Stochastic Analysis and Applications

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

Fractional Brownian motion is a family of Gaussian stochastic processes indexed by the Hurst parameter in the interval (0, 1) that have been empirically verified as suitable for models for many physical phenomena. The initial empirical verification was made for the occurrence of rainfall along the Nile River. Subsequent empirical verifications have been made for economic data (e.g. stock prices), telecommunications (e.g. ATM traffic), and medicine (e.g. the occurrence of epileptic seizures). In this project a stochastic calculus for fractional Brownian motion with H in (1/2, 1) is used to investigate the solutions of stochastic differential equations with a fractional Brownian motion. Physical phenomena are often modeled by stochastic differential equations. Bilinear stochastic differential equations have been used extensively in modeling so the solutions of finite dimensional bilinear equations with a fractional Brownian motion will be investigated to determine explicit solutions in a variety of cases with noncommuting linear operators appearing in the equations by using a stochastic calculus and some Lie theoretic methods. Furthermore, bilinear stochastic differential equations in an infinite dimensional Hilbert space will be investigated because they serve as models for some important stochastic partial differential equations. Parameter identification for stochastic systems is a basic component of the modeling problem. For linear stochastic differential equations with a fractional Brownian motion, a weighted pseudo least squares method for estimation will be investigated to verify the convergence of the family of estimators. The effect of time discretizations of the continuous time least squares estimation algorithm for the parameters of a linear stochastic system is important because typically the observations of the system state are sampled. This effect will be investigated for linear systems with a fractional Brownian motion to determine if biases persist as the sampling intervals approach zero. The determination and the application of the question of absolute continuity for the measure of a fractional Brownian motion will be addressed by a method of martingales that are obtained as stochastic integrals of a fractional Brownian motion. The Radon-Nikodym derivative for this absolute continuity will be applied to problems of stochastic control, filtering, and the calculation of mutual information. Stochastic models provide useful descriptions of physical phenomena. The stochastic models are used to describe random or unknown perturbations of a system or unmodeled dynamics. Fractional Brownian motion is a class of stochastic processes whose usefulness has been empirically verified for many physical phenomena that occur in a wide variety of fields, such as, hydrology, economic data, telecommunications and medicine. Stochastic differential equations with a fractional Brownian motion that are formally differential equations with an additive fractional Brownian motion, are an important class of stochastic models for physical phenomena. Some collections of these stochastic differential equations will be investigated. The estimation of parameters of a stochastic model will be investigated because of its importance for modeling. Some other questions for these stochastic differential equations will also be investigated.