  A microscopic model is established for financial Brownian motion from the
direct observation of the dynamics of high-frequency traders (HFTs) in a
foreign exchange market. Furthermore, a theoretical framework parallel to
molecular kinetic theory is developed for the systematic description of the
financial market from microscopic dynamics of HFTs. We report first on a
microscopic empirical law of traders' trend-following behavior by tracking the
trajectories of all individuals, which quantifies the collective motion of HFTs
but has not been captured in conventional order-book models. We next introduce
the corresponding microscopic model of HFTs and present its theoretical
solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like
equations are derived from the microscopic dynamics via the
Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first
microscopic model that has been directly validated through data analysis of the
microscopic dynamics, exhibiting quantitative agreements with mesoscopic and
macroscopic empirical results.
