Stochastic Analysis of Gaussian Fractional Noises

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

The aim of this project is to establish original results and open new research directions in stochastic analysis. A first objective is to derive new central limit theorems applying techniques of Malliavin calculus in a variety of settings including additive functionals of the fractional Brownian motion, spacial moments of the Brownian local time increments, winding number of the complex fractional Brownian motion, and diffusion approximations in a space-time Gaussian velocity field. Rates of convergence of the Kolmogorov and related distances, and convergence of probability densities will be also investigated in these problems. A second objective of the project is to study stochastic partial differential equations driven by a Gaussian noise which is white in time and it has the covariance of a fractional Brownian motion in the space variable. The existence and uniqueness of a solution, the regularity of its trajectories and its probability distribution, and Feynman-Kac formulas for the moments, are concrete goals of the project. The project also aims to establish the rate of convergence of Euler-type numerical approximation schemes for stochastic differential equations driven by a fractional Brownian motion using techniques of fractional calculus and variational properties of the fractional Brownian motion. Stochastic analysis provides suitable mathematical tools to study ordinary and partial differential equations perturbed by a random noise, which are useful models in many areas of physics, telecommunications and economics. The application of these equations requires efficient numerical approximation schemes, and convenient estimates for the probability distribution of the solution. This project will make a significant progress in these topics by developing powerful mathematical techniques like the Malliavin calculus. On the other hand, the project will broaden the range of applications of Malliavin calculus, establishing new fundamental asymptotic results in probability and statistics. Motivated by applications in telecommunications and mathematical finance, the project will focus on input noises possessing long memory property such as the fractional Brownian motion.