SZABO Zoltan

The treatment of uncertainties is a fundamental problem in the financial context. The variables studied are often time-dependent, with thick distribution tails. In this thesis, we are interested in tools to take into account uncertainties in its main forms: statistical uncertainties, parametric uncertainties and model error, keeping in mind that we wish to apply them to this context. The first part is devoted to the establishment of concentration inequalities in the context of variables with thick tails. The objective of these inequalities is to quantify the confidence that can be given to an estimator based on a finite size of observations. In this thesis, we establish new concentration inequalities, which cover in particular the case of estimators with lognormal distribution.In the second part, we deal with the impact of the model error for the estimation of the covariance matrix on stock returns, under the assumption that there is an instantaneous covariance process between the returns whose present value depends on its past value. We can then explicitly construct the best estimate of the covariance matrix for a given time and investment horizon, and show that it provides the smallest realized variance with high probability in the minimum variance portfolio framework.In the third part, we propose an approach to estimate the Sharpe ratio and the portfolio allocation when they depend on parameters considered uncertain. Our approach involves the adaptation of a stochastic approximation technique for the computation of the polynomial decomposition of the quantity of interest.Finally, in the last part of this thesis, we focus on portfolio optimization with target distribution. This technique can be formalized without any model assumptions on the returns. We propose to find these portfolios by minimizing divergence measures based on kernel functions and optimal transport theory.