The study of expected utility maximization in continuous-time stochastic market models dates back to the seminal work of Merton 1969 and has since been central to the area of Mathematical Finance. The associated stochastic optimization problems have been extensively studied. The problem formulation relies on two strong underlying assumptions: the ability to specify the underpinning market model and the knowledge of the investor's risk preferences. However, neither of these inputs is easily available, if at all. Resulting issues have attracted continuous attention and prompted very active and diverse lines of research. This thesis seeks to contribute towards this literature and questions related to both of the above issues are studied. Specifically, we study the implications of certain qualitative properties of the utility function; we introduce, and study various aspects of, the notion of robust forward investment criteria; and we study the investment problem associated with risk- and ambiguity-averse preference criteria defined in terms of quasiconcave utility functionals.

This dissertation consists of two essays in international macroeconomics. The first essay shows that optimal fiscal and monetary policy is time consistent in a standard small open economy. Further, there exist many maturity structures of public debt capable of rendering the optimal policy time consistent. This result is in sharp contrast with that obtained in the context of closed-economy models. In the closed economy, the time consistency of optimal monetary and fiscal policy imposes severe restrictions on public debt in the form of a unique term structure of public debt that governments can leave to their successors at each point in time. The time consistent result is robust: optimal policy is time consistent when both real and nominal bonds have finite horizons. While in a closed economy, governments must have both nominal and real bonds, and have at least real bonds over an infinite horizon to render optimal policy time consistent. The second essay uses a dynamic stochastic general equilibrium model to theoretically rationalize the empirical finding that sudden stops have weaker effects on outputs when the small open economy is more open to trade. First, welfare costs of sudden stops are decreasing in trade openness. The reason is that when the economy is more open to trade, the economy will have less volatile capital, which leads to less volatile output. In terms of welfare, when the small open economy is more open to trade, the welfare costs of sudden stops will be smaller. Second, sudden stops may be welfare improving to the small open economy. This is because when the representative household is a net borrower in the international capital market, its consumption will be negatively correlated with country spread. Since utility is a concave function of consumption, it must be a convex function of country spread. That is, when the country spread is more volatile, the mean utility is higher. The two findings are robust: they hold with one sector economy model, and two sector economy models with homogenous capital and heterogenous capital. In addition, this paper shows that a counter-cyclical tariff rate policy is not welfare-improving. / Dissertation

A central bank is usually assigned two functions: the control of inflation and the maintenance of a safetybanking sector. What are the precise conditions under which trigger strategies from the private sector can solve the time inconsistency problem and induce the central bank to choose zero inflation under a nonstationary natural rate? Can an optimal contract be used together with reputation forces to implement a desired socially optimal monetary policy rule? How to design a truthtelling contract to control the risk taking behaviors of the bank? My dissertation attempts to deal with these issues using three primary methodologies: monetary economics, game theory and optimal stochastic control theory.

A central bank is usually assigned two functions: the control of inflation and the maintenance of a safetybanking sector. What are the precise conditions under which trigger strategies from the private sector can solve the time inconsistency problem and induce the central bank to choose zero inflation under a nonstationary natural rate? Can an optimal contract be used together with reputation forces to implement a desired socially optimal monetary policy rule? How to design a truthtelling contract to control the risk taking behaviors of the bank? My dissertation attempts to deal with these issues using three primary methodologies: monetary economics, game theory and optimal stochastic control theory.

