Studies on Dynamics of Financial Markets and Reacting Flows
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
One of the central problems in financial markets analysis is to understand the nature of the underlying stochastic dynamics. Several intraday behaviors are analyzed to study trading day ensemble averages of both high frequency foreign exchange and stock markets data. These empirical results indicate that the underlying stochastic processes have nonstationary increments. The three most liquid foreign exchange markets and five most actively traded stocks each contains several time intervals during the day where the mean square fluctuation and variance of increments can be fit by power law scaling in time. The fluctuations in return within these intervals follow asymptotic bi-exponential distributions. Based on these empirical results, an intraday stochastic model with linear variable diffusion coefficient is proposed to approximate the real dynamics of financial markets to the lowest order, and to test the effects of time averaging techniques typically used for financial time series analysis. The proposed model replicates major statistical characteristics of empirical financial time series and only ensemble averaging techniques deduce the underlying dynamics correctly. The proposed model also provides new insight into the modeling of financial markets' dynamics in microscopic time scales.
Also discussed are analytical and computational studies of reacting flows. Many dynamical features of the flows can be inferred from modal decompositions and coupling between modes. Both proper orthogonal (POD) and dynamic mode (DMD) decompositions are conducted on high-frequency, high-resolution empirical data and their results and strengths are compared and contrasted. In POD the contribution of each mode to the flow is quantified using the latency only, whereas each DMD mode can be associated a latency as well as a unique complex growth rate. By comparing DMD spectra from multiple nominally identical experiments, it is possible to identify "reproducible" modes in a flow. A similar differentiation cannot be made using POD. Time-dependent coefficients of DMD modes are complex. Even in noisy experimental data, it is found that the phase of these coefficients (but not their magnitude) exhibits repeatable dynamics. Hence it is suggested that dynamical characterizations of complex flows are best analyzed through the phase dynamics of reproducible DMD modes.