Stochastic Systems and Control
Project Award Date: 08-01-2002
The major focus of this proposal is the applications of a stochastic calculus for a fractional Brownian motion to stochastic systems, identification and control. The importance of a fractional Brownian motion as a stochastic model has been emphasized in a wide variety of applications such as hydrology, economics, and telecommunications, so it is natural to expect that this work should have wide applicability. Since a stochastic calculus is available now for a fractional Brownian motion, it is important to focus on the applications of this new calculus.
The goal is to develop further the stochastic calculus for a fractional Brownian motion similar to the way the stochastic calculus for Brownian motion was developed to provide the tools for solving problems of stochastic systems with a fractional Brownian motion. Motivated by the needs of stochastic control, filtering, arid stochastic adaptive control, some tools are necessary for solving the problems of identification, filtering, and control of stochastic systems with a fractional Brownian motion.
There is a logical hierarchy in the development of this stochastic systems theory. Since the systems are described by stochastic differential equations and stochastic partial equations with a.noise modeled by a fractional Brownian motion, the proposed work investigates differential equations. It is proposed to study existence and uniqueness of solutions of stochastic differential equations, the absolute continuity of measures described by differential equation transformations of a fractional Brownian motion, and the martingales formed from a fractional Brownian motion.
The study of models with a fractional Brownian motion is a major component of this proposal. To formulate useful models, parameter identification for these models is basic. A least squares type method has been used for some stochastic systems with a fractional Brownian motion, and this work is proposed for expansion. Filtering for stochastic processes obtained from a fractional Brownian motion is proposed for further development. Some numerical analysis questions for the discretization of continuous time identification algorithms are proposed for investigation.
Fractional Brownian motion in an infinite dimensional Hilbert space is important for the study of stochastic partial differential equations with a fractional Brownian motion. Various questions are proposed for study of these equations that include existence and uniqueness of solutions, limiting measures, boundary noise and control. These topics are necessary for the development of applications of a fractional Brownian motion to stochastic systems described by stochastic partial differential equations.
Another important class of stochastic models is one that is described by hidden Markov processes. This class of models has natural use in many applications. One subclass of these models is stochastic differential equations where the coefficients are functions of a continuous time Markov chain. Adaptive control problems for these models where the coefficients are also functions of some unknown parameters are proposed for investigation because little work has been done on this topic though it is important for applications, in particular financial models.
This work is expected to provide an important contribution to stochastic systems, identification, and control. It is expected to have a significant impact on graduate education.
Faculty Investigator(s): Tyrone Duncan (PI), Bozenna Pasik-Duncan
Student Investigator(s): Ian Lewis, Joshua Meyers, Sarah Feldt, Yasong Jin, Jianhui Liu, Jiangxia Dong, Xiaobo Liu, Olga Nemon, Kristen Hughes, Jeffrey Mazzapica, Megan O'Byrne, John (Yiannis) Zachariou, Naomi Smith, Jennifer Guerra, Andrew Womack, Dominique Duncan, Fei Huang, Maryann Huey, Jennifer Kensler, Qiaoke Wu, Guanqun Tao, Erin Carmody, Xiaosong Hu, Kyle Chauvin
Primary Sponsor(s): NSF