Discretization of processes with stopping times and uncertainty quantification for stochastic algorithms.

Summary This thesis contains two parts that study two different topics. Chapters 1-4 are devoted to problems of discretization of processes with stopping times. In Chapter 1 we study the optimal discretization error for stochastic integrals with respect to a continuous multidimensional Brownian semimartingale. In this framework we establish a trajectory lower bound for the renormalized quadratic variation of the error. We provide a sequence of stopping times that gives an asymptotically optimal discretization. This sequence is defined as the output time of random ellipsoids by the semimartingale. Compared to the previous results we allow a rather large class of semimartingales. We prove that the lower bound is exact. In Chapter 2 we study the adaptive version of the model of the optimal discretization of stochastic integrals. In Chapter 1 the construction of the optimal strategy uses the knowledge of the diffusion coefficient of the considered semimartingale. In this work we establish an asymptotically optimal discretization strategy that is adaptive to the model and does not use any information about the model. We prove the optimality for a rather general class of discretization grids based on kernel techniques for adaptive estimation. In Chapter 3 we study the convergence of renormalized discretization error laws of Itô processes for a concrete and rather general class of discretization grids given by stopping times. Previous works on the subject consider only the case of dimension 1. Moreover they concentrate on particular cases of grids, or prove results under abstract assumptions. In our work the boundary distribution is given explicitly in a clear and simple form, the results are shown in the multidimensional case for the process and for the discretization error. In Chapter 4 we study the parametric estimation problem for diffusion processes based on time-lapse observations. Previous works on the subject consider deterministic, strongly predictable or random observation times independent of the process. Under weak assumptions, we construct a suite of consistent estimators for a large class of observation grids given by stopping times. An asymptotic analysis of the estimation error is performed. Furthermore, for the parameter of dimension 1, for any sequence of estimators that verifies an unbiased LCT, we prove a uniform lower bound for the asymptotic variance. We show that this bound is exact. Chapters 5-6 are devoted to the uncertainty quantification problem for stochastic approximation bounds. In Chapter 5 we analyze the uncertainty quantification for stochastic approximation limits (SA). In our framework the limit is defined as a zero of a function given by an expectation. This expectation is taken with respect to a random variable for which the model is supposed to depend on an uncertain parameter. We consider the limit of SA as a function of this parameter. We introduce an algorithm called USA (Uncertainty for SA). It is a procedure in increasing dimension to compute the basic chaos expansion coefficients of this function in a basis of a well chosen Hilbert space. The convergence of USA in this Hilbert space is proved. In Chapter 6 we analyze the convergence rate in L2 of the USA algorithm developed in Chapter 5. The analysis is non-trivial because of the infinite dimension of the procedure. The rate obtained depends on the model and the parameters used in the USA algorithm. Its knowledge allows to optimize the rate of growth of the dimension in USA.