# American Institute of Mathematical Sciences

March  2017, 10(1): 239-261. doi: 10.3934/krm.2017010

## 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

Received  January 2016 Revised  May 2016 Published  November 2016

Fund Project: The first author is supported by the Leverhulme Trust research project grant "Novel discretisations for higher-order nonlinear PDE" (RPG-2015-69). The second and third authors acknowledge partial support from the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617 and from the Austrian Science Fund (FWF), grants P22108, P24304, and W1245

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.

Citation: Bertram Düring, Ansgar Jüngel, Lara Trussardi. A kinetic equation for economic value estimation with irrationality and herding. Kinetic & Related Models, 2017, 10 (1) : 239-261. doi: 10.3934/krm.2017010
##### References:

show all references

##### References:
Left: Percentage of bubbles (left) and crashes (right) depending on the choice of $\alpha$ and $\beta$. The parameters are $R=0.025$, $W=0.5$, $\delta=\kappa=1$
Left: percentage of bubbles for varying $\alpha$ ($\beta=0.25$) and constant $\alpha$ ($\beta=0.5$ and $\beta=0.05$). Right: probability distribution function for the logarithmic return (in blue) and normal distribution (in magenta) with the same mean and variance as the return
Mean asset value $m_w(f(t))$ versus time $t$ for $\alpha=0.05$ (left) and $\alpha=0.5$ (right). The function $W(t)$ is represented by the solid line in between the dashed lines which represent the functions $W(t)+R$ and $W(t)-R$. The parameters are $\beta=0.25$, $R=0.025$, $\delta=2$, $\kappa=1$
Mean asset value $m_w(f(t))$ versus time for $\delta=0.01$ (left) and $\delta=100$ (right) with $\alpha=0.25$, $\beta=0.2$, $R=0.025$, $\kappa=1$
Mean asset value $m_w(f(t))$ and Bollinger bands $R^\pm(t)$ versus time for $\alpha=0.05$ (left) and $\alpha=0.2$ (right). The parameters are $\beta=0.25$, $W=0.5$, $\delta=1$, $\kappa=1$
Mean asset value $m_w(f(t))$ (left column) and Bollinger bands $R^\pm(t)$ (right column) versus time. The function $W(t)$ has a jump at $t=0.2$. The parameters are $\alpha=0.05$, $\beta=0.25$, $R=0.025$, $\delta=\kappa=1$. Upper row: $\eta=\pm 0.06$, lower row: $\eta=\pm 0.18$
 [1] Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169 [2] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [3] Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485 [4] Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016 [5] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [6] Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65 [7] Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056 [8] Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008 [9] Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 [10] Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028 [11] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 [12] John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371 [13] Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079 [14] Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120 [15] Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028 [16] Marcello Delitala, Tommaso Lorenzi. A mathematical model for value estimation with public information and herding. Kinetic & Related Models, 2014, 7 (1) : 29-44. doi: 10.3934/krm.2014.7.29 [17] Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044 [18] Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215 [19] Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165 [20] Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124

2018 Impact Factor: 1.38