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Deterministic particle approximation of the Hughes model in one space dimension
A kinetic equation for economic value estimation with irrationality and herding
1. | Department of Mathematics, University of Sussex, Pevensey Ⅱ, Brighton BN1 9QH, United Kingdom |
2. | Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10,1040 Wien, Austria |
A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.
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Asset Pricing Under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding Oxford University Press, Oxford, 2001.
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Continuous limit of a crowd motion and herding model: Analysis and numerical simulations, Kinetic Related Models, 4 (2011), 1025-1047.
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A mathematical model for value estimation with public information and herding, Kinet. Relat. Models, 7 (2014), 29-44.
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Trend following algorithms for technical trading in stock markets, J. Emerging Tech. Web Intell., 3 (2011), 136-145.
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[21] |
W. Hamilton,
Geometry for the selfish herd, J. Theor. Biol., 31 (1971), 295-311.
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[22] |
T. Hillen, K. Painter and M. Winkler,
Anisotropic diffusion in oriented environments can lead to singularity formation, Europ. J. Appl. Math., 24 (2013), 371-413.
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[23] |
D. Hirshleifer,
Investor psychology and asset pricing, J. Finance, 56 (2001), 1533-1597.
|
[24] |
P. Jahangiri, A. Nejat, J. Samadi and A. Aboutalebi,
A high-order Monte Carlo algorithm for the direct simulation of Boltzmann equation, J. Comput. Phys., 231 (2012), 4578-4596.
doi: 10.1016/j.jcp.2012.02.029. |
[25] |
T. Lux and M. Marchesi,
Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.
|
[26] |
D. Maldarella and L. Pareschi,
Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730.
doi: 10.1016/j.physa.2011.08.013. |
[27] |
L. Pareschi and G. Toscani,
Interacting Multiagent Systems, Kinetic Equations and Monte Carlo methods, Oxford University Press, Oxford, 2014. |
[28] |
R. Raafat, N. Chater and C. Frith,
Herding in humans, Trends Cognitive Sci., 13 (2009), 420-428.
doi: 10.1016/j.tics.2009.08.002. |
[29] |
L. Rook,
An economic psychological approach to herd behavior, J. Econ., 40 (2006), 75-95.
doi: 10.1080/00213624.2006.11506883. |
[30] |
R. Shiller,
Irrational Exuberance Princeton University Press, Princeton, 2015.
doi: 10.1515/9781400865536. |
[31] |
G. Toscani,
Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.
doi: 10.4310/CMS.2006.v4.n3.a1. |
[32] |
E. Zeidler,
Nonlinear Functional Analysis and Applications Vol. ⅡA. Springer, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
show all references
References:
[1] |
E. Altshuler, O. Ramos, Y. Núñez, J. Fernández, A. Batista-Leyva and C. Noda,
Symmetry breaking in escaping ants, Aner. Natur, 166 (2005), 643-649.
doi: 10.1086/498139. |
[2] |
A. Amadori and R. Natalini,
Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory, J. Math. Anal. Appl., 284 (2003), 511-531.
doi: 10.1016/S0022-247X(03)00339-1. |
[3] |
C. Avery and P. Zemsky,
Multidimensional uncertainty and herd behavior in financial markets, Amer. Econ. Rev., 88 (1998), 724-748.
|
[4] |
A. Banerjee,
A simple model of herd behavior, Quart. J. Econ., 107 (1992), 797-817.
doi: 10.2307/2118364. |
[5] |
S. Bickhchandani, D. Hirshleifter and I. Welch,
A theory of fads, fashion, custim, and cultural change as informational cascades, J. Polit. Econ., 100 (1992), 992-1026.
|
[6] |
G. Bird,
Molecular Gas Dynamics and the Direct Simulation of Gas Flows Oxford University Press, Oxford, 1995. |
[7] |
L. Boudin and F. Salvarani,
A kinetic approach to the study of opinion formation, ESAIM Math. Mod. Anal. Num., 43 (2009), 507-522.
doi: 10.1051/m2an/2009004. |
[8] |
M. Brunnermeier,
Asset Pricing Under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding Oxford University Press, Oxford, 2001.
doi: 10.1093/0198296983.001.0001. |
[9] |
M. Burger, P. Markowich and J.-F. Pietschmann,
Continuous limit of a crowd motion and herding model: Analysis and numerical simulations, Kinetic Related Models, 4 (2011), 1025-1047.
