Diese Dissertation analysiert BSDEs und PDEs mit singulären Endbedingungen, welche in Problemen der optimalen Portfolioliquidierung auftreten. In den vergangenen Jahren haben Portfolioliquidierungsprobleme in der Literatur zur Finanzmathematik große Aufmerksamkeit erhalten. Ihre wichtigste Eigenschaft ist die singuläre Endbedingung der durch die Liquidierungsbedingung induzierten Wertfunktion, welche eine singuläre Endbedingung der zugehörigen BSDE oder PDE impliziert. Diese Arbeit besteht aus drei Kapiteln. Das erste Kapitel analysiert ein Portfolioliquidierungsproblem für mehrere Wertpapiere mit sofortigem und anhaltendem Preiseinfluss und stochastischer Resilienz. Wir zeigen, dass die Wertfunktion durch eine mehrdimensionale BSRDE mit singulärer Endbedingung beschrieben werden kann. Wir weisen die Existenz einer Lösung dieser BSRDE nach und zeigen, dass diese durch eine Folge von Lösungen von BSRDEs mit endlicher und wachsender Endbedingung approximiert werden kann. Eine neue a priori-Abschätzung für die approximierenden BSRDEs wird für den Nachweis hergeleitet. Das zweite Kapitel betrachtet ein Portfolioliquidierungsproblem mit unbeschränkten Kostenkoeffizienten. Wir weisen die Existenz einer eindeutigen nichtnegativen Viskositätslösung der HJB-Gleichung nach. Das Existenzresultat basiert auf einem neuartigen Vergleichsprinzip für semi-stetige Viskositätssub-/-superlösungen für singuläre PDEs. Stetigkeit der Viskositätslösung ist hinreichend für das Verifikationsargument. Im dritten Kapitel untersuchen wir ein optimales Liquidierungsproblem unter Mehrdeutigkeit der Parameter des Preiseinflusses. In diesem Fall kann die Wertfunktion durch die Lösung einer semilinearen PDE mit superlinearem Gradienten beschrieben werden. Zuerst zeigen wir die Existenz einer Viskositätslösung indem wir unser Vergleichsprinzip für singuläre PDEs erweitern. Sodann weisen wir die Regularität mit einer asymptotischen Entwicklung der Lösung am Endzeitpunkt nach. / This dissertation analyzes BSDEs and PDEs with singular terminal condition arising in models of optimal portfolio liquidation. Portfolio liquidation problems have received considerable attention in the financial mathematics literature in recent years. Their main characteristic is the singular terminal condition of the value function induced by the liquidation constraint, which translates into a singular terminal state constraint on the associated BSDE or PDE. The dissertation consists of three chapters. The first chapter analyzes a multi-asset portfolio liquidation problem with instantaneous and persistent price impact and stochastic resilience. We show that the value function can be described by a multi-dimensional BSRDE with a singular terminal condition. We prove the existence of a solution to this BSRDE and show that it can be approximated by a sequence of the solutions to BSRDEs with finite increasing terminal condition. A novel a priori estimate for the approximating BSRDEs is established for the verification argument. The second chapter considers a portfolio liquidation problem with unbounded cost coefficients. We establish the existence of a unique nonnegative continuous viscosity solution to the HJB equation. The existence result is based on a novel comparison principle for semi-continuous viscosity sub-/supersolutions for singular PDEs. Continuity of the viscosity solution is enough to carry out the verification argument. The third chapter studies an optimal liquidation problem under ambiguity with respect to price impact parameters. In this case the value function can be characterized by the solution to a semilinear PDE with superlinear gradient. We first prove the existence of a solution in the viscosity sense by extending our comparison principle for singular PDEs. Higher regularity is then established using an asymptotic expansion of the solution at the terminal time.

