| This proposal is aimed at developing a new line of research in the area of general second order random processes with stationary increments (si-processes). Such processes arise in many fields of stochastic modelling, either directly, or as building blocks for more complicated models. A central example is the fractional Brownian motion (fBm), which has been at the focus of scienctific interest in recent years. Other important special cases are the integrated stationary processes. New applications of si-processes, for instance in the analysis of large telecommunications systems, mathematical finance and other areas where long-range dependence phenomena occur, require the development of prediction formulae, stochastic integration theories, large deviation bounds, series expansions, etcetera. For the fBm many of these have been obtained in the last decade, but the methods do often not generalize to arbitrary si-processes. The central point of this proposal is the observation that the spectral theory of vibrating strings provides a flexible and powerful methodology to study the structure of si-processes. In this setting the fBm is just a special example in a much wider framework. The goal of the proposed project is to use the relation with vibrating strings to develop the spectral theory of general si-processes, and to apply it to tackle the probabilistic and statistical problems that occur in new application areas. |
