Stochastic Analysis & Applications
Project Award Date: 08-01-2005
Fractional Brownian motion is a family of Gaussian processes indexed by the Hurst parameter H in the interval (0, 1) whose applicability, especially for H in the interval (1/2, 1), in describing a wide variety of physical phenomena has been empirically verified. Since these physical phenomena are often described by stochastic differential equations, therefore, it is important to investigate stochastic differential equations with a fractional Brownian motion. Bilinear stochastic differential equations have been used extensively in modeling, e.g., stock prices, so a study of finite dimensional bilinear equations with a fractional Brownian motion is proposed to determine explicit solutions in a variety of cases with noncommuting operators by using a stochastic calculus for a fractional Brownian motion and some methods from Lie theory. Furthermore, bilinear stochastic differential equations in an infinite dimensional Hilbert space are proposed for study because they model stochastic partial differential equations. Parameter identification in stochastic systems is a basic modeling problem. For linear stochastic differential equations with a fractional Brownian motion, a weighted least squares method for estimation is proposed to ensure the convergence of the family of estimators. The effect of discretizations of continuous time least squares estimation algorithms on linear systems is important because typically the observations of the system are sampled. It is proposed to investigate this effect for linear systems with a fractional Brownian motion to determine if biases persist as the sampling intervals approach zero. The determination and application of the absolute continuity of the measure for a fractional Brownian motion is proposed by a method of martingales that are obtained as stochastic integrals of a fractional Brownian motion. The Radon-Nikodym derivatives from this absolute continuity will be applied to problems of stochastic control, filtering and the calculation of mutual information.
Faculty Investigator(s): Tyrone Duncan (PI), Bozenna Pasik-Duncan
Student Investigator(s): Guanqun Tao, Jennifer Kensler, Yasong Jin, Tingting Wang, John (Yiannis) Zachariou, James Melbourne, Xiaobo Liu, Lina Zhao, Ranting Yao, Xiaorong Yang, Vinay Kalyankar, Benjamin Pera, Xianping Li, Jason Shea
Primary Sponsor(s): NSF