金融布朗运动的微观动力学理论
Kinetic Theory for Finance Brownian Motion from Microscopic Dynamics
摘要：
最近的技术发展使研究人员能够详细地科学地研究社会现象，金融市场特别吸引了物理学家，因为布朗运动在物理学中发挥了关键作用。在我们之前的报告中（arXiv：1703.06739;出现在物理评论快报），我们提出了一个趋势跟踪高频交易者（HFT）的微观模型及其与金融布朗运动动力学的理论关系，直接得到了金融市场中单个HFT跟踪轨迹的数据分析的支持。在这里，我们展示了HFT模型的数学基础平行于传统的统计物理动力学理论。我们首先精确地推导出HFT模型相空间分布的时间演化方程，它对应于传统分析力学中的Liouville方程。通过对HFT模型的Liouville方程进行系统化约，得到了金融布朗运动的Bogoliubov-Born-Green-Kirkwood-Yvon方程.然后，我们推导出玻尔兹曼和Langevin的订单簿和价格动态的假设分子混沌的方程。定性行为的模型进行了渐近研究，通过求解玻尔兹曼和Langevin类方程的大量的HFTs，这是通过Monte-Carlo模拟的数值验证。我们的动力学描述突出了金融布朗运动和物理布朗运动之间的并行数学结构。
英文摘要：
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.
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