Statistical analysis of multifractal random walk processes.

Summary We study some properties of a class of real continuous-time random processes, the multifractal random walks. A remarkable feature of these processes is their self-similarity property: the law of the small-scale process is identical to the large-scale one with a multiplicative random factor independent of the process. The first part of the thesis is devoted to the question of the convergence of the empirical moment of the process increase in a rather general asymptotic, where the step of the increase can tend to zero at the same time as the observation horizon tends to infinity. The second part proposes a family of non-parametric tests that distinguish between multifractal and semi-martingale Itô random walks. After showing the consistency of these tests, we study their behavior on simulated data. In the third part, we construct an asymmetric multifractal random walk process such that the past growth is negatively correlated with the square of the future growth. This type of leverage effect is notably observed on stock prices and financial indices. We compare the empirical properties of the process obtained with real data. The fourth part concerns the estimation of the parameters of the process. We start by showing that under certain conditions, two of the three parameters cannot be estimated. We then study the theoretical and empirical performances of different estimators of the third parameter, the intermittency coefficient, in a Gaussian case.