A statistical approach to the spectral decomposition of the conditional expectation operator: applications to Markov processes.

Summary This thesis consists of five chapters, and focuses on the statistical analysis of the spectral decomposition of the conditional expectation operator. The first chapter introduces the notion of Markov processes, then their construction from the semigroup of conditional expectation operators and the associated infinitesimal generator. In the second chapter, we consider the problem of the nonparametric estimation of the trend and volatility functions of a scalar diffusion process, from discrete time observed data. We introduce the notion of a truncated process and study its dynamic properties. The functions belonging to the generator domain of the truncated process verify explicit constraints that can then be used in the estimation phase. The third chapter deals with the extension of the spectral methods used in the framework of scalar diffusions to the case of non-reversible processes. This extension is a generalization of the usual linear canonical analysis. We consider a nonlinear canonical analysis based on a kernel estimator of the density, which allows us to obtain the convergence and the asymptotic normality of the estimators of the canonical correlations and of the canonical variables. The fourth chapter focuses on the intraday dynamics of transaction prices in financial markets operating with continuous matching. Transaction prices have two major characteristics: they are discrete in level and exist only at random trading dates. We propose a model that takes into account these two characteristics and then compare it to a structural model in which there is an underlying asset value in continuous time. The fifth chapter proposes an application of the spectral decomposition of the conditional expectation operator to the approach of the price of a derivative asset.