Le contrôle optimal stochastique (en temps discret) s'intéresse aux problèmes de décisions séquentielles sous incertitude. Les applications conduisent à des problèmes d'optimisation degrande taille. En réduisant leur taille, les méthodes de décomposition permettent le calcul numérique des solutions. Nous distinguons ici deux formes de décomposition. La emph{décomposition chaînée}, comme la Programmation Dynamique, résout successivement, des sous-problèmes de petite taille. La décomposition parallèle, comme le Progressive Hedging, consiste à résoudre itérativement et parallèlement les sous-problèmes, coordonnés par un algorithme maître. Dans la première partie de ce manuscrit, Dynamic Programming: Risk and Convexity, nous nous intéressons à la décomposition chaînée, en particulier temporelle, connue sous le nom de Programmation Dynamique. Dans le chapitre 2, nous étendons le cas traditionnel, risque-neutre, de la somme en temps des coûts, à un cadre plus général pour lequel nous établissons des résultats de cohérence temporelle. Dans le chapitre 3, nous étendons le résultat de convergence de l'algorithme SDDP (Stochastic Dual Dynamic Programming Algorithm) au cas où les fonctions de coûts (convexes) ne sont plus polyhédrales. Puis, nous nous tournons vers la décomposition parallèle, en particulier autour des méthodes de décomposition obtenues en dualisant les contraintes (contraintes spatiales presque sûres, ou de non-anticipativité). Dans la seconde partie de ce manuscrit, Duality in Stochastic Optimization, nous commençons par souligner que de telles contraintes peuvent soulever des problèmes de dualité délicats (chapitre 4).Nous établissons un résultat de dualité dans les espaces pairés $Bp{mathrm{L}^infty,mathrm{L}^1}$ au chapitre 5. Finalement, au chapitre 6, nous montrons un résultat de convergence de l'algorithme d'Uzawa dans $mathrm{L}^inftyp{Omega,cF,PP;RR^n}$, qui requière l'existence d'un multiplicateur optimal. La troisième partie de ce manuscrit, Stochastic Spatial Decomposition Methods, est consacrée à l'algorithme connu sous le nom de DADP (Dual Approximate Dynamic Programming Algorithm). Au chapitre 7, nous montrons qu'une suite de problèmes d'optimisation où une contrainte presque sûre est relaxée en une contrainte en espérance conditionnelle épi-converge vers le problème original si la suite des tribus converge vers la tribu globale. Finalement, au chapitre 8, nous présentons l'algorithme DADP, des interprétations, et des résultats de convergence basés sur la seconde partie du manuscrit / Stochastic optimal control addresses sequential decision-making under uncertainty. As applications leads to large-size optimization problems, we count on decomposition methods to tackle their mathematical analysis and their numerical resolution. We distinguish two forms of decomposition. In chained decomposition, like Dynamic Programming, the original problemis solved by means of successive smaller subproblems, solved one after theother. In parallel decomposition, like Progressive Hedging, the original problemis solved by means of parallel smaller subproblems, coordinated and updated by amaster algorithm. In the first part of this manuscript, Dynamic Programming: Risk and Convexity, we focus on chained decomposition; we address the well known time decomposition that constitutes Dynamic Programming with two questions. In Chapter 2, we extend the traditional additive in time and risk neutral setting to more general ones for which we establish time-consistency. In Chapter 3, we prove a convergence result for the Stochastic Dual Dynamic Programming Algorithm in the case where (convex) cost functions are no longer polyhedral. Then, we turn to parallel decomposition, especially decomposition methods obtained by dualizing constraints (spatial or non-anticipative). In the second part of this manuscript, Duality in Stochastic Optimization, we first point out that such constraints lead to delicate duality issues (Chapter 4).We establish a duality result in the pairing $Bp{mathrm{L}^infty,mathrm{L}^1}$ in Chapter 5. Finally, in Chapter 6, we prove the convergence of the Uzawa Algorithm in~$mathrm{L}^inftyp{Omega,cF,PP;RR^n}$.The third part of this manuscript, Stochastic Spatial Decomposition Methods, is devoted to the so-called Dual Approximate Dynamic Programming Algorithm. In Chapter 7, we prove that a sequence of relaxed optimization problems epiconverges to the original one, where almost sure constraints are replaced by weaker conditional expectation ones and that corresponding $sigma$-fields converge. In Chapter 8, we give theoretical foundations and interpretations to the Dual Approximate Dynamic Programming Algorithm

