Aggregation tests, convergence and limits in continuous time of GARCH models.

Summary In this thesis we investigate the continuous time limit of conditionally heteroscedastic GARCH models. In the first part we study the aggregation of GARCH models and the genesis of weak GARCHs as well as the temporal aggregation of stochastic volatility specifications (SR-SARV) bypassing several limitations of weak GARCHs. The second part examines contemporaneous or individual aggregation of GARCH models and proposes methods for estimating weak GARCH representations. In the third part, an analysis of the weak convergence and limit of a sequence of semi-martingales is proposed. Four relevant applications in finance involving semi-martingales and weak convergence illustrate the analysis: AOA, Asian option pricing, dynamic programming, local risk minimization strategies. The diffusion limit of weak GARCH is then comfortably addressed in the fourth part. This limit can take several forms depending on the convergence conditions considered. The asymptotic equivalence of the statistical experiments of GARCH, stochastic volatility and diffusion processes is also clarified.