In this work, hybrid systems with regional dynamics are considered. These are systems where transitions between different dynamical regimes occur as the continuous state of the system reaches given switching surfaces. In particular, the attention is focused on the optimal control problem associated with such systems. More precisely, given a specific cost function, the goal is to determine the optimal path of going from a given starting point to a fixed final state during an a priori specified time horizon. The key characteristic of the approach presented in this thesis is a hierarchical decomposition of the hybrid optimal control problem, yielding to a framework which allows a solution on different levels of control. On the highest level of abstraction, the regional structure of the state space is taken into account and a discrete representation of the connections between the different regions provides global accessibility relations between regions. These are used on a lower level of control to formulate the main theorem of this work, namely, the Hybrid Bellman Equation for multimodal systems, which, in fact, provides a characterization of global optimality, given an upper bound on the number of transitions along a hybrid trajectory. Not surprisingly, the optimal solution is hybrid in nature, in that it depends on not only the continuous control signals, but also on discrete decisions as to what domains the system's continuous state should go through in the first place. The main benefit with the proposed approach lies in the fact that a hierarchical Dynamic Programming algorithm can be used to representing both a theoretical characterization of the hybrid solution's structural composition and, from a more application-driven point of view, a numerically implementable calculation rule yielding to globally optimal solutions in a regional dynamics framework. The operation of the recursive algorithm is highlighted by the consideration of numerous examples, among them, a heterogeneous multi-agent problem.

Cette thèse se développe sur deux fronts: nous nous concentrons sur les méthodes numériques des problèmes de contrôle optimal, en particulier sur le Principe de la Programmation Dynamique et sur le Model Predictive Control (MPC) et nous présentons des applications de techniques de contrôle en biologie. Dans la première partie, nous considérons l'approximation d'un problème de contrôle optimal avec horizon infini, qui combine une première étape, basée sur MPC permettant d'obtenir rapidement une bonne approximation de la trajectoire optimal, et une seconde étape, dans la quelle l¿équation de Bellman est résolue dans un voisinage de la trajectoire de référence. De cette façon, on peux réduire une grande partie de la taille du domaine dans lequel on résout l¿équation de Bellman et diminuer la complexité du calcul. Le deuxième sujet est le contrôle des méthodes Level Set: on considère un problème de contrôle optimal, dans lequel la dynamique est donnée par la propagation d'un graphe à une dimension, contrôlé par la vitesse normale. Un état finale est fixé, l'objectif étant de le rejoindre en minimisant une fonction coût appropriée. On utilise la programmation dynamique grâce à une réduction d'ordre de l'équation utilisant la Proper Orthogonal Decomposition. La deuxième partie est dédiée à l'application des méthodes de contrôle en biologie. On présente un modèle décrit par une équation aux dérivées partielles qui modélise l'évolution d'une population de cellules tumorales. On analyse les caractéristiques du modèle et on formule et résout numériquement un problème de contrôle optimal concernant ce modèle, où le contrôle représente la quantité du médicament administrée. / This thesis is divided in two parts: in the first part we focus on numerical methods for optimal control problems, in particular on the Dynamic Programming Principle and on Model Predictive Control (MPC), in the second part we present some applications of the control techniques in biology. In the first part of the thesis, we consider the approximation of an optimal control problem with an infinite horizon, which combines a first step based on MPC, to obtain a fast but rough approximation of the optimal trajectory and a second step where we solve the Bellman equation in a neighborhood of the reference trajectory. In this way, we can reduce the size of the domain in which the Bellman equation can be solved and so the computational complexity is reduced as well. The second topic of this thesis is the control of the Level Set methods: we consider an optimal control, in which the dynamics is given by the propagation of a one dimensional graph, which is controlled by the normal velocity. A final state is fixed and the aim is to reach the trajectory chosen as a target minimizing an appropriate cost functional. To apply the Dynamic Programming approach we firstly reduce the size of the system using the Proper Orthogonal Decomposition. The second part of the thesis is devoted to the application of control methods in biology. We present a model described by a partial differential equation that models the evolution of a population of tumor cells. We analyze the mathematical and biological features of the model. Then we formulate an optimal control problem for this model and we solve it numerically.

