Estimation, Sampling, and Simulation for Long-Range Dependent Processes

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

In a variety of phenomena in science and engineering, such as those studied in hydrology, signal processing, computer science, operations research, economics, finance, and biology, it is often the case that observations that are far apart in time or space are strongly correlated. Such phenomena can be modeled by stochastic systems that allow for long-range dependence. For such models to be relevant, it is important to be able to make reliable inferences about the inherent degree of memory in the system, as well as other model parameters. Moreover, it is critical to have efficient simulation and sampling schemes, tailored to the task of interest and incorporating the system memory. This project aims to develop novel, computationally feasible, rigorous, rate-optimal inferential and sampling methods that overcome limitations of existing methodologies. The results of this research can contribute to a more widespread and reliable use of stochastic systems that account for long-range dependence. This research aims to resolve various problems regarding the inference, design, and simulation of long-range dependent processes, bridging the gap between the continuous-time nature of these models and the discrete-time nature of their observations. Specifically, the goals are to: (i) develop statistically- and computationally-efficient parameter estimators in a coupled stochastic system where the noise of the observed diffusion is correlated with the noise of a hidden long-memory process; (ii) develop and analyze optimal non-uniform sampling methods for tracking a fractional stochastic differential equation (SDE) with a constraint on the number of observations; (iii) develop efficient simulation algorithms for fractional SDEs with long memory; and (iv) analyze the approximation error for reflected additive noise fractional SDEs. The research will rely to a considerable extent on the theories of fractional and Malliavin calculus. 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.