From random markets to random markets. Statistical modeling of financial markets: empirical studies and theoretical approaches.

Summary This thesis is a multidisciplinary study of the statistical regularities in the price variations of stock market assets, with three objectives: the identification of these regularities from empirical market data, their representation in probabilistic form and finally the elaboration of theoretical models linking the existence of these regularities to underlying market mechanisms. We use techniques from stochastic process theory, econometrics, economic theory and statistical mechanics. The first part is devoted to mathematical tools such as stochastic processes, levy processes, sums of random variables, cumulants, long dependence processes. The second part deals with the study of the statistical properties of empirical data from various markets and countries, trying to identify universal statistical regularities common to all these time series. Particular attention is given to the properties of temporal aggregation (scaling laws), distribution tails and the serial dependence of price increments. We then compare the empirical regularities with the predictions of the different models proposed in the literature: random walk, diffusion processes, garch, subordinate processes and deterministic chaos. The third part attempts to develop theoretical models linking the statistical properties of aggregate variables (price, supply and demand) to economic mechanisms and to the behavior of agents. The first model shows how mimetic behavior in financial markets leads to a leptokurtic distribution of aggregate demand and returns. The second model shows how, starting from a realistic level of volatility, taking into account the feedback between the dynamics of price and excess demand can lead to the appearance of stock market crashes whose frequency and dynamic form correspond to those observed empirically.