Les problèmes de contrôle stochastique optimal à horizon fini forment une classe de problèmes de contrôle optimal où interviennent des processus stochastiques considérés sur un intervalle de temps borné. Tout comme beaucoup de problème de contrôle optimal, ces problèmes sont résolus en utilisant le principe de la programmation dynamique qui induit une équation aux dérivées partielles (EDP) appelée équation d'Hamilton-Jacobi-Bellman. Les méthodes basées sur la discrétisation de l’espace sous forme de grille, les méthodes probabilistes ou plus récemment les méthodes max-plus peuvent alors être utilisées pour résoudre cette équation. Cependant, le premier type de méthode est mis en défaut quand un espace à dimension grande est considéré à cause de la malédiction de la dimension tandis que le deuxième type de méthode ne permettait jusqu'ici que de résoudre des problèmes où la non linéarité de l'équation aux dérivées partielles par rapport à la Hessienne n'est pas trop forte. Quant au troisième type de méthode, il entraine une explosion de la complexité de la fonction valeur. Nous introduisons dans cette thèse deux nouveaux schémas probabilistes permettant d'agrandir la classe des problèmes pouvant être résolus par les méthodes probabilistes. L'une est adaptée aux EDP à coefficients bornés tandis que l'autre peut être appliqué aux EDP à coefficients bornés ou non bornés. Nous prouvons la convergence des deux schémas probabilistes et obtenons des estimées de l'erreur de convergence dans le cas d'EDP à coefficients bornés. Nous donnons également quelques résultats sur le comportement du deuxième schéma dans le cas d'EDP à coefficients non bornés. Ensuite, nous introduisons une méthode complètement nouvelle pour résoudre les problèmes de contrôle stochastique optimal à horizon fini que nous appelons la méthode max-plus probabiliste. Elle permet d'utiliser le caractère non linéaire des méthodes max-plus dans un contexte probabiliste tout en contrôlant la complexité de la fonction valeur. Une application au calcul du prix de sur-réplication d'une option dans un modèle de corrélation incertaine est donnée dans le cas d’un espace à dimension 2 et 5. / Stochastic optimal control problems with finite horizon are a class of optimal control problems where intervene stochastic processes in a bounded time. As many optimal control problems, they are often solved using a dynamic programming approach which results in a second order Partial Differential Equation (PDE) called the Hamilton-Jacobi-Bellman equation. Grid-based methods, probabilistic methods or more recently max-plus methods can be used then to solve this PDE. However, the first type of methods default in a space of high dimension because of the curse of dimensionality while the second type of methods allowed till now to solve only problems where the nonlinearity of the PDE with respect to the second order derivatives is not very high. As for the third type of method, it results in an explosion of the complexity of the value function. We introduce two new probabilistic schemes in order to enlarge the class of problems that can be solved with probabilistic methods. One is adapted to PDE with bounded coefficients while the other can be applied to PDE with bounded or unbounded coefficients. We prove the convergence of the two probabilistic scheme and obtain error estimates in the case of a PDE with bounded coefficients. We also give some results about the behavior of the second probabilistic scheme in the case of a PDE with unbounded coefficients. After that, we introduce a completely new type of method to solve stochastic optimal control problems with finite horizon that we call the max-plus probabilistic method. It allows to add the non linearity feature of max-plus methods to a probabilistic method while controlling the complexity of the value function. An application to the computation of the optimal super replication price of an option in an uncertain correlation model is given in a 5 dimensional space.

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Networked embedded control systems are present almost everywhere. A recent trend is to introduce wireless sensor networks in these systems, to take advantage of the added mobility and flexibility offered by wireless solutions. In such networks, the sensor observations are typically quantized and transmitted over noisy links. Concerning the problem of closed-loop control over such non-ideal communication channels, relatively few works have appeared so far. This thesis contributes to this field, by studying some fundamentally important problems in the design of joint source--channel coding and optimal control. The main part of the thesis is devoted to joint design of the coding and control for scalar linear plants, whose state feedbacks are transmitted over binary symmetric channels. The performance is measured by a finite-horizon linear quadratic cost function. The certainty equivalence property of the studied systems is utilized, since it simplifies the overall design by separating the estimation and the control problems. An iterative optimization algorithm for training the encoder--decoder pairs, taking channel errors into account in the quantizer design, is proposed. Monte Carlo simulations demonstrate promising improvements in performance compared to traditional approaches. Event-triggered control strategies are a promising solution to the problem of efficient utilization of communication resources. The basic idea is to let each control loop communicate only when necessary. Event-triggered and quantized control are combined for plants affected by rarely occurring disturbances. Numerical experiments show that it is possible to achieve good control performance with limited control actuation and sensor communication. / QC 20101109

The world bond market is nearly twice as large as the equity market. The goal of this dissertation is to study the dynamics of bond price. Among the liquidity risk, interest rate risk and default risk, this dissertation will focus on the liquidity risk and trading strategy. Under the mathematical frame of stochastic control, we model price setting in U.S. bond markets where dealers have multiple instruments to smooth inventory imbalances. The difficulty in obtaining the optimal trading strategy is that the optimal strategy and value function depend on each other, and the corresponding HJB equation is nonlinear. To solve this problem, we derived an approximate optimal explicit trading strategy. The result shows that this trading strategy is better than the benchmark central symmetric trading strategy.

