Communications in Mathematical Sciences

Volume 9 (2011)

Number 4

A kinetic model on portfolio in finance

Pages: 1073 – 1096

DOI: http://dx.doi.org/10.4310/CMS.2011.v9.n4.a7

Author

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

Abstract

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.

Keywords

econophysics, portfolio, Boltzmann equation

2010 Mathematics Subject Classification

35B40, 91B60

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