1007/s002850100095.

[39]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[40]

J. Park, H. J. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Cont., 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.

[41]

M. R. Roussel, The use of delay differential equations in chemical kinetics, J. Phys. Chem., 100 (1996), 8323-8330. 

[42]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.

[43]

R. Sipahi, F. M. Atay and S. I. Niculescu, Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers, SIAM J. Appl. Math., 68 (2008), 738-759.  doi: 10.1137/060673813.

[44]

J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A., 366 (2007), 397-402.  doi: 10.1016/j.physleta.2007.01.083.

[45]

G. Stoica, A stochastic delay financial model, Proc. Amer. Math. Soc., 133 (2005), 1837-1841.  doi: 10.1090/S0002-9939-04-07765-2.

[46]

T. Vicsek, A. Cziròk, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[47]

J. Zhang, J. Zhu and C. Qian, On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyn. Syst., 17 (2018), 1716-1741.  doi: 10.1137/17M112765X.

Figure 1.  Verification of Theorem 3.3: Time evolution of position(top), velcocity(middle) and variance(bottom) with two types of communication (2)(left) and (3)(right). Each line show the results with various time delays. History data and other parameter values are given in Section 5.1.1
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Figure 2.  Violation of condition (6): Time evolution of position(top), velcocity(middle) and variance(bottom) with two types of communication (2)(left) and (3)(right). Each line show the results with various time delays. History data and other parameter values are given in Section 5.1.2
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Figure 3.  Simulation Results for $ N = 3 $: Time evolution of position(top), velcocity(middle) and variance(bottom) with two types of communication (2)(left) and (3)(right). Each line show the results with various time delays. History data and other parameter values are given in (18)
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Figure 4.  Simulation Results with $ \gamma = -1 $ in (4): Time evolution of position(top), velcocity(middle) and variance(bottom) with two types of communication (2)(left) and (3)(right). Each line show the results with various time delays. History data and other parameter values are given in (18)
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Figure 5.  Simulation Results with $ \gamma = -1 $ in (4): Time evolution of position(top), velcocity(middle) and variance(bottom) with two types of communication (2)(left) and (3)(right). Each line show the results with various time delays. History data and other parameter values are given in (19), violating the condition (15)
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Figure 6.  Comparison of real and simulated volatility data of General Electric (GE)

Left: Real volatility data
Center: Simulated data based on (1)
Right: Simulated data based on (21)
Note: Volatilities are drawn over the sample period. We use thirty firms listed on DJIA and $ \lambda = 10 $. See text for details of data sources and values of other parameters

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Figure 7.  Historical and Simulated Volatilities with different $ \lambda $: DIS and CSCO

Left-top: Real volatility data
Right-top: Simulated data based on $ \textbf{CS}(\tau) $ with $ \lambda = -10 $
Left-bottom: Simulated data based on $ \textbf{CS}(\tau) $ with $ \lambda = -30 $
Right-bottom: Simulated data based on $ \textbf{CS}(\tau) $ with $ \lambda = -100 $
Note: Volatilities are drawn over the sample period. We use thirty firms listed on DJIA. See text for details of data sources and values of other parameters.

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[1]

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