Some contributions of Bayesian and computational learning methods to portfolio selection problems.

Summary This thesis is a study of various optimal portfolio allocation problems where the rate of appreciation, called drift, of the Brownian motion of asset dynamics is uncertain. We consider an investor with a belief about drift in the form of a probability distribution, called a priori. Uncertainty about drift is taken into account by a Bayesian learning approach that updates the a priori probability distribution of drift. The thesis is divided into two independent parts. The first part contains two chapters: the first one develops the theoretical results, and the second one contains a detailed application of these results on market data. The first part of the thesis is devoted to the Markowitz portfolio selection problem in the multidimensional case with drift uncertainty. This uncertainty is modeled via an arbitrary a priori law that is updated using Bayesian filtering. We first transform the Bayesian Markowitz problem into a standard auxiliary control problem for which dynamic programming is applied. Then, we show the existence and uniqueness of a regular solution to the associated semi-linear partial differential equation (PDE). In the case of an a priori Gaussian distribution, the multidimensional solution is explicitly computed. Moreover, we study the quantitative impact of learning from the progressively observed data, by comparing the strategy that updates the drift estimate, called learning strategy, to the one that keeps it constant, called nonlearning strategy. Finally, we analyze the sensitivity of the learning gain, called information value, to different parameters. We then illustrate the theory with a detailed application of the previous results to historical market data. We highlight the robustness of the added value of learning by comparing the optimal learning and non-learning strategies in different investment universes: indices of different asset classes, currencies and smart beta strategies. The second part deals with a discrete time portfolio optimization problem. Here, the investor's objective is to maximize the expected utility of the terminal wealth of a portfolio of risky assets, assuming an uncertain drift and a maximum drawdown constraint satisfied. In this section, we formulate the problem in the general case, and we numerically solve the Gaussian case with the constant relative risk aversion (CRRA) utility function, via a deep learning algorithm. Finally, we study the sensitivity of the strategy to the degree of uncertainty surrounding the drift estimate and empirically illustrate the convergence of the unlearned strategy to a constrained Merton problem, without short selling.