Motiviert durch das Problem der optimalen Strategie beim Handel einer großen Aktienposition, behandelt diese Arbeit ein stochastisches Kontrollproblem mit zwei besonderen Eigenschaften. Zum einen wird davon ausgegangen, dass das Kontrollproblem eine exponentielle Verzögerung in der Kontrollvariablen beinhaltet, zum anderen nehmen wir an, dass die Koeffizienten des Kontrollproblems linear in der Kontrollvariablen sind. Wir erhalten ein degeneriertes stochastisches Kontrollproblem, dessen Lösung - sofern sie existiert - Bang-Bang-Charakter hat. Die resultierende Unstetigkeit der optimalen Kontrolle führt dazu, dass die Existenz einer optimalen Lösung nicht selbstverständlich ist und bewiesen werden muss. Es wird eine Folge von stochastischen Kontrollproblemen mit Zustandsprozessen konstruiert, deren jeweilige Diffusionsmatrix invertierbar ist und die ursprüngliche degenerierte Diffusionsmatrix approximiert. Außerdem stellen die Kostenfunktionale der Folge eine konvexe Approximation des ursprünglichen linearen Kostenfunktionals dar. Um die Konvergenz der Lösungen dieser Folge zu zeigen, stellen wir die Kontrollprobleme in Form von stochastischen Vorwärts-Rückwärts-Differential-gleichungen (FBSDEs) dar. Wir zeigen, dass die zu der konstruierten Folge von Kontrollproblemen gehörigen Lösungen der Vorwärts-Rückwärtsgleichungen – zumindest für eine Teilfolge - in Verteilung konvergieren. Mit Hilfe einer Konvexitätsannahme der Koeffizienten ist es möglich, einen Kontroll-prozess auf einem passenden Wahrscheinlichkeitsraum zu konstruieren, der optimal für das ursprüngliche stochastische Kontrollproblem ist. Neben der damit bewiesenen Existenz einer optimalen (Bang-Bang-) Lösung, wird damit auch eine glatte Approximation der unstetigen Bang-Bang-Lösung erreicht, welche man für die numerische Approximation des Problems verwenden kann. Die Ergebnisse werden schließlich dann in Form von numerischen Simulationen auf das Problem der optimalen Handels¬ausführung angewendet. / Motivated by the problem of how to optimally execute a large stock position, this thesis considers a stochastic control problem with two special properties. First, the control problem has an exponential delay in the control variable, and so the present value of the state process depends on the moving average of past control decisions. Second, the coefficients are assumed to be linear in the control variable. It is shown that a control problem with these properties generates a mathematically challenging problem. Specifically, it becomes a stochastic control problem whose solution (if one exists) has a bang-bang nature. The resulting discontinuity of the optimal solution creates difficulties in proving the existence of an optimal solution and in solving the problem with numerical methods. A sequence of stochastic control problems with state processes is constructed, whose diffusion matrices are invertible and approximate the original degenerate diffusion matrix. The cost functionals of the sequence of control problems are convex approximations of the original linear cost functional. To prove the convergence of the solutions, the control problems are written in the form of forward-backward stochastic differential equations (FBSDEs). It is then shown that the solutions of the FBSDEs corresponding to the constructed sequence of control problems converge in law, at least along a subsequence. By assuming convexity of the coefficients, it is then possible to construct from this limit an admissible control process which, for an appropriate reference stochastic system, is optimal for our original stochastic control problem. In addition to proving the existence of an optimal (bang-bang) solution, we obtain a smooth approximation of the discontinuous optimal bang-bang solution, which can be used for the numerical solution of the problem. These results are then applied to the optimal execution problem in form of numerical simulations.