Portfolio selection has always been a fundamental challenge in the field of finance and captured the attention of researchers in the financial area. Merton's portfolio problem is an optimization problem in finance and aims to maximize an investor's portfolio. This thesis studies Merton's Optimal Investment-Consumption Problem under the Jourdain--Sbai stochastic volatility model and seeks to maximize the expected discounted utility of consumption and terminal wealth. The results of our study can be split into three main parts. First, we derived the Hamilton--Jacobi--Bellman equation related to our stochastic optimal control problem.  Second, we simulated the optimal controls, which are the weight of the risky asset and consumption. This has been done for all the three models within the scope of the Jourdain--Sbai model: Quadratic Gaussian, Stein & Stein, and Scott's model. Finally, we developed the system of equations after applying the Crank-Nicolson numerical scheme when solving our HJB partial differential equation.

L'objectif de cette thèse est de développer la théorie du contrôle stochastique et ses applications en dynamique des populations. D'un point de vue théorique, nous présentons l'étude de problèmes de contrôle stochastique à horizon fini sur des processus de diffusion, de branchement non linéaire et de branchement-diffusion. Dans chacun des cas, nous raisonnons par la méthode de la programmation dynamique en veillant à démontrer soigneusement un argument de conditionnement analogue à la propriété de Markov forte pour les processus contrôlés. Le principe de la programmation dynamique nous permet alors de prouver que la fonction valeur est solution (régulière ou de viscosité) de l'équation de Hamilton-Jacobi-Bellman correspondante. Dans le cas régulier, nous identifions également un contrôle optimal markovien par un théorème de vérification. Du point de vue des applications, nous nous intéressons à la modélisation mathématique du cancer et de ses stratégies thérapeutiques. Plus précisément, nous construisons un modèle hybride de croissance de tumeur qui rend compte du rôle fondamental de l'acidité dans l'évolution de la maladie. Les cibles de la thérapie apparaissent explicitement comme paramètres du modèle afin de pouvoir l'utiliser comme support d'évaluation de stratégies thérapeutiques. / The main objective of this thesis is to develop stochastic control theory and applications to population dynamics. From a theoritical point of view, we study finite horizon stochastic control problems on diffusion processes, nonlinear branching processes and branching diffusion processes. In each case we establish a dynamic programmic principle by carefully proving a conditioning argument similar to the strong Markov property for controlled processes. Then we deduce that the value function is a (viscosity or regular) solution of the associated Hamilton-Jacobi-Bellman equation. In the regular case, we further identify an optimal control in the class of markovian strategies thanks to a verification theorem. From a pratical point of view, we are interested in mathematical modelling of cancer growth and treatment. More precisely, we build a hybrid model of tumor growth taking into account the essential role of acidity. Therapeutic targets appear explicitly as model parameters in order to be able to evaluate treatment strategies.

Economic uncertainty may affect significantly people’s behavior and hence macroeconomic variables. It is thus important to understand how people behave in presence of different kinds of economic risk. The present dissertation focuses therefore on the impact of the uncertainty in capital and labor income on the individual saving behavior. The underlying uncertain variables are here modeled as stochastic processes that each obey a specific stochastic differential equation, where uncertainty stems either from Poisson or Lévy processes. The results on the optimal behavior are derived by maximizing the individual expected lifetime utility. The first chapter is concerned with the necessary mathematical tools, the change-of-variables formula and the Hamilton-Jacobi-Bellman equation under Poisson uncertainty. We extend their possible field of application in order make them appropriate for the analysis of the dynamic stochastic optimization problems occurring in the following chapters and elsewhere. The second chapter considers an optimum-saving problem with labor income, where capital risk stems from asset prices that follow geometric L´evy processes. Chapter 3, finally, studies the optimal saving behavior if agents face not only risk but also uncertain spells of unemployment. To this end, we turn back to Poisson processes, which here are used to model properly the separation and matching process.