<p dir="ltr">The future of the global economy lies in space. As the economic and scientific benefits from space become more accessible and apparent to the public, the demand for more spacecrafts will only increase. However, simply using the current space architecture to sustain any major activities past low Earth orbit is infeasible. The limiting factor of relying on ground operators via the Deep Space Network will blunt future growth in cislunar space traffic as the bandwidth is insufficient to satisfy the needs of every spacecraft in this domain. For this reason, spacecrafts must begin to operate autonomously or semi-autonomously for operators to be able to manage more missions at a given time. This thesis focuses on the guidance and navigation policies that could help vehicles such as logistical or resupply spacecrafts perform their rendezvous autonomously. It is found that using GNSS signals and Moon-based optical navigation has the potential to help spacecrafts perform autonomous orbit determination in near-Moon trajectories. The estimations are high enough quality such that a stochastic controller can use this navigation solution to confidently guide the spacecraft to a target within a tolerance before proximity operations commence. As the reliance on the ground is shifted away, spacecrafts would be able to operate in greater numbers outside of Earth's lower orbits, greatly assisting humanity's presence in space. </p>

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本研究探討面臨顯著背景風險（諸如核保等風險）金融機構之投資策略，考量加入國外投資風險下，該金融機構如何決定最適動態資產配置策略，為充分反映市場風險、匯率風險及核保風險，本研究以隨機方程式描述資產價值及核保經驗之變動，並以假想之人壽保險公司作為討論對象，預估未來現金流量並建構公司財務資訊相關之隨機模型，給定最低資本限制下，於指定投資期限內達到全期淨值(盈餘)最佳效用值為目標。本文依照給定之背景風險建構隨機控制模型，利用動態規劃法求出最適資產配置。結果顯示最適投資組合將由三項要素組成：1.極小化盈餘變化之變異數之部位；2.類似於短期投資組合策略之避險部位；以及3.用以規避背景風險之避險部位。因為模型複雜性之限制，以逼近馬可夫理論之數值方法計算最適投資策略。 / This paper analyzes the optimal asset allocation for insurers and investors who are required to cope with significant background risks due to underwriting uncertainties and interest rate risks among a set of stochastic investment opportunities. In order to hedge properly the country risks due to local volatile financial market, the foreign investment opportunities are included in the optimal portfolio decision. In this study, detailed formulation using the projected cash flows of a hypothetical life insurance company and its related stochastic phenomena are constructed. The insurers are assumed to maximize the expected discounted utility of their surplus over the investment horizon under the minimal capital requirement. Our problem is formulated as a stochastic control framework. According to the optimal solution, the optimal portfolio can be characterized by three components: a hedging component minimizing the variance of the change in surplus, a hedging component familiar to myopic portfolio rule, and a risk hedging component against the background risks. Since the explicit solutions cannot be achieved due to model complexity, the Markov chain approximation methods are employed to obtain the optimal control solutions in our numerical illustration.

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.

Die kumulative Dissertation beschäftigt sich mit stochastischen und deterministischen dynamischen Verkaufsmodellen für langlebige sowie verderbliche Güter. Die analysierten dynamischen Modelle sind durch die Möglichkeit der simultanen Variation von Preis und Werbung in stetiger Zeit charakterisiert und folgen den aktuellen Entwicklungen der Dynamischen Preissetzung. Dabei steht die Berücksichtigung und Analyse von (i) Zeitinhomogenitäten, (ii) Adoptionseffekten, (iii) Oligopolwettbewerb und (iv) der Risikoaversion des Entscheiders im Zentrum der Arbeit. Für die Spezialfälle isoelastischer und exponentieller Nachfrage in Verbindung mit isoelastischer Werbewirkung gelingt es explizite Lösungen der optimalen Preis- und Werbekontrollen herzuleiten. Die optimal gesteuerten Verkaufsprozesse können analytisch beschrieben und ausgewertet werden. Insbesondere werden neben erwarteten Preis- und Restbestandsentwicklungen auch assoziierte Gewinnverteilungen untersucht und Sensitivitätsresultate hergeleitet. Darüber hinaus wird analysiert unter welchen Bedingungen monopolistische Strategien sozial effizient sind und welche Besteuerungs- und Subventionsmechanismen geeignet sind um Effizienz herzustellen. Die Ergebnisse sind in sechs Artikel gefasst und bieten ökonomische Einsichten in verschiedene praktische Verkaufsanwendungen, speziell im Bereich des elektronischen Handels. / The cumulative dissertation deals with stochastic and deterministic dynamic sales models for durable as well as perishable products. The models analyzed are characterized by simultaneous dynamic pricing and advertising controls in continuous time and are in line with recent developments in dynamic pricing. They include the modeling of multi-dimensional decisions and take (i) time dependencies, (ii) adoption effects (iii), competitive settings and (iv) risk aversion, explicitly into account. For special cases with isoelastic demand functions as well as with exponential ones explicit solution formulas of the optimal pricing and advertising feedback controls are derived. Moreover, optimally controlled sales processes are analytically described. In particular, the distribution of profits, the expected evolution of prices as well as inventory levels are analyzed in detail and sensitivity results are obtained. Furthermore, we consider the question whether or not monopolistic policies are socially efficient; in special cases, we propose taxation/subsidy mechanisms to establish efficiency. The results are presented in six articles and provide economic insights into a variety of dynamic sales applications of the business world, especially in the area of e-commerce.