Non-Markovian Stochastic Processes and Their Applications : From Anomalous Diffusion To Time Series Analysis

This book represents a forward step in the comprehension of the relationships between certain non-Markovian processes and many integral-partial differential equations usually used to model systems manifesting long memory properties. The author made the book the more self consistent as possible by presenting all the advanced mathematical tools needed to understand the original parts. In particular, fractional Brownian motion and fractional Gaussian noise are presented as elementary examples of non-Markovian processes. These processes, together with FARIMA processes, can be used to model and estimate Long-Range Dependence (or long memory) in many contexts: physics, meteorology, hydrology, but also finance, economy, etc. Within the book LRD is studied, statistics and parametric methods of estimation are presented and many real data examples are provided. Then, the theory of fractional integrals and derivatives, which results very appropriate to model long-memory systems, is introduced. Finally, generalizations of the normal diffusion are investigated and, in order to find a connection with LRD, grey Brownian motion and more general non-Markovian processes are defined and studied.