Problems involving optimal sequential making in uncertain dynamic systems arise in domains such as engineering, science and economics. Such problems can often be cast in the framework of Markov Decision Process (MDP). Solving an MDP requires computing the optimal value function and the optimal policy. The idea of dynamic programming (DP) and the Bellman equation (BE) are at the heart of solution methods. The three important exact DP methods are value iteration, policy iteration and linear programming. The exact DP methods compute the optimal value function and the optimal policy. However, the exact DP methods are inadequate in practice because the state space is often large and in practice, one might have to resort to approximate methods that compute sub-optimal policies. Further, in certain cases, the system observations are known only in the form of noisy samples and we need to design algorithms that learn from these samples. In this thesis we study interesting theoretical questions pertaining to approximate and learning algorithms, and also present an interesting application of MDPs in the domain of crowd sourcing. Approximate Dynamic Programming (ADP) methods handle the issue of large state space by computing an approximate value function and/or a sub-optimal policy. In this thesis, we are concerned with conditions that result in provably good policies. Motivated by the limitations of the PBE in the conventional linear algebra, we study the PBE in the (min, +) linear algebra. It is a well known fact that deterministic optimal control problems with cost/reward criterion are (min, +)/(max, +) linear and ADP methods have been developed for such systems in literature. However, it is straightforward to show that infinite horizon discounted reward/cost MDPs are neither (min, +) nor (max, +) linear. We develop novel ADP schemes namely the Approximate Q Iteration (AQI) and Variational Approximate Q Iteration (VAQI), where the approximate solution is a (min, +) linear combination of a set of basis functions whose span constitutes a subsemimodule. We show that the new ADP methods are convergent and we present a bound on the performance of the sub-optimal policy. The Approximate Linear Program (ALP) makes use of linear function approximation (LFA) and offers theoretical performance guarantees. Nevertheless, the ALP is difficult to solve due to the presence of a large number of constraints and in practice, a reduced linear program (RLP) is solved instead. The RLP has a tractable number of constraints sampled from the original constraints of the ALP. Though the RLP is known to perform well in experiments, theoretical guarantees are available only for a specific RLP obtained under idealized assumptions. In this thesis, we generalize the RLP to define a generalized reduced linear program (GRLP) which has a tractable number of constraints that are obtained as positive linear combinations of the original constraints of the ALP. The main contribution here is the novel theoretical framework developed to obtain error bounds for any given GRLP. Reinforcement Learning (RL) algorithms can be viewed as sample trajectory based solution methods for solving MDPs. Typically, RL algorithms that make use of stochastic approximation (SA) are iterative schemes taking small steps towards the desired value at each iteration. Actor-Critic algorithms form an important sub-class of RL algorithms, wherein, the critic is responsible for policy evaluation and the actor is responsible for policy improvement. The actor and critic iterations have deferent step-size schedules, in particular, the step-sizes used by the actor updates have to be generally much smaller than those used by the critic updates. Such SA schemes that use deferent step-size schedules for deferent sets of iterates are known as multitimescale stochastic approximation schemes. One of the most important conditions required to ensure the convergence of the iterates of a multi-timescale SA scheme is that the iterates need to be stable, i.e., they should be uniformly bounded almost surely. However, the conditions that imply the stability of the iterates in a multi-timescale SA scheme have not been well established. In this thesis, we provide veritable conditions that imply stability of two timescale stochastic approximation schemes. As an example, we also demonstrate that the stability of a widely used actor-critic RL algorithm follows from our analysis. Crowd sourcing (crowd) is a new mode of organizing work in multiple groups of smaller chunks of tasks and outsourcing them to a distributed and large group of people in the form of an open call. Recently, crowd sourcing has become a major pool for human intelligence tasks (HITs) such as image labeling, form digitization, natural language processing, machine translation evaluation and user surveys. Large organizations/requesters are increasingly interested in crowd sourcing the HITs generated out of their internal requirements. Task starvation leads to huge variation in the completion times of the tasks posted on to the crowd. This is an issue for frequent requesters desiring predictability in the completion times of tasks specified in terms of percentage of tasks completed within a stipulated amount of time. An important task attribute that affects the completion time of a task is its price. However, a pricing policy that does not take the dynamics of the crowd into account might fail to achieve the desired predictability in completion times. Here, we make use of the MDP framework to compute a pricing policy that achieves predictable completion times in simulations as well as real world experiments.

