A Kinetic Description for the Herding Behavior in Financial Market

As a continuation of the study of the herding model proposed in (Bae et al. in arXiv:1712.01085, 2017), we consider in this paper the derivation of the kinetic version of the herding model, the existence of the measure-valued solution and the corresponding herding behavior at the kinetic level. We first consider the mean-field limit of the particle herding model and derive the existence of the measure-valued solutions for the kinetic herding model. We then study the herding phenomena of the solutions in two different ways by introducing two different types of herding energy functionals. First, we derive a herding phenomena of the measure-valued solutions under virtually no restrictions on the parameter sets using the Barbalat’s lemma. We, however, don’t get any herding rate in this case. On the other hand, we also establish a Grönwall type estimate for another herding functional, leading to the exponential herding rate, under comparatively strict conditions. These results are then extended to smooth solutions.

[1]  E. Tadmor,et al.  From particle to kinetic and hydrodynamic descriptions of flocking , 2008, 0806.2182.

[2]  Nicola Bellomo,et al.  Toward a Mathematical Theory of Behavioral-Social Dynamics for Pedestrian Crowds , 2014, 1411.0907.

[3]  A. Figalli,et al.  A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment , 2017, Analysis & PDE.

[4]  P. Nelson,et al.  The prigogine-herman kinetic model predicts widely scattered traffic flow data at high concentrations , 1998 .

[5]  Massimo Fornasier,et al.  Particle, kinetic, and hydrodynamic models of swarming , 2010 .

[6]  Lara Trussardi,et al.  A kinetic equation for economic value estimation with irrationality and herding , 2016, 1601.03244.

[7]  D. Burini,et al.  Collective learning modeling based on the kinetic theory of active particles. , 2016, Physics of life reviews.

[8]  Jos'e A. Carrillo,et al.  A well-posedness theory in measures for some kinetic models of collective motion , 2009, 0907.3901.

[9]  Hyeong-Ohk Bae,et al.  A Particle Model for the Herding Phenomena Induced by Dynamic Market Signals , 2017, Journal of Statistical Physics.

[10]  Robert Herman,et al.  Kinetic theory of vehicular traffic , 1971 .

[11]  P. Degond,et al.  Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior , 2007, 0710.4996.

[12]  Lorenzo Pareschi,et al.  On a Kinetic Model for a Simple Market Economy , 2004, math/0412429.

[13]  Axel Klar,et al.  SELF-PROPELLED INTERACTING PARTICLE SYSTEMS WITH ROOSTING FORCE , 2010 .

[14]  A. Bertozzi,et al.  Self-propelled particles with soft-core interactions: patterns, stability, and collapse. , 2006, Physical review letters.

[15]  Lorenzo Pareschi,et al.  Reviews , 2014 .

[16]  Nicola Bellomo,et al.  On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives , 2011, SIAM Rev..

[17]  Pierre Auger,et al.  Spatial structures in simulations of animal grouping , 2007 .

[18]  Eitan Tadmor,et al.  A New Model for Self-organized Dynamics and Its Flocking Behavior , 2011, 1102.5575.

[19]  G. Toscani,et al.  Kinetic models of opinion formation , 2006 .

[20]  Seung-Yeal Ha,et al.  Time-asymptotic interaction of flocking particles and an incompressible viscous fluid , 2012 .

[21]  Jianhong Shen,et al.  Cucker–Smale Flocking under Hierarchical Leadership , 2006, q-bio/0610048.

[22]  Seung-Yeal Ha,et al.  How Do Cultural Classes Emerge from Assimilation and Distinction? An Extension of the Cucker-Smale Flocking Model , 2014 .

[23]  Seung-Yeal Ha,et al.  Emergent dynamics of the Cucker-Smale flocking model and its variants , 2016, 1604.04887.

[24]  Seung-Yeal Ha,et al.  A simple proof of the Cucker-Smale flocking dynamics and mean-field limit , 2009 .

[25]  Antoine Mellet,et al.  Hydrodynamic limit of the kinetic Cucker–Smale flocking model , 2012, 1205.6831.

[26]  Lorenzo Pareschi,et al.  Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences , 2010 .

[27]  J. Carrillo,et al.  Double milling in self-propelled swarms from kinetic theory , 2009 .

[28]  Pierre Degond,et al.  Continuum limit of self-driven particles with orientation interaction , 2007, 0710.0293.

[29]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[30]  Seung-Yeal Ha,et al.  ASYMPTOTIC FLOCKING DYNAMICS OF CUCKER-SMALE PARTICLES IMMERSED IN COMPRESSIBLE FLUIDS , 2014 .

[31]  I. Couzin,et al.  Collective memory and spatial sorting in animal groups. , 2002, Journal of theoretical biology.

[32]  Jesús Rosado,et al.  Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model , 2010, SIAM J. Math. Anal..

[33]  Felipe Cucker,et al.  Emergent Behavior in Flocks , 2007, IEEE Transactions on Automatic Control.

[34]  Eitan Tadmor,et al.  Critical thresholds in 1D Euler equations with nonlocal forces , 2014, 1411.1791.

[35]  M Dolfin,et al.  Modeling human behavior in economics and social science. , 2017, Physics of life reviews.

[36]  C. Villani.  Optimal Transport: Old and New , 2008 .

[37]  Yongsik Kim,et al.  APPLICATION OF FLOCKING MECHANISM TO THE MODELING OF STOCHASTIC VOLATILITY , 2013 .

[38]  Nicola Bellomo,et al.  Stochastic Evolving Differential Games Toward a Systems Theory of Behavioral Social Dynamics , 2015, 1506.05699.

[39]  P. Degond,et al.  A Hierarchy of Heuristic-Based Models of Crowd Dynamics , 2013, 1304.1927.

[40]  Nicola Bellomo,et al.  From the Microscale to Collective Crowd Dynamics , 2013, Multiscale Model. Simul..

[41]  D. Helbing,et al.  Self-Organization Phenomena in Pedestrian Crowds , 1998, cond-mat/9806152.

[42]  Yongsik Kim,et al.  A mathematical model for volatility flocking with a regime switching mechanism in a stock market , 2015 .

[43]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.