Probabilistic analysis in dynamical systems

This research program contains four interrelated projects regarding the modeling of stochastic behavior in time-discrete dynamical systems. The dynamical systems under consideration have applications to option pricing in finance via local limit theory, estimation in nonlinear dynamics, molecular evolution via random substitution models, and brain activity via dynamical modeling. Randomness in dynamical systems is expressed by fluctuations of ergodic averages around the mean, which is subject to central limit theorems, local limit theorems and large deviations. This leads to important questions of data representation and evaluation for stationary observations: here density estimation and U-statistics via martingale approximation and improved Black-Scholes formulas in the theory of financial markets and securities, using extensions of the Cox-Ross-Rubinstein model and local limit theorems. Random substitutions, which are a particular class of time-discrete dynamical systems, are used to model molecular evolution. A finer analysis of these systems, their properties and analytic tractability is a main objective of the project using probabilistic boundary theory and subadditive ergodic theory. Random external input creates stochastic behavior for neural activities. The objectives here are ergodicity and fluctuation in the long time evolution of Levina's discrete time "integrate and fire" model for neurons and the short time dynamics in the avalanche sizes during the firing period using branching and Poisson processes. Time evolution of natural phenomena often can only be understood, modeled, and analyzed through the conceptual use of random input. Uncertainty, which limits the ability to precisely forecast events, is dominant in many applications, for example, biological processes, evolution of financial securities and markets, and data-based prediction in statistics. Uncertainty may originate in modeling inaccuracy, in measurement error in experiments and data collection, or inherently in a system subjected to external influences. This project aims to develop techniques to reduce the negative influence of uncertainty in specific situations: (1) in statistics to improve information recovery from data using advanced mathematical tools (martingale approximation and conjugacy in dynamical system theory); (2) in molecular evolution by enhancing models for DNA evolution analysis using random substitutions, thus leading to better diagnostic methods; (3) in financial markets by introducing correct and verifiable models that differentiate deterministic features of the market from its remaining stochastic risk, leading to improved forecasts and thus smaller risk; and (4) in neuroscience by clarifying aspects of neuronal signal transmission, which exhibits irregular behavior that currently is not well understood.