Parameter estimation for stochastic differential equations driven by fractional Brownian motion

Abstract

This dissertation systematically considers the inference problem for stochastic differential equations (SDE) driven by fractional Brownian motion. For the volatility parameter and Hurst parameter, the estimators are constructed using iterated power variations. To prove the strong consistency and the central limit thoerems of the estimators, the asymptotics of the power variatons are studied, which include the strong consistency, central limit theorem, and the convergence rate for the iterated power variations of the Skorohod integrals with respect to fractional Brownian motion. The iterated logarithm law of the power variations of fractional Brownian motion is proved. The joint convergence along different subsequence of power variations of Skorohod integrals is also studied in order to derive the central limit theorem for the estimators. Another important topic considered in this dissertation is the estimation of drift parameters of the SDEs. A least squares estimator (LSE) is proposed and the strong consistency is proved for the fractional Ornstein-Uhlenbeck process that is the solution to the linear SDE. The fourth moment theorem is applied to obtain the central limit theorems. Then the LSE is considered for the drift parameter of the multi-dimensional nonlinear SDE. While proving the strong consistency of LSE, the regularity structure of the SDE’s solution is explored and the maximal inequality for the Skorohod integrals is derived. The main tools used in this research are Malliavin calculus and some Gaussian analysis elements.