Studies in stochastic processes: Adaptive wavelet decompositions and operator fractional Brownian motions.

摘要

The thesis is centered around the themes of wavelet methods for stochastic processes, and of operator self-similarity. It comprises three parts. The first two parts concern particular wavelet-based decompositions of stationary processes, in either continuous or discrete time. The decompositions are essentially characterized by uncorrelated detail coefficients and possibly correlated approximation coefficients. This is of interest, for example, in simulation and maximum likelihood estimation. In discrete time, the focus is somewhat on long memory time series. The last part of the thesis concerns operator fractional Brownian motions. These are Gaussian operator self-similar processes with stationary increments, and are multivariate analogues of the one-dimensional fractional Brownian motion. We establish integral representations of operator fractional Brownian motions, study their basic properties and examine questions of uniqueness.