In this thesis we have analyzed the Auroral Electrojet (AE) Index over the years 2000 to 2005, a time series consisting of over 3 000 000 data points. This data is described as a multi- fractal stochastic process. We first introduce a class of random multiplicative measures, which provide the multi-fractality in the stochastic processes. We also review the theory of fractal dimensions and scaling functions, before introducing the Multifractal Model of Asset Returns (MMAR), . The scaling properties of various versions of the MMAR model are compared with the scaling function of the AE Index, and through this we describe the multi-fractal properties of the AE Index. Additionally, we have studied probability density functions (pdf) at different time scales, and used this to compare the stochastic models with the AE data. Finally we have tested our diagnostic tools on simulated multi-fractal models. These experiments show that the methods are capable of detecting multi-fractality.