La thèse, intitulée « Essai sur la Théorie de l’Actualisation : Utilité Escomptée Subjective et Sensibilité à la Variation », s’inscrit dans le périmètre de recherche associé à la théorie de la décision intertemporelle et s’engage précisément dans deux sentiers qui ont été, jusqu’à présent, relativement peu exploités. D’une part, elle s’interroge sur la diversité observée – et modélisée – des mécanismes d’actualisation en cherchant à expliquer une telle diversité sur la base d’une structure comportementale unique (Partie I. Une investigation de l’actualisation). D’autre part, face à cette diversité croissante des mécanismes d’actualisation recensés, la thèse s’interroge sur la condition de cohérence temporelle en cherchant à caractériser des préférences temporelles cohérentes sur la base de la même structure comportementale définie dans la première partie (Partie II. Une investigation de la cohérence temporelle). / The thesis, entitled “Essay on the Discounting Theory: Subjective Discounted Utility and Variation Sensibility”, gathers a set of theoretical works related to Intertemporal Decision Theory. In particular, two unexploited fields are investigated. On one hand, the thesis explains the great heterogeneity in discounting by a single behavioral pattern called time perception. A general axiomatic model, the Subjective Discounted Utility, generalizes any Discounted Utility model by associating each discount mechanism to a unique time perception (Part I. An Investigation of Discounting). Many applications are dedicated to extreme time horizons. On the other hand, the Subjective Discounted Utility model is applied to a general study of time consistent preferences. Axiomatic conditions are defined on time preferences to characterize time consistent and time inconsistent preferences (Part II. An Investigation of Time Consistency).

[pt] Tradicionalmente, em indústrias de produção de peças discretas, no nível tático do planejamento da produção, é calculado o plano mestre de produção (Master Production Scheduling – MPS), que estabelece a quantidade de cada bem a ser produzida por período. Com o MPS em mãos, a necessidade de matéria-prima é levantada e o requerimento de material é realizado levandose em consideração o lead time de chegada das peças, que está relacionado com o modal de transporte previamente definido pela empresa. Mais próximo da operação, o sequenciamento dos jobs é feito com o objetivo de atender ao planejamento do MPS. Normalmente, esses quatro problemas - definição do modal de transporte, elaboração do plano mestre de produção, definição do momento de compra de materiais e sequenciamento da produção - são tratados em momentos diferentes e, muitas vezes, de forma determinística. Neste trabalho é avaliado o impacto financeiro e operacional de realizar o planejamento de forma determinística e segregada. Em particular, avaliase: (i) o impacto da estocasticidade e co-otimização do planejamento da produção e das decisões de compra e (ii) a viabilidade do sequenciamento. Para (i) é proposto um modelo de otimização estocástica de dois estágios que co-otimiza as decisões de volume de produção, momentos de compra e modal de transporte. Para (ii) restrições de sequenciamento de jobs são adicionadas através de uma heurística e de um modelo de programação matemática. Avaliações empíricas são feitas por meio de dois experimentos numéricos com dados realistas de uma empresa do setor automobilístico. Para (i) observamos uma redução de custo de 7 por cento na operação para o ano de 2018 (ano do planejamento) e cerca de 3,5 por cento para 5000 cenários simulados (out-ofsample), comparando o modelo de dois estágios proposto com o procedimento normalmente adotado na indústria. Para (ii) também observamos uma redução de 8 por cento no custo de operação de 2018, e de 9,6 por cento para 50 cenários simulados (outof- sample), em relação ao modelo proposto em (i) e 13 por cento no custo de operação de 2018 e 15,6 por cento para 50 cenários simulados (out-of-sample), em comparação com o modelo típico da indústria. / [en] Traditionally, in discrete manufacturing industries, at the tactical level of production planning, the master production scheduling (MPS) is calculated, which establishes the quantity of each good to be produced per period. With the MPS in hand, the need for raw material is raised and the material requirement is carried out taking into account the lead time arrival of the parts, which is related to the transport mode previously defined by the company. Closer to the operation, the jobs scheduling is done with the purpose of meeting MPS planning. Typically, these four problems - definition of the transportation mode, preparation of master production scheduling, definition of the time to purchase materials and job scheduling - are dealt with at different times and often in a deterministic way. In this work we evaluate the financial and operational impact of carrying out the planning in a deterministic and segregated way. In particular, we assess: (i) the impact of stochasticity and co-optimization of production planning and purchasing decisions and (ii) the feasibility of job scheduling. For (i) a two-stage stochastic optimization model is proposed that co-optimizes production volume decisions, purchase moments and transportation mode. For (ii) sequencing constraints of jobs are added through a heuristic and a mathematical programming model. Empirical assessments are made through two numerical experiments with realistic data from a discrete manufacturing company. For (i) we observed 7 percent cost reduction in the operation for the year 2018 (planning year) and approximately 3.5 percent for 5000 simulated scenarios (out-of-sample), comparing the proposed two-stage model with the procedure typically adopted in the industry. For (ii) we also observed a reduction of 8 percent in the 2018 operation cost, and 9.6 percent for 50 simulated scenarios (out-of-sample), in relation to the model proposed in (i) and 13 percent in the 2018 operation cost and 15.6 percent for 50 simulated scenarios (out-of-sample), compared to the typical industry model.

