  Recent technological development has enabled researchers to study social
phenomena scientifically in detail and financial markets has particularly
attracted physicists since the Brownian motion has played the key role as in
physics. In our previous report (arXiv:1703.06739; to appear in Phys. Rev.
Lett.), we have presented a microscopic model of trend-following high-frequency
traders (HFTs) and its theoretical relation to the dynamics of financial
Brownian motion, directly supported by a data analysis of tracking trajectories
of individual HFTs in a financial market. Here we show the mathematical
foundation for the HFT model paralleling to the traditional kinetic theory in
statistical physics. We first derive the time-evolution equation for the
phase-space distribution for the HFT model exactly, which corresponds to the
Liouville equation in conventional analytical mechanics. By a systematic
reduction of the Liouville equation for the HFT model, the
Bogoliubov-Born-Green-Kirkwood-Yvon hierarchal equations are derived for
financial Brownian motion. We then derive the Boltzmann-like and Langevin-like
equations for the order-book and the price dynamics by making the assumption of
molecular chaos. The qualitative behavior of the model is asymptotically
studied by solving the Boltzmann-like and Langevin-like equations for the large
number of HFTs, which is numerically validated through the Monte-Carlo
simulation. Our kinetic description highlights the parallel mathematical
structure between the financial Brownian motion and the physical Brownian
motion.
