Kinetic Theory for Finance Brownian Motion from Microscopic Dynamics

Abstract

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.

arXiv comment36 pages, 15 figures

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• Inferring the Langevin Equation with Uncertainty via Bayesian Neural Networks
• Quantitative statistical analysis of order-splitting behaviour of individual trading accounts in the Japanese stock market over nine years
• Exact solution to a generalised Lillo-Mike-Farmer model with heterogeneous order-splitting strategies
• Anomalous diffusion and price impact in the fluid-limit of an order book
• Exact solution to two-body financial dealer model: revisited from the viewpoint of kinetic theory