A Kinetic Description for the Herding Behavior in Financial Market - Bae, Hyeong Ohk; Cho, Seung Yeon; Kim, Jeongho; Yun, Seok Bae (Journal of Statistical Physics, 2019)
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.
Keywords: Barbalat’s theorem; Collective behavior; Herding model; Mean-field limit; Measure-valued solutions