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In dieser Arbeit werden verschiedene Eigenschaften von dynamischen konvexen Risikomaßen für beschränkte Zufallsvariablen untersucht. Dabei gehen wir vor allem der Frage nach, wie die Risikobewertungen in verschiedenen Zeitpunkten von einander abhängen, und wie sich solche Zeitkonsistenzeigenschaften in der Dynamik der Penalty-Funktionen und Risikoprozesse widerspiegeln. Im Kapitel 2 widmen wir uns zunächst der starken Zeitkonsistenz und charakterisieren diese mithilfe von Akzeptanzmengen, Penalty-Funktionen und einer gemeinsamen Supermartingaleigenschaft des Risikoprozesses und seiner Penalty-Funktion. Die Charakterisierung durch Penalty-Funktionen liefert eine explizite Form der Doob- und der Riesz-Zerlegung des Prozesses der Penalty-Funktionen. Anschließend führen wir einen schwächeren Begriff der Zeitkonsistenz ein, den wir Besonnenheit nennen. In Analogie zu dem zeitkonsistenten Fall charakterisieren wir Besonnenheit durch Akzeptanzmengen, Penalty-Funktionen und eine bestimmte Supermartingaleigenschaft. Diese Supermartingaleigenschaft gilt allgemeiner für alle beschränkten adaptierten Prozesse, die sich ohne zusätzliches Risiko aufrechterhalten lassen. Wir nennen solche Prozesse nachhaltig und beschreiben Nachhaltigkeit durch eine gemeinsame Supermartingaleigenschaft des Prozesses und der schrittweisen Penalty-Funktionen. Dieses Resultat kann als eine verallgemeinerte optionale Zerlegung unter konvexen Restriktionen gesehen werden. Mithilfe der Supermartingaleigenschaft identifizieren wir das stark zeitkonsistente dynamische Risikomaß, das aus jedem beliebigen Risikomaß rekursiv konstruiert werden kann, als den kleinsten Prozeß, der nachhaltig ist und den Endverlust minimiert. Diese Beschreibung liefert ein neues Argument für den Einsatz von zeitkonsistenten Risikomaßen. Im Kapitel 3 diskutieren wir das asymptotische Verhalten von zeitkonsistenten und von besonnenen Risikomaßen hinsichtlich der asymptotischen Sicherheit und der asymptotischen Präzision. Im Kapitel 4 werden die allgemeinen Ergebnisse aus den Kapiteln 2 und 3 anhand des entropischen Risikomaßes und des Superhedging-Preisprozesses veranschaulicht. / In this thesis we study various properties of a dynamic convex risk measure for bounded random variables. The main subject is to investigate possible interdependence of conditional risk assessments at different times and the manifestation of these time consistency properties in the dynamics of corresponding penalty functions and risk processes. In Chapter 2 we focus first on the strong notion of time consistency and characterize it in terms of penalty functions, acceptance sets and a joint supermartingale property of the risk measure and its penalty function. The characterization in terms of penalty functions provides the explicit form of the Doob and of the Riesz decomposition of the penalty function process for a time consistent risk measure. Then we introduce and study a weaker notion of time consistency, that we call prudence. Similar to the time consistent case, we characterize prudent dynamic risk measures in terms of acceptance sets, of penalty functions and by a certain supermartingale property. This supermartingale property holds more generally for any bounded adapted process that can be upheld without any additional risk. We call such processes sustainable, and we give an equivalent characterization of sustainability in terms of a combined supermartingale property of a process and one-step penalty functions. This result can be viewed as a generalized optimal decomposition under convex constraints. The supermartingale property allows us to characterize the strongly time consistent risk measure arising from any dynamic risk measure via recursive construction as the smallest process that is sustainable and covers the final loss. Thus our discussion provides a new reason for using strongly time consistent risk measures. In Chapter 3 we discuss the limit behavior of time consistent and of prudent risk measures in terms of asymptotic safety and of asymptotic precision. In the final Chapter 4 we illustrate the general results of Chapter 2 and Chapter 3 by examples. In particular we study the entropic dynamic risk measure and the superhedging price process under convex constraints.

