Statistical Aspects of the fractional stochastic calculus

Abstract

We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the Gaussian theory of regularity and supremum estimation to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of Hölder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a basic discretized version of the MLE, is still a strongly consistent estimator. Key Words and Phrases: maximum likelihood estimator, fractional Brownian motion, strong consistency, stochastic differential equation, Gaussian regularity theory,