Au cours de cette thèse, nous nous sommes d'abord intéressés à l'optimisation des ressources et à la modulation adaptative dans les systèmes SC-FDMA (Single Carrier Frequency Division Multiple Access). Ce problème d'optimisation est combinatoire à complexité de calcul exponentielle. Afin de pallier à cette difficulté, nous avons utilisé la théorie de la dualité canonique, grâce à laquelle, la complexité du problème d'optimisation devient polynômiale et cela en constitue une amélioration remarquable. L'approche proposée est très proche de la solution optimale. Nous avons ensuite étudié la problématique complexe de l'allocation de ressources pour le "Streaming Vidéo" dans les réseaux sans fil, où il est nécessaire d'assurer une transmission vidéo de haute qualité en présence de canaux et de brouillages variables au cours du temps. Dans ce contexte, nous avons proposé une nouvelle méthode d'allocation de puissance conjointement à l'adaptation du débit vidéo. Pour ce faire, nous avons adopté une approche de la théorie de contrôle, intitulée "Risk-Sensitive Control". Nous avons dédié la troisième partie de la thèse à la conception d'une nouvelle stratégie "best-M" pour le renvoi du CQI (Channel Quality Indicator) pour les systèmes multi-utilisateurs et multi-porteuses. En générale, l'erreur d'estimation du CQI ainsi que son délai de renvoi sont gérés au niveau de la station de base. Notre nouvelle stratégie "best-M" suppose que la gestion de ces problèmes est confiée aux utilisateurs. De ce fait, la performance du système se trouve améliorée sans que son débit de signalisation ne soit augmenté en voix montante. / Adaptive resource allocation in wireless communication systems is crucial in order to support the diverse QoS needs of the services and optimize resource utilization. The design of resource allocation schemes should consider the service type for which it is intended. Moreover, due to feedback delay and channel estimation error, the Channel Quality Indicator (CQI) reported to the transmitter may not be a perfect measure of the channel quality and its use for resource allocation may severely degrade the systems performance. In this thesis, we study resource allocation and CQI reporting for wireless networks while taking the aforementioned factors into consideration. First, we consider resource allocation and adaptive modulation in uplink SC-FDMA systems. This is a combinatorial problem whose optimal solution is exponentially complex. We use canonical duality theory to derive a polynomial complexity resource allocation algorithm that provides a nearly optimal solution to the problem. Then, we focus on resource allocation for video streaming in wireless networks with time-varying interference. To this end, by using risk-sensitive control approach, we develop a cross-layer optimization framework that performs power control at the PHY/MAC layer and rate adaptation at the APPLICATION layer jointly and provides fairness among nodes. Finally, by using stochastic control and game theory, we design a robust best-M CQI reporting scheme for multi-carrier and multi-user systems which takes into account the impact of feedback delay and error in CQI computation. Performing resource allocation on the basis of the proposed CQI reporting can significantly improve the system performance.

Les prévisions relatives trafic de données au sein des systèmes de communications sans-fil suggèrent une croissance exponentielle, principalement alimentée par l’essor de transferts vidéo mobiles. Etant donné la nature soudaine et fluctuante des demandes de transfert vidéo, il faut dès à présent réfléchir à de nouveaux algorithmes d’allocation de ressources performants. En effet, les algorithmes en couche physique traditionnels, qui réalisent de l’allocation de ressources sous l’hypothèse classique que les transmetteurs sont toujours saturés avec des bits d’information, risquent à l’avenir de s’avérer inefficients. Pour cette raison, les algorithmes de demain se doivent d’être dynamiques, dans le sens où ils seront capables de prendre en compte la nature stochastique des fluctuations du trafic de données et qu’ils intégreront des informations issus de processus de couches supérieures.L’idée centrale de cette thèse est de développer des algorithmes, travaillant avec des informations issues de la couche PHY et de la couche NET, dans un scénario Multi-cells et MIMO (Multiple Inputs, Multiple Outputs).Plus particulièrement, nous considérons un réseau de stations de base (BS) équipés avec plusieurs antennes, chargés de servir plusieurs terminaux mobiles équipés d’une seule antenne (UT) dans leurs cellules respectives. Ce qui nous différencie des travaux précédents, c’est que nous tenons compte de l’aléa avec lequel des demandes de transferts peuvent arriver et que, pour cette raison, nous modélisons la formation de queue de données au niveau des stations de base. Dans cette disposition, nous développons plusieurs algorithmes multicouches, réalisant de l’allocation de ressources décentralisée, et ce, dans une optique d’efficacité énergétique. En particulier, il s’agit ici de réaliser des algorithmes réalisant du beamforming de façon décentralisée et capables de contrôler des fluctuations de trafic, des algorithmes optimisant l’efficacité énergétique sous une contrainte de qualité de service moyenne, des algorithmes de planification décentralisés dans des scénarios multi-cellulaires. Dans cette perspective, nous choisissons de recourir non seulement à des outils d’optimisation de la théorie de Lyapunov, mais également à la théorie des matrices aléatoires et à la théorie du contrôle stochastique. / Future wireless communication systems are expected to see an explosion in the wireless traffic which is mainly fueled by mobile video traffic. Due to the time varying and bursty nature of video traffic, wireless systems will see a widerrange of fluctuations in their traffic patterns. Therefore, traditional physical layer based algorithms which perform resource allocation under the assumption that the transmitters are always saturated with information bits, might no longer be efficient. It is, thus, important to design dynamic resource allocation algorithms which can incorporate higher layer processes and account for the stochastic nature of the wireless traffic.The central idea of this thesis is to develop cross-layer design algorithmsbetween the physical and the network layer in a multiple input multiple output (MIMO) multi-cell setup. Specifically, we consider base stations (BSs) equipped with multiple antennas serving multiple single antenna user terminals (UTs) in their respective cells. In contrast to the previous works, we consider the randomness in the arrival of information bits and hence account for the queuing at the BSs. With this setup, we develop various cross-layer based resource allocation algorithms. We incorporate two important design considerations namely decentralized design and energy efficiency. In particular, we focus on developing decentralized beamforming and traffic flow controller design, energy efficient design under time average QoS constraints and decentralized scheduling strategy in a multi-cell scenario. To this end, we use tools from Lyapunov optimization, random matrix theory and stochastic control theory.

