Book Description

The present thesis is a study of different optimal portfolio allocation problems in the case where the appreciation rate, named the drift, of the Brownian motion driving the dynamics of the assets is uncertain. We consider an investor having a belief on the drift in the form of a probability distribution, called a prior. The uncertainty about the drift is managed through a Bayesian learning approach which allows for the update of the drift's prior probability distribution. The thesis is divided into two self-contained parts; the first part being split into two chapters: the first develops the theory and the second contains a detailed application to actual market data. A third part constitutes an Appendix and details the data used in the applications. The first part of the thesis is dedicated to the multidimensional Markowitz portfolio selection problem in the case of drift uncertainty. This uncertainty is modeled via an arbitrary prior law which is updated using Bayesian filtering. We first embed the Bayesian-Markowitz problem into an auxiliary standard control problem for which dynamic programming is applied. Then, we show existence and uniqueness of a smooth solution to the related semi-linear partial differential equation (PDE). In the case of a Gaussian prior probability distribution, the multidimensional solution is explicitly computed. Additionally, we study the quantitative impact of learning from the progressively observed data, by comparing the strategy which updates the initial estimate of the drift, i.e. the learning strategy, to the one that keeps it constant, named the non-learning strategy. Ultimately, we analyze the sensitivity of the gain from learning, called value of information or informative value, with respect to different parameters. Next, we illustrate the theory with a detailed application of the previous results on actual market data. We emphasize the robustness of the value added of learning by comparing learning to non-learning optimal strategies in different investment universes: indices of various asset classes, currencies and smart beta strategies. The second part tackles a discrete-time portfolio optimization problem. Here, the goal of the investor 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. In this part, we formulate the problem in the general case, and we solve numerically the Gaussian case with the Constant Relative Risk Aversion (CRRA) type utility function via a deep learning resolution. Ultimately, we study the sensitivity of the strategy to the degree of uncertainty of the drift and, as a byproduct, give empirical evidence of the convergence of the non-learning strategy towards a no short-sale constrained Merton problem.