Quantifying uncertainties in asset management: kernel methods and statistical fluctuations.

Summary The treatment of uncertainties is a fundamental problem in the financial context, and more precisely in portfolio optimisation. The variables studied are often time dependent, with heavy tails. In this thesis, we are interested in tools allowing 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 the financial context.The first part is devoted to the establishment of concentration inequalities for variables with heavy tailed distributions. The objective of these inequalities is to quantify the confidence that can be given to an estimator based on observations of finite size. In this thesis, we establish new concentration inequalities which include the case of estimators with log-normal distribution.In the second part, we discuss 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 values. One can then explicitly construct the best estimate of the covariance matrix for a given time and investment horizon, and we show that this estimate gives the best performance 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 formalised without the need for any model assumptions on returns. We propose to find these portfolios by minimizing divergence measures based on kernels or optimal transport. Since these divergence measures can be unbounded and have not been studied much yet in the unbounded kernel case, we establish new convergence guarantees based on concentration inequalities.