# Discrete-time portfolio optimization under maximum drawdown constraint with drift uncertainty

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## Optimal stochastic control problems: formulation and resolution

We begin by outlining the basic structure of stochastic optimization problems in continuous time and finite horizon while describing the main approach to solve them.
This theory and the proofs of the results can be found in Pham (2009). Consider a dynamic system formalized by its state at any time and evolving in an uncertain environment represented by a probability space ( ;F; P). Usually, we denote by Xt(!) the state of the system at time t in case ! 2 is the state of the world. The dynamics of the system t 7! Xt(!) is described by a stochastic differential equation (SDE). The dynamics of the system is influenced by a control, expressed as a process = (t)t, whose value is decided at each time t in function of the available information. The control should respect some constraints assumed in the problem, and as such, is called admissible.

### Markowitz problem: formulation and continuous-time solution

In Chapters 3 and 4, we deal with an innovative version of the Markowitz portfolio selection problem which implies that we review briefly the standard framework and some results.
Markowitz portfolio analysis is a mathematical procedure to determine the optimal portfolios in which to invest. The theory, initiated by Markowitz (1952), is a major step forward in finance. The objective of portfolio analysis is to find the set of efficient portfolios, i.e. the ones that:
have the greatest expected return for a given level of risk, or conversely.
offer the lowest level of risk for a given level of expected return.
These efficient portfolios are located on the so-called efficient frontier, represented by the dotted line in Figure 2.1, starting at the minimum variance portfolio and going upwards.

#### Filtering theory and the Bayesian learning approach

The present thesis focuses on portfolio optimization with parameters uncertainty and thus, concentrates on a framework in which the investor cannot observe directly the drift of the Brownian motion driving the dynamics of the assets: she can only observe past and present assets’ prices. We assume the unknown drift is an unobservable random variable, independent of the driving Brownian motion and with known probability distribution. This case is refered to as the partial observations case.
Since we use filtering theory to manage drift uncertainty, we exhibit in this section important standard results of this theory proved in Bain and Crisan (2009).

Portfolio optimization under parameters uncertainty – contributions

Motivation. In its standard form, the parameters of the Markowitz model, drifts and volatilities, are known and constant. This assumption raises the question of giving a value to these parameters. Typically, they are estimated from past data or calculated based on predictive analyses at inception, and are considered constant afterwards. This convenient solution does not look appropriate in practice since it does not adapt to evolving market regimes, materialized through the information contained in the changing prices. Not considering this information is detrimental in the optimization process, because it does not allow for an update of the model, given the most recent available information. In our approach, we incorporate uncertainty in the drift parameter while keeping the covariance parameter known and constant. This choice is supported, for instance by Merton (1980), who showed that estimates of variances and covariances of assets’ returns are more accurate than estimates of their expected returns. Furthermore, being wrong in the estimation of expected returns can result in a very suboptimal portfolio a posteriori. Indeed, Best and Grauer (1991) argued that mean-variance optimal portfolios are very sensitive to the level of expected returns.
Precisely, our approach considers that the drift has a known probability distribution which will be updated using the Bayesian learning approach.

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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

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