In dieser Arbeit wird erstmalig der Einfluss stochastischer Schwankungen auf endoreversible Modelle untersucht. Hierfür wird die Novikov-Maschine mit drei verschieden Wärmetransportgesetzen (Newton, Fourier, asymmetrisch) betrachtet. Während die maximale verrichtete Arbeit und der dazugehörige Wirkungsgrad recht einfach im Falle konstanter Wärmebadtemperaturen hergeleitet werden können, ändern sich dies, falls die Temperaturen stochastisch fluktuieren können. Im letzteren Fall muss die stochastische optimale Kontrolltheorie genutzt werden, um das Maximum der zu erwartenden Arbeit und die dazugehörige Kontrollstrategie zu ermitteln. Im Allgemeinen kann die Lösung derartiger Probleme auf eine nichtlineare, partielle Differentialgleichung, welche an eine Optimierung gekoppelt ist, zurückgeführt werden. Diese Gleichung wird stochastische Hamilton-Jacobi-Bellman-Gleichung genannt. Allerdings können, wie in dieser Arbeit dargestellt, die Berechnungen vereinfacht werden, wenn man annimmt, dass die Fluktuationen unabhängig von der betrachteten Kontrollvariablen sind. In diesem Fall zeigen analytische Betrachtungen, dass die Gleichungen für die verrichtete Arbeit and den Wirkungsgrad ihre ursprüngliche Form behalten, aber manche Terme müssen durch entsprechende Zeitmittel bzw. Erwartungswerte ersetzt werden, jeweils abhängig von der betrachteten Art der Kontrolle. Basierend auf einer Analyse der Leistungsparameter im Falle einer Gleichverteilung der heißen Temperatur der Novikov-Maschine können Schlussfolgerungen auf deren Monotonieverhalten gezogen werden. Der Vergleich verschiedener, zeitunabhängiger, symmetrischer Verteilungen führt zu einer bis dato unbekannten Erweiterung des Curzon-Ahlborn-Wirkungsgrades im Falle kleiner Schwankungen. Weiterhin wird eine Analyse einer Novikov-Maschine mit asymmetrischen Wärmetransport, bei der das Verhalten der heißen Temperatur durch einen Ornstein-Uhlenbeck-Prozess beschrieben wird, durchgeführt. Abschließend wird eine Novikov-Maschine mit Fourierscher Wärmeleitung, bei der die Dynamik der heißen Temperatur von der Kontrollvariable abhängt, betrachtet. Durch das Lösen der Hamilton-Jacobi-Bellman-Gleichung können neuartige Schlussfolgerungen gezogen werden, wie derartige Systeme optimal zu steuern sind. / In this thesis, the influence of stochastic fluctuations on the performance of endoreversible engines is investigated for the first time. For this, a Novikov-engine with three different heat transport laws (Newtonian, Fourier, asymmetric) is considered. While the maximum work output and corresponding efficiency can be deduced easily in the case of constant heat bath temperatures, this changes, if these temperatures are allowed to fluctuate stochastically. In the latter case, stochastic optimal control theory has to be used to find the maximum of the expected work output and the corresponding control policy. In general, solving such problems leads to a non-linear, partial differential equation coupled to an optimization, called the stochastic Hamilton-Jacobi-Bellman equation. However, as presented in this thesis, calculations can be simplified, if one assumes that the fluctuations are independent of the considered control variable. In this case, analytic considerations show that the equations for performance measures like work output and efficiency keep their original form, but terms have to be replaced by appropriate time averages and expectation values, depending on the considered control type. Based on an analysis of the performance measures in the case of a uniform distribution of the hot temperature of the Novikov engine, conclusions on their monotonicity behavior are drawn. The comparison of several, time independent, symmetric distributions reveals a to date unknown extension to the Curzon-Ahlborn efficiency in the case of small fluctuations. Furthermore, an analysis of a Novikov engine with asymmetric heat transport, where the behavior of the hot temperature is described by an Ornstein-Uhlenbeck process, is performed. Finally, a Novikov engine with Fourier heat transport is considered, where the dynamics of the hot temperature depends on the control variable. By solving the corresponding Hamilton-Jacobi-Bellman equation, new conclusions how to optimally control such systems are drawn.

