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Table of contents
Résumé
Abstract
List of Figures
Notations
1 Introduction (in French)
1.1 Revue de la littérature
1.1.1 Problème de contrôle optimal stochastique : formulation et résolution
1.1.2 Formulation et solution en temps continu du problème de Markowitz
1.1.3 Théorie du filtrage et apprentissage bayésien
1.1.4 Maximum Drawdown
1.2 Optimisation de portefeuilles avec incertitude de paramètres – contributions
1.2.1 Chapitre 3
1.2.2 Chapitre 4
1.3 Optimisation de portefeuilles avec contraintes de risques – contributions
1.3.1 Chapitre 5
2 Introduction
2.1 Literature Review
2.1.1 Optimal stochastic control problems: formulation and resolution
2.1.2 Markowitz problem: formulation and continuous-time solution .
2.1.3 Filtering theory and the Bayesian learning approach
2.1.4 Maximum drawdown
2.2 Portfolio optimization under parameters uncertainty – contributions .
2.2.1 Chapter 3
2.2.2 Chapter 4
2.3 Portfolio optimization under risk measure constraints – contributions .
2.3.1 Chapter 5
I The Markowitz portfolio selection problem with drift uncertainty
3 Bayesian learning for the Markowitz portfolio selection problem
3.1 Introduction
3.2 Markowitz problem with prior law on the uncertain drift
3.3 Bayesian learning
3.4 Solution to the Bayesian-Markowitz problem
3.4.1 Main result
3.4.2 On existence and smoothness of the Bayesian risk premium
3.4.3 Examples
3.4.3.1 Prior discrete law
3.4.3.2 The Gaussian case
3.5 Impact of learning on the Markowitz strategy
3.5.1 Computation of the Sharpe ratios
3.5.2 Value of Information
3.5.2.1 Standard deviation of the drift
3.5.2.2 Sharpe ratio of the asset
3.5.2.3 Time
3.5.2.4 Investment horizon
3.6 Conclusion
3.A Appendices
3.A.1 Proof of Lemma 3.2.3
3.A.2 Proof of Lemma 3.2.4
3.A.3 Proof of Theorem 3.4.1
3.A.4 Proofs of Theorem 3.4.3
3.A.5 Proof of Lemma 3.4.5
4 Dealing with drift uncertainty: a Bayesian learning approach
4.1 Introduction
4.2 The model
4.3 Market data
4.4 The Base Case result
4.5 Sensitivity analysis
4.5.1 Impact of uncertainty
4.5.2 Impact of leverage
4.5.3 Impact of the review frequency
4.5.4 Impact of the rebalancing frequency
4.6 Investing in foreign currencies
4.7 Investing in factor strategies
4.A Appendices
4.A.1 Dataset of Indices
4.A.1.1 Commodity
4.A.1.2 Bond
4.A.1.3 Equity
4.A.1.4 Cash
4.A.2 Dataset of currencies
4.A.3 Dataset of Smart Beta strategies
4.A.3.1 Smart Beta strategies
II Discrete-time portfolio optimization under maximum drawdown constraint with drift uncertainty
5 Discrete-time portfolio optimization under maximum drawdown constraint with partial information and deep learning resolution
5.1 Introduction
5.2 Problem setup
5.3 Dynamic programming system
5.3.1 Change of measure and Bayesian filtering
5.3.2 The static set of admissible controls
5.3.3 Derivation of the dynamic programming equation
5.3.4 Special case: CRRA utility function
5.4 The Gaussian case
5.4.1 Bayesian Kalman filtering
5.4.2 Finite-dimensional dynamic programming equation
5.5 Deep learning numerical resolution
5.5.1 Architectures of the deep neural networks
5.5.2 Hybrid-Now algorithm
5.5.3 Numerical results
5.5.3.1 Learning and non-learning strategies
5.5.3.2 Learning, non-learning and constrained equally-weighted strategies
5.5.3.3 Non-learning and Merton strategies
5.5.4 Sensitivities analysis
5.6 Conclusion
5.A Appendices
5.A.1 Proof of Proposition 5.3.1
5.A.2 Proof of Proposition 5.3.2
5.A.3 Proof of Lemma 5.3.3
5.A.4 Proof of Lemma 5.3.4
5.A.5 Proof of Lemma 5.3.5
5.A.6 Proof of Lemma 5.3.6
Bibliography
