Communications in Mathematical Sciences

Volume 9 (2011)

Number 4

A kinetic model on portfolio in finance

Pages: 1073 – 1096



Jiahang Che (Department of Mathematical Sciences, Tsinghua University, Beijing, China)


In this paper a multi-dimensional simplified kinetic model (following the ideas of onedimensional model by Cordier, Pareschi, Toscani [S. Cordier, L. Pareschi, and G. Toscani, J. Stat. Phys., 120, 253–277, 2005]) which uses Mossin’s expression for portfolio [J. Mossin, Econometrica: Journal of the Econometric Society, 768–783, 1966] is established to describe the time evolution of the portfolio distribution for several risky assets in the market. The existence and uniqueness of L¹-solutions of the model and the L¹-weak compactness of the time-scaled solutions are proved. Furthermore, the limit of the time-scaled solutions is proved to satisfy a weak form of the multidimensional Fokker–Planck equation under some assumptions on the parameters in the trading rule.


econophysics, portfolio, Boltzmann equation

2010 Mathematics Subject Classification

35B40, 91B60

Published 29 July 2011