In this thesis a mathematical description and analysis of the Cumulative Prospect Theory is presented. Conditions that ensure well-posedness of the problem are provided, as well as existence results concerning optimal policies for discrete-time incomplete market models and for a family of diffusion market models. A brief outline of how this work is organised follows. In Chapter 2 important results on weak convergence and discrete time finance models are described, these facts form the main background to introduce in Chapter 3 the problem of optimal investment under the CPT theorem in a discrete time setting. We describe our model, present some assumptions and main results are derived. The second part of this work comprises the description of the martingale problem formulation of diffusion processes in Chapter 4. A key result on the limits and topological properties of the set of laws of a class of Itô processes is described in Chapter 5. Finally, we introduce a factor model that includes a class of stochastic volatility models, possibly with path-depending coefficients. Under this model, the problem of optimal investment with a behavioural investor is analysed and our main results on well-posedness and existence of optimal strategies are described under the framework of weak solutions. Further research and challenges when applying the techniques developed in this work are described.

Our aim is to examine the problem of optimal asset allocation for investors exhibiting a behaviour in the face of uncertainty which is not consistent with the usual axioms of Expected Utility Theory. This thesis is divided into two main parts. In the first one, comprising Chapter II, we consider an arbitrage-free discrete-time financial model and an investor whose risk preferences are represented by a possibly nonconcave utility function (defined on the non-negative half-line only). Under straightforward conditions, we establish the existence of an optimal portfolio. As for Chapter III, it consists of the study of the optimal investment problem within a continuous-time and (essentially) complete market framework, where asset prices are modelled by semi-martingales. We deal with an investor who behaves in accordance with Kahneman and Tversky's Cumulative Prospect Theory, and we begin by analysing the well-posedness of the optimisation problem. In the case where the investor's utility function is not bounded above, we derive necessary conditions for well-posedness, which are related only to the behaviour of the distortion functions near the origin and to that of the utility function as wealth becomes arbitrarily large (both positive and negative). Next, we focus on an investor whose utility is bounded above. The problem's wellposedness is trivial, and a necessary condition for the existence of an optimal trading strategy is obtained. This condition requires that the investor's probability distortion function on losses does not tend to zero faster than a given rate, which is determined by the utility function. Provided that certain additional assumptions are satisfied, we show that this condition is indeed the borderline for attainability, in the sense that, for slower convergence of the distortion function, there does exist an optimal portfolio. Finally, we turn to the case of an investor with a piecewise power-like utility function and with power-like distortion functions. Easily verifiable necessary conditions for wellposedness are found to be sufficient as well, and the existence of an optimal strategy is demonstrated.

Cette thèse étudie la distorsion de probabilité dans le jugement clinique afin de comparer le jugement des médecins à des modèles statistiques. Nous supposons que les médecins forment leur jugement clinique en intégrant une composante analytique et une composante intuitive. Dans ce cadre, les médecins peuvent souffrir de plusieurs biais dans la façon dont ils évaluent et intègrent les deux composantes. Cette thèse rassemble les résultats obtenus sur le terrain et en laboratoire. À partir de données médicales, nous avons constaté que les médecins n'étaient pas aussi bons que les modèles statistiques à intégrer des évidences médicales. Ils surestimaient les petites probabilités que le patient soit malade et sous­-estimaient les probabilités élevées. Nous avons constaté que leur jugement biaisé pourrait entraîner un sur­-traitement. Comment améliorer leur jugement? Premièrement, nous avons envisagé de remplacer le jugement du médecin par la probabilité de notre modèle statistique. Pour améliorer la décision, il était nécessaire d'élaborer un score statistique qui combine le modèle analytique, la composante intuitive du médecin et sa déviation observée par rapport à la décision attendue. Deuxièmement, nous avons testé en laboratoire des facteurs qui peuvent influencer le traitement de l'information. Nous avons trouvé que la capacité des participants à apprendre la valeur de la composante analytique, sans feedback externe, dépend de la qualité de leur composante intuitive et de leur mémoire de travail. Nous avons aussi trouvé que la capacité des participants à intégrer les deux composantes dépend de leur mémoire de travail, mais pas de leur évaluation de la composante intuitive. / This thesis studies probability distortion in clinical judgment to compare physicians’ judgment with statistical models. We considered that physicians form their clinical judgment by integrating an analytical component and an intuitive component. We documented that physicians may suffer from several biases in the way they evaluate and integrate the two components. This dissertation gathers findings from the field and the lab. With actual medical data practice, we found that physicians were not as good as the statistical models at integrating consistently medical evidence. They over­estimated small probabilities that the patient had the disease and under­ estimated large probabilities. We found that their biased probability judgment might cause unnecessary health care treatment. How then can we improve physician judgment? First, we considered to replace physician judgment by the probability generated from our statistical model. To actually improve decision it was necessary to develop a statistical score that combines the analytical model, the intuitive component of the physician and his observed deviation from the expected decision. Second, we tested in the lab factors that may affect information processing. We found that participants’ ability to learn about the value of the analytical component, without external feedback, depends on the quality of their intuitive component and their working memory. We also found that participants’ ability to integrate both components together depends on their working memory but not their evaluation of the intuitive component.