doi: 10.3934/krm.2011.4.1025. |
[10] |
V. Comincioli, L. Della Croce and G. Toscani,
A Boltzmann-like equation for choice formation, Kinet. Relat. Models, 2 (2009), 135-149.
doi: 10.3934/krm.2009.2.135. |
[11] |
S. Cordier, L. Pareschi and G. Toscani,
On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.
doi: 10.1007/s10955-005-5456-0. |
[12] |
P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz,
A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068.
doi: 10.1007/s10955-013-0805-x. |
[13] |
M. Delitala and T. Lorenzi,
A mathematical model for value estimation with public information and herding, Kinet. Relat. Models, 7 (2014), 29-44.
doi: 10.3934/krm.2014.7.29. |
[14] |
A. Devenow and I. Welch,
Rational herding in financial economics, Europ. Econ. Rev., 40 (1996), 603-615.
doi: 10.1016/0014-2921(95)00073-9. |
[15] |
G. Dimarco and L. Pareschi,
High order asymptotic-preserving schemes for the Boltzmann equation, Comptes Rendus Methématique, 350 (2012), 481-486.
doi: 10.1016/j.crma.2012.05.010. |
[16] |
B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram,
Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708.
doi: 10.1098/rspa.2009.0239. |
[17] |
B. Düring and M. -T. Wolfram, Opinion dynamics: Inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 471 (2015), 20150345, 21pp.
doi: 10.1098/rspa.2015.0345. |
[18] |
J. Dyer, A. Johansson, D. Helbing, I. Couzin and J. Krause,
Leadership, consensus decision making and collective behaviour in humans, Phil. Trans. Roy. Soc. B: Biol. Sci., 364 (2009), 781-789.
doi: 10.1098/rstb.2008.0233. |
[19] |
M. Escobedo, J.-L. Vázquez and E. Zuazua,
Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842.
doi: 10.1090/S0002-9947-1994-1225573-2. |
[20] |
S. Fong, J. Tai and Y.W. Si,
Trend following algorithms for technical trading in stock markets, J. Emerging Tech. Web Intell., 3 (2011), 136-145.
doi: 10.4304/jetwi.3.2.136-145. |
[21] |
W. Hamilton,
Geometry for the selfish herd, J. Theor. Biol., 31 (1971), 295-311.
doi: 10.1016/0022-5193(71)90189-5. |
[22] |
T. Hillen, K. Painter and M. Winkler,
Anisotropic diffusion in oriented environments can lead to singularity formation, Europ. J. Appl. Math., 24 (2013), 371-413.
doi: 10.1017/S0956792512000447. |
[23] |
D. Hirshleifer,
Investor psychology and asset pricing, J. Finance, 56 (2001), 1533-1597.
|
[24] |
P. Jahangiri, A. Nejat, J. Samadi and A. Aboutalebi,
A high-order Monte Carlo algorithm for the direct simulation of Boltzmann equation, J. Comput. Phys., 231 (2012), 4578-4596.
doi: 10.1016/j.jcp.2012.02.029. |
[25] |
T. Lux and M. Marchesi,
Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.
|
[26] |
D. Maldarella and L. Pareschi,
Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730.
doi: 10.1016/j.physa.2011.08.013. |
[27] |
L. Pareschi and G. Toscani,
Interacting Multiagent Systems, Kinetic Equations and Monte Carlo methods, Oxford University Press, Oxford, 2014. |
[28] |
R. Raafat, N. Chater and C. Frith,
Herding in humans, Trends Cognitive Sci., 13 (2009), 420-428.
doi: 10.1016/j.tics.2009.08.002. |
[29] |
L. Rook,
An economic psychological approach to herd behavior, J. Econ., 40 (2006), 75-95.
doi: 10.1080/00213624.2006.11506883. |
[30] |
R. Shiller,
Irrational Exuberance Princeton University Press, Princeton, 2015.
doi: 10.1515/9781400865536. |
[31] |
G. Toscani,
Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.
doi: 10.4310/CMS.2006.v4.n3.a1. |
[32] |
E. Zeidler,
Nonlinear Functional Analysis and Applications Vol. ⅡA. Springer, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |






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