[pt] Esta tese é composta por quatro artigos que descrevem diferentes formas de inclusão de aversão a risco em problemas dinâmicos, ressaltando seus aspectos teóricos e consequências práticas envolvidas em técnicas de otimização sob incerteza aplicadas a problemas financeiros. O primeiro artigo propões uma interpretação econômica e analisa as consequencias práticas da consistência temporal, em que particular para o problema de seleção de portfólio. No segunfo artigo, também aplicado à seleção de portfólio, é proposto um modelo que considera empréstimo como variável de decisão e uma função convexa e linear por partes que representa a existência de diversos credores com diferentes limites de crédito e taxas de juros. A performance do modelo proposto é melhor que as aproximações existentes e garante otimalidade para a situação de vários credores. No terceiro artigo, desenvolve-se um modelo de emissão de títulos de dívida de uma empresa que seja financiar um conjunto pré-determinado de projetos. Trata-se de um modelo de otimização dinâmico sob incerteza que considera títulos pré e pós-fixados com diferentes maturidades e formas de amortização. As principais contribuições são o tratammento de um horizonte longuíssimo prazo através de uma estrutura híbrida dos cenários; a modelagem detalhada do pagamento de cupons e amortizações; o desenvolvimento de uma função objetivo multi-critério que reflete o trade-off entre risco-retorno além de outras medidas de performance financeiras como a taxa de alavancagem (razão passivos sobre ativos). No quarto artigo é desenvolvido um modelo de programação estocástica multi-estágio para obter a política ótima de caixa de uma empresa cujo custo de investimento e o custo da dívida são incertos e modelados em diferentes regimes. As contribuições são a extensão de metodologia de equilíbrio dual para um modelo estocástico; a proposição de uma regra de decisão baseada na estrutura de regime dos fatores de risco que aproxima de forma satisfatória o modelo original. / [en] This PhD Thesis is composed of four working papers, each one with a respective chapter on this thesis, with contributions on risk averse stochastic programming models. In particular, it focuses on analyzing the practical consequences of certain theoretical concepts of decision theory, finance and optimization. The first working paper analyzes the practical consequences and the economic interpretation of time consistent optimal policies, in particular for well known portfolio selection problem. The second paper has also a contribution to the portfolio selection literature. Indeed, we develop leverage optimal strategy considering a single-period debt with a piecewise linear borrowing cost function, which represents the actual situation faced by investors, and show a significant gap in comparison to the suboptimal solutions obtained by the usual linear approximation. Moreover, we develop a multistage extension where our cost function indirectly penalizes the excess of leverage, which is closely related to the contribution of the next working paper. The contribution of the third working paper is to penalize excess of leverage in a debt issuance multistage model that optimizes over several types of bonds with fixed or floating rate, different maturities and amortization patterns. For the sake of dealing with the curse of dimensionality of a long term problem, we divide the planning horizon into a detailed part at the beginning followed by a policy rule approximation for the remainder. Indeed, our approximation mitigates the end effects of a truncated model which is closely related to the contributions of the forth working paper. The forth paper develops a multistage model that seeks to obtain the optimal cash holding policy of a firm. The main contributions are a methodology to end effect treatment for a multistage model with infinite horizon and the development of a policy rule as approximation of the optimal solution.