A characterization of self-affine processes in finance through the scaling function

This work focuses on a method to characterize stochastic processes by way of their scaling function tau(·). The kernel idea is that, notwithstanding the properties of the stochastic generating process of raw data, a minimum set of conditions is required to provide empirical estimation of the corresponding scaling function. Additionally, if the generating process belongs to the class of self-affine processes it is possible to express tau in a very simple and elegant way. Starting from the defnition of self-affinity (self-similarity) given By Castaing in the feld of fully developed turbulence, a general closed Form for the scaling function of self-similar processes will be derived, using the relationships between the probability density functions of the increments of self-similar processes at finer and coarser scales through a self-affinity kernel. In particular, the attention will be focused onto two main aspects of Castaing’s formalism: a) Starting from the the Castaing’s formula it is possible to find the scaling function as the Laplace transform of a proper self–affinity kernel; b)conversely, if the functional form of function tau is already known, it is possible to uniquely find the corresponding self-affinity kernel. In order to prove the generality of the results obtained, two examples will be provided (in the mono-fractal and in the multi-fractal case). The latter result is quite interesting, for its practical (econometric) applications, when the behaviour of real data can be replicated by means of self-affine processes. In such case, the generalization provided to the Castaing’s formula makes possible to link through a closed form the (given) probability density function of the increments at some larger time scale together to the (ungiven) probability density function at a smaller scale, with proper parameters,according to the desired scaling function. Obviously this assumption holds if we assume the self-affinity of process under examination. However, this appear to be not too much conditioning. To such purpose, the shape of the scaling function tau of Dow Jones daily returns will be estimated, and hence compared to those derived from time series generated by different stochastic processes (mostly self-affine). first set of conclusions will be hence drawn, since the analysis of the shape of scaling functions suggests a possible criterion to choose the best matching stochastic process to model empirical data.