MODELING FRACTAL DYNAMICAL SYSTEMS IN THE COMPLEX SPACE. FROM MACRO-OBSERVATION TO MICRO-OBSERVATION

The thesis presented here describes the dynamics of fractal systems subject to fractional Brownian motions (of order n ) with independent increments. One has to define the problem in the complex plane and consider the complexvalued state. After some preliminaries intended to support the paper it is described how dynamics with real-valued states can be continued in the complex plane, and three modeling axioms are proposed. A short background on fractional Brownian motion is displayed for the convenience of the reader, and, as an illustration, the approach is applied to stock market dynamics.