Special Issue: Fractional calculus and its applications

Scaling defines a relative measure with respect to which a model, computations or statistical analyses are made. For example, a time scale defines a measure of time, on the basis of which, models, data and statistical information are defined, recorded and analyzed. A model based on discrete time day data, defining for example a stock price from day to day differs necessarily from intraday data and of course, they differ from the theoretical definition of continuous time and data models. As a result applying fractional operators to models and data has a particular meaning. While day measurements are discrete, time recorded in a given time interval, a day, intraday measurement time intervals can vary from milliseconds to minutes and hours. Both are discrete while fractional models are continuous. They differ from classical continuous (Riemanian) models in their computational and fractional time intervals by the numerical speed of convergence that calculations are defined by. Furthermore, most complex and dynamic systems are defined over multiple time scales, contributing to systems complexity as well as to a randomness summarized by their mixtures which may call “multi-fractal”. In this view, the same notion of time demands a more accurate consideration. Is the physical time appropriate to describe financial phenomena? Or could it be possible that many apparent irregularities we observe vanish under more appropriate timedeformations capable to account for the diverse intensity with whom financial events occur? The statistical implications to the many problems associated to time scales in financial modeling, in natural and social sciences are confronted raise both opportunities and challenges and a multitude of research papers and books that have approached fractional calculus and randomness from different vantage points. The origins and the interpretation of fractional models are many and not new. There is an extensive history and developments with celebrated names such as Cauchy, Liebniz, Liouville, Abel, Caputo, Riesz and so many others that have raised questions that challenged mathematicians, physicists and applied mathematicians over the last few hundreds years. There is an extensive bibliography on fractional calculus and many applications spanning physics, calculus, data analysis, stochastic and Brownian Motion, the Brownian Bridge and α-stable distributions as they have appeared in many research areas. Doctoral theses and books have also been written and provide a broad and varied perspective to the relevance and applicability of fractional calculus. Although there are many theoretical and applied fractional problems, it requires additional research and empirical study to assess the effects of fractional models relative to conventional (Riemanian) models. For example, consider the speed at which a train travels. A fast train that records images as it travels at high speed has relatively small informative and granular detail when compared to a “slower” moving train, that records all exists in its path. Yet, they are both observing the same landscape, each defined by the granularity of their records, that define in fact a vantage point for their analysis. Discrete time models differ from one another by the time intervals defining the resolution of the data (whether deterministic or stochastic). Each discrete time defining a snapshot of an instant which we seek to reconcile to a theoretical and granular free model (i.e. a continuous time model). Similarly, a picture taken by a pixels endowed camera may reveal far more information relative to a pixel poor camera. Fractional models, unlike discrete time models of various granularity, commonly represented by granular time series, provide therefore a continuous time interpretation of fractional models based on a parametric fractional granularity. As a result, they provide a means to reconcile theoretically the relationship between model of different granularities. Such a transformation has important implications to how information is recorded and processed as well as how fractional operators alter our measures, what we see and how we may reconcile what we see with what is. A simple and intuitive example will highlight some of the problems and applications that fractional calcu-