Abstract
In this thesis we are interested in the study and the modeling of the phenomenon of complexity emerging from financial series according to different time scales and spaces. We propose a statistical method to measure the level of complexity of any financial timeseries without any a priori on the deterministic or random nature of the latter. The spatio-temporal study is based on a statistical approach known under the name of Dynamic Symbolic, a branch of information theory allowing to simplify the analysis of the evolution of complex dynamic systems through the transformation of the observed data in a series of symbols. We propose an application of it on the Standard & Poor500 index and note through the generalized entropy of Shannon that its dynamics vary strongly according to the time scale of study, showing more predictable information on higher temporal frequencies and characterized by a more Markovian process on lower scales. We show by numerical experiments that the complex phenomenon of this financial timeseries is the consequence of a superposition of several local statistics on different time scales resulting on the global scale by the emergence of a non-constant variance.We thus propose a method of modeling this non-constant variance by a generalization of the Boltzmann's statistics through a Bayesian inference. We show that the inverse gamma and log normal family’s characteristic of a power-law and the presence of long memory allow us to significantly model the process of stochastic volatility on high frequency scales. On low frequency scales, the gamma family is preferred. We also show that this transition of statistics according to the temporal scale makes it possible, among other things, to explain the change in the complex behavior of the studied series, such as the presence of a strong correlation between the returns observed on high frequencies and the more random and less predictable characteristics on lower scales.