Fractional Integration Calculus

In order to model randomness in any stochastic model, one may do so by asserting a distribution of the random component. The somewhat more sophisticated approach—especially when modeling dynamical issues—is defining a suitable stochastic process. The overwhelming majority of treatable models based on stochastic processes deals with classical Brownian motion as the source of randomness. This is mainly due to the two main properties of this process, which are its Gaussian character, on the one hand, and its lack of serial correlation, on the other hand. However, though being easy to manage, these features often do not map things as they truly are. Real time series often fluctuate in a non-Gaussian fashion and/or are by all means serially correlated. A great deal of research effort has been invested to get a grip on the first problem; from the onset by introducing random jumps. Currently, researchers suggest so-called alpha-stable processes which are a special group of Levy processes. With the classical Brownian motion, these processes share the property of self-similarity. However, in the literature of financial mathematics, few extensions have beenroposed to overcome the assumption of independent increments for the stochastic processes. The most popular model was introduced by Mandelbrot and van Ness (1968). They hold true the Gaussian character of the process but allow for dependence over the line of time. Figure 2.1 by Cont and Tankov (2004) depicts the relations between important sets of stochastic processes. We see that while the intersection of all three sets is classical Brownian motion, fractional Brownian motion is still Gaussian and self-similar but no longer has independent increments.