Dans cette thèse, on étudie le problème de la consommation et de l’investissement pour le marché financier de "spread" (différence entre deux actifs) défini par le processus Ornstein-Uhlenbeck (OU). Ce manuscrit se compose de sept chapitres. Le chapitre 1 présente une revue générale de la littérature et un bref résumé des principaux résultats obtenus dans cetravail où différentes fonctions d’utilité sont considérées. Dans le chapitre 2, on étudie la stratégie optimale de consommation / investissement pour les fonctions puissances d’utilité pour un intervalle de temps réduit a 0 < t < T < T0. Dans ce chapitre, nous étudions l’équation de Hamilton–Jacobi–Bellman (HJB) par la méthode de Feynman - Kac (FK). L’approximation numérique de la solution de l’équation de HJB est étudiée et le taux de convergence est établi. Il s’avère que dans ce cas, le taux de convergencedu schéma numérique est super–géométrique, c’est-à-dire plus rapide que tous ceux géométriques. Les principaux théorèmes sont énoncés et des preuves de l’existence et de l’unicité de la solution sont données. Un théorème de vérification spécial pour ce cas des fonctions puissances est montré. Le chapitre 3 étend notre approche au chapitre précédent à la stratégie de consommation/investissement optimale pour tout intervalle de temps pour les fonctions puissances d’utilité où l’exposant γ doit être inférieur à 1/4. Dans le chapitre 4, on résout le problème optimal de consommation/investissement pour les fonctions logarithmiques d’utilité dans le cadre du processus OU multidimensionnel en se basant sur la méthode de programmation dynamique stochastique. En outre, on montre un théorème de vérification spécial pour ce cas. Le théorème d’existence et d’unicité pour la solution classique de l’équation de HJB sous forme explicite est également démontré. En conséquence, les stratégies financières optimales sont construites. Quelques exemples sont donnés pour les cas scalaires et pour les cas multivariés à volatilité diagonale. Le modèle de volatilité stochastique est considéré dans le chapitre 5 comme une extension du chapitre précédent des fonctions logarithmiques d’utilité. Le chapitre 6 propose des résultats et des théorèmes auxiliaires nécessaires au travail.Le chapitre 7 fournit des simulations numériques pour les fonctions puissances et logarithmiques d’utilité. La valeur du point fixe h de l’application de FK pour les fonctions puissances d’utilité est présentée. Nous comparons les stratégies optimales pour différents paramètres à travers des simulations numériques. La valeur du portefeuille pour les fonctions logarithmiques d’utilité est également obtenue. Enfin, nous concluons nos travaux et présentons nos perspectives dans le chapitre 8. / This thesis studies the consumption/investment problem for the spread financial market defined by the Ornstein–Uhlenbeck (OU) process. Recently, the OU process has been used as a proper financial model to reflect underlying prices of assets. The thesis consists of 8 Chapters. Chapter 1 presents a general literature review and a short view of the main results obtained in this work where different utility functions have been considered. The optimal consumption/investment strategy are studied in Chapter 2 for the power utility functions for small time interval, that 0 < t < T < T0. Main theorems have been stated and the existence and uniqueness of the solution has been proven. Numeric approximation for the solution of the HJB equation has been studied and the convergence rate has been established. In this case, the convergence rate for the numerical scheme is super geometrical, i.e., more rapid than any geometrical ones. A special verification theorem for this case has been shown. In this chapter, we have studied the Hamilton–Jacobi–Bellman (HJB) equation through the Feynman–Kac (FK) method. The existence and uniqueness theorem for the classical solution for the HJB equation has been shown. Chapter 3 extended our approach from the previous chapter of the optimal consumption/investment strategies for the power utility functions for any time interval where the power utility coefficient γ should be less than 1/4. Chapter 4 addressed the optimal consumption/investment problem for logarithmic utility functions for multivariate OU process in the base of the stochastic dynamical programming method. As well it has been shown a special verification theorem for this case. It has been demonstrated the existence and uniqueness theorem for the classical solution for the HJB equation in explicit form. As a consequence the optimal financial strategies were constructed. Some examples have been stated for a scalar case and for a multivariate case with diagonal volatility. Stochastic volatility markets has been considered in Chapter 5 as an extension for the previous chapter of optimization problem for the logarithmic utility functions. Chapter 6 proposed some auxiliary results and theorems that are necessary for the work. Numerical simulations has been provided in Chapter 7 for power and logarithmic utility functions. The fixed point value h for power utility has been presented. We study the constructed strategies by numerical simulations for different parameters. The value function for the logarithmic utilities has been shown too. Finally, Chapter 8 reflected the results and possible limitations or solutions

