Limit theorems and statistical inference for solutions of some stochastic (partial) differential equations

Abstract

The starting point for the thesis is an Ornstein-Uhlenbeck type stochastic differential equation dXt=(L(t)-cXt)dt+dBt with a real number value at time zero. The driving process B is a fractional Brownian motion with Hurst parameter H between 0.5 and 1, L is a bounded periodic deterministic function and c is a real valued parameter. In the thesis the task of estimation of L and c is solved when the solution process X is observed continuously. In the first part L is considered to be a linear combination of known functions with the same period, and the vector of the coefficients together with the parameter c is estimated using the least squares method for c<0, i.e. in the non-ergodic setting. We show strong consistency of the estimator as well as its asymptotic normality in the first p components. For the last component a noncentral limit theorem is proved. In the second part we consider the ergodic setting, i.e. c>0, and construct a nonparametric estimator for the function L. The idea for construction is to decompose L with respect to a known orthonormal basis and to estimate coefficients of a finite number of summands, letting this number tend to infinity afterwards. For this estimator L2 consistency is shown and its rate of convergence is calculated. The next part is dealing with a similar equation, this time driven by a Rosenblatt process, a non-Gaussian process with the covariance structure of a fractional Brownian motion with Hurst parameter H between 0.5 and 1. We consider again the setting in which L is a linear combination of known functions and c is positive, i.e. the ergodic case. We construct three estimators for the vector of parameters and study their first and second order asymptotics. The last part of the thesis deals with a different problem. There the stochastic wave equation is considered driven by a random field that is white in space and fractional in time with Hurst parameter H between 0.5 and 1. The objects we investigate are so-called empirical power variations of the solution, i.e. averages over the weighted differences of the solution process raised to some integer powers, where the weights fulfil certain conditions. We show central and noncentral limit theorems for these averages, determine in some cases the third order asymptotics in terms of the Wasserstein distance and use these results to define strongly consistent and asymptotically normal estimators for H.