Self-similar random processes: applications in turbulence and finance.

Summary Within the framework of self-similar random processes, we propose a stationary multifractal process model with continuous scale invariance. After introducing the notion of fractal applied to measurements, functions and random processes, we recall the properties of the main paradigm of multifractal processes: the multiplicative cascades built on orthogonal wavelet bases. The will to generalize these models leads us to the construction of a multifractal process based on a radically different philosophy than the cascades, in the form of a multifractal random walk (the MRW process). The main message that emerges from this study is the complete characterization of the multifractality of the process by the long-range correlation structure of the amplitude of its variations. In a second step, we analyze experimental velocity signals recorded in hydrodynamic flows of fully developed turbulence. Eulerian signals from different experimental configurations are exhaustively analyzed and, for the first time, we present a study of Lagrangian velocity signals (recorded by Nicolas Mordant and Jean-François Pinton at ENS Lyon). The good modeling of the Lagrangian turbulence by the MRW process leads us to propose an original research track which could provide a microscopic explanation to the intermittency phenomenon. Finally, in a third part, we analyze financial signals with respect to our theoretical results. The MRW model seems to be relevant again in this context and we propose two practical and direct applications of our observations : optimization of dynamic portfolio management and volatility prediction.