Information and Stochastic Differential Equations in Financial Markets

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Ichiba DMS-1615229 The project sheds light on mathematical aspects of information in financial markets. Here information signifies the data that are observations of a random process, for example the price of a stock over time. In the era of information technology more financial information is available about markets and each investor needs to process the flooding information. Precise mathematical understanding of information provides some criteria to decide which information is more important than other information in the financial market. Sharing this mathematical idea among investors would provide efficiency of information processing, enhance transparency of financial trading, and possibly increase the stability of the financial markets. The project also has potential consequences in other fields of research where data arise from observations of random processes, such as economics, statistics, engineering, and physics. In approximating random phenomena in financial markets, financial information is understood as a set of filtrations generated by continuous-time stochastic processes. The investigator studies rank-based diffusion market models -- these are systems of stochastic partial differential equations with singularities of various sorts -- that may better represent market information. In order to quantify and better understand the information in these processes, he examines them in two different information settings, non-Brownian filtrations and information drifts, In the first setting he considers existence of solutions, from which he can work to separate non-Browwnian noise from observable information. In the second he examines the information drift when the filtration is enlarged, which can introduce arbitrage opportunities. The consequences of emerging non-Brownian filtration and information drifts are examined in the context of stochastic portfolio theory. These two problems are closely related.