In dieser Arbeit wird erstmalig der Einfluss stochastischer Schwankungen auf endoreversible Modelle untersucht. Hierfür wird die Novikov-Maschine mit drei verschieden Wärmetransportgesetzen (Newton, Fourier, asymmetrisch) betrachtet. Während die maximale verrichtete Arbeit und der dazugehörige Wirkungsgrad recht einfach im Falle konstanter Wärmebadtemperaturen hergeleitet werden können, ändern sich dies, falls die Temperaturen stochastisch fluktuieren können. Im letzteren Fall muss die stochastische optimale Kontrolltheorie genutzt werden, um das Maximum der zu erwartenden Arbeit und die dazugehörige Kontrollstrategie zu ermitteln. Im Allgemeinen kann die Lösung derartiger Probleme auf eine nichtlineare, partielle Differentialgleichung, welche an eine Optimierung gekoppelt ist, zurückgeführt werden. Diese Gleichung wird stochastische Hamilton-Jacobi-Bellman-Gleichung genannt. Allerdings können, wie in dieser Arbeit dargestellt, die Berechnungen vereinfacht werden, wenn man annimmt, dass die Fluktuationen unabhängig von der betrachteten Kontrollvariablen sind. In diesem Fall zeigen analytische Betrachtungen, dass die Gleichungen für die verrichtete Arbeit and den Wirkungsgrad ihre ursprüngliche Form behalten, aber manche Terme müssen durch entsprechende Zeitmittel bzw. Erwartungswerte ersetzt werden, jeweils abhängig von der betrachteten Art der Kontrolle. Basierend auf einer Analyse der Leistungsparameter im Falle einer Gleichverteilung der heißen Temperatur der Novikov-Maschine können Schlussfolgerungen auf deren Monotonieverhalten gezogen werden. Der Vergleich verschiedener, zeitunabhängiger, symmetrischer Verteilungen führt zu einer bis dato unbekannten Erweiterung des Curzon-Ahlborn-Wirkungsgrades im Falle kleiner Schwankungen. Weiterhin wird eine Analyse einer Novikov-Maschine mit asymmetrischen Wärmetransport, bei der das Verhalten der heißen Temperatur durch einen Ornstein-Uhlenbeck-Prozess beschrieben wird, durchgeführt. Abschließend wird eine Novikov-Maschine mit Fourierscher Wärmeleitung, bei der die Dynamik der heißen Temperatur von der Kontrollvariable abhängt, betrachtet. Durch das Lösen der Hamilton-Jacobi-Bellman-Gleichung können neuartige Schlussfolgerungen gezogen werden, wie derartige Systeme optimal zu steuern sind. / In this thesis, the influence of stochastic fluctuations on the performance of endoreversible engines is investigated for the first time. For this, a Novikov-engine with three different heat transport laws (Newtonian, Fourier, asymmetric) is considered. While the maximum work output and corresponding efficiency can be deduced easily in the case of constant heat bath temperatures, this changes, if these temperatures are allowed to fluctuate stochastically. In the latter case, stochastic optimal control theory has to be used to find the maximum of the expected work output and the corresponding control policy. In general, solving such problems leads to a non-linear, partial differential equation coupled to an optimization, called the stochastic Hamilton-Jacobi-Bellman equation. However, as presented in this thesis, calculations can be simplified, if one assumes that the fluctuations are independent of the considered control variable. In this case, analytic considerations show that the equations for performance measures like work output and efficiency keep their original form, but terms have to be replaced by appropriate time averages and expectation values, depending on the considered control type. Based on an analysis of the performance measures in the case of a uniform distribution of the hot temperature of the Novikov engine, conclusions on their monotonicity behavior are drawn. The comparison of several, time independent, symmetric distributions reveals a to date unknown extension to the Curzon-Ahlborn efficiency in the case of small fluctuations. Furthermore, an analysis of a Novikov engine with asymmetric heat transport, where the behavior of the hot temperature is described by an Ornstein-Uhlenbeck process, is performed. Finally, a Novikov engine with Fourier heat transport is considered, where the dynamics of the hot temperature depends on the control variable. By solving the corresponding Hamilton-Jacobi-Bellman equation, new conclusions how to optimally control such systems are drawn.

Economic uncertainty may affect significantly people’s behavior and hence macroeconomic variables. It is thus important to understand how people behave in presence of different kinds of economic risk. The present dissertation focuses therefore on the impact of the uncertainty in capital and labor income on the individual saving behavior. The underlying uncertain variables are here modeled as stochastic processes that each obey a specific stochastic differential equation, where uncertainty stems either from Poisson or Lévy processes. The results on the optimal behavior are derived by maximizing the individual expected lifetime utility. The first chapter is concerned with the necessary mathematical tools, the change-of-variables formula and the Hamilton-Jacobi-Bellman equation under Poisson uncertainty. We extend their possible field of application in order make them appropriate for the analysis of the dynamic stochastic optimization problems occurring in the following chapters and elsewhere. The second chapter considers an optimum-saving problem with labor income, where capital risk stems from asset prices that follow geometric L´evy processes. Chapter 3, finally, studies the optimal saving behavior if agents face not only risk but also uncertain spells of unemployment. To this end, we turn back to Poisson processes, which here are used to model properly the separation and matching process.