Large-Scale Interactions in Financial Markets

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

One of the essential roles of the global financial markets is to generate future investment opportunities and to meet various needs from real economic activities, where participants in the financial markets interact both locally and globally through trades. Their interactions are understood as dynamic consequences of participants’ trading behaviors. The interactions yield market growth over time and sometimes trigger systemic risk events in unexpected ways. The main goal of the project is to understand mathematically interactions and their consequences in large financial markets. In order to analyze both homogeneous interactions with the market averages and heterogeneous interactions among similar participants, graphical interactions are modeled and analyzed with the methods in the system of stochastic partial differential equations with probability distribution constraints. The analysis of complicated financial markets in the project will provide mathematical tools for comprehensive understanding of financial networks among investors and can be applied to the similar network structures observed in other fields such as Biology, Data Science, Physics and Social Sciences. The award will support the opportunity for research experiences for graduate students and postdoctoral scholars. The focus of this project is to understand the dependence structures in the stochastic processes in the system of nonlinear stochastic differential equations of mean-field type and of directed chain interactions in large financial markets. With the directed chain interactions, the local dependence structure is preserved in the limit and the stochastic chaos does not necessarily propagate. Four subproblems are considered. First, the directed chain interactions are extended to the interactions through random tree or network structures. The outputs are sufficient conditions on drift and diffusion functional for the existence and uniqueness of weak solutions to the stochastic differential equations. Second, network detection problems are examined by a class of newly introduced, stochastic partial differential equations through stochastic filtering. Third, the dependence of stochastic volatility of multiple financial securities are analyzed and compared with rough volatility models driven by the fractional Brownian motion. Fourth, the optimization problem of market participants is analyzed in the mean-field stochastic game with directed chain interactions. The goal of solving this problem is to derive the master equation for the large population equilibria and provide understandings of competition and cooperation among investors in financial markets. The results of the project will shed some light on the family of joint probability laws in the space of continuous functions which are not of the product form yet mathematically tractable. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.