Small and Large Scale Limits of Multifractal Stochastic Processes with Applications

Various classes of multifractal processes, that is processes that display different properties at different scales, are studied. Most of the processes examined in this work exhibit stable trends at small scales and Gaussian trends at large scales, although the opposite can also occur. Many natural phenomena exhibit a fractal structure depending on some scaling factor, such as space or time. Thus, these types of processes have many useful modeling applications, including Biology and Economics. First, generalized tempered stable processes are defined and studied, following the original work on tempered stable processes by Jan Rosinski [16]. Generalized tempered stable processes encompass the modern variations on tempered stable distributions that have been introduced in the field, including "Modified tempered stable distributions [10]," "Layered stable distributions [8]," and "Lamperti stable processes [2]." This work shows generalized tempered stable processes exhibit multifractal properties at different scales in the space of cadlag functions equipped with the Skorokhod topology and investigates other properties, such as series representations and absolute continuity. Next, processes driven by generalized tempered stable processes involving a certain Volterra kernel are defined and short and long term behavior is established, following the work of Houdré and Kawai [7]. Finally, inspired by the work of Pipiras and Taqqu [13], the multifractal behavior of more general infinitely divisible processes is established, based on the Lévy-Itô representation of infinitely divisible processes. Numerous examples are given throughout the entire text to exemplify the strong presence of processes of this type in current literature.