Two optimisation problems were considered, in which market exposure is indirectly controlled. The first one models the capital of a company and an independent portfolio of new businesses, each one represented by a Cram\'r-Lundberg process. The company can choose the proportion of new business it wants to take on and can alter this proportion over time. Here the objective is to find a strategy that maximises the survival probability. We use a point processes framework to deal with the impact of an adapted strategy in the intensity of the new business. We prove that when Cram\'{e}r-Lundberg processes with exponentially distributed claims, it is optimal to choose a threshold type strategy, where the company switches between owning all new businesses or none depending on the capital level. For this type of processes that change both drift and jump measure when crossing the constant threshold, we solve the one and two-sided exit problems. This optimisation problem is also solved when the capital of the company and the new business are modelled by spectrally positive L\'vy processes of bounded variation. Here the one-sided exit problem is solved and we prove optimality of the same type of threshold strategy for any jump distribution. The second problem is a stochastic variation of the work done by Taylor about underwriting in a competitive market. Taylor maximised discounted future cash flows over a finite time horizon in a discrete time setting when the change of exposure from one period to the next has a multiplicative form involving the company's premium and the market average premium. The control is the company's premium strategy over a the mentioned finite time horizon. Taylor's work opened a rich line of research, and we discuss some of it. In contrast with Taylor's model, we consider the market average premium to be a Markov chain instead of a deterministic vector. This allows to model uncertainty in future conditions of the market. We also consider an infinite time horizon instead of finite. This solves the time dependency in Taylor's optimal strategies that were giving unrealistic results. Our main result is a formula to calculate explicitly the value function of a specific class of pricing strategies. Further we explore concrete examples numerically. We find a mix of optimal strategies where in some examples the company should follow the market while in other cases should go against it.

"January 2003" Includes bibliographical references (leaves 120-129) Presents performance analyses of dynamic, stochastic task scheduling policies for a real- time-communications system where tasks lose value as they are delayed in the system.

提撥原則是固定給付退休基金所必須特別重視的經營策略，提撥率為基金贊助者定期提撥於未來成員退休給付的準備金。過高的提撥率會造成基金管理上的財務壓力，退休金計劃採行相對提撥方式，將同時加重基金贊助者與基金成員的財務負擔，而過低的提撥率則會造成退休給付的準備金不足，將使退休基金未來面臨無力清償成員退休給付的困境，因此適當且長期穩定的提撥原則成為退休基金決策者的經營目標，而必須特別重視基金的財務風險管理。 本研究著重於探討如何數量化退休基金經營的穩定性與安全性，陳述隨機控制理論的觀點，應用動態規劃的發展結果，建立基金於離散時間的動態回饋控制模型，仔細探討並說明Haberman與Sung (1994)及Chang (1999a)所定義基金管理者的風險測度，使退休基金最主要的兩種經營風險，亦即提撥穩定性的風險（contribution rate risk）和財務清償的風險（solvency risk）能夠於基金財務規劃的期限內達到最小值，風險測度可提供決策者客觀衡量基金經營時的風險指標，表達有別於會計帳面之財務數字外有效的財務資訊。利用最適化的概念與給定參數及遞迴條件的限制下，計算基金於財務規劃期間內的最適提撥金額。 最後，我們以台灣公務人員退撫基金為研究對象進行數值分析，由實例分析的流程我們詳細探討最適化理論與實際財務評估的應用過程，且最適的結果可以提供退休基金決策者更詳盡且及時的財務資訊，輔助退休基金管理者於多期決策的擬定過程。 / Funding policy is the crucial management decision in the defined benefit pension schemes. The plan sponsor is required to calculate the contribution rate and accumulate in advance as he reserve for the future contingent retirement and ancillary obligations for the plan members. High contribution results an accelerating financial burden for the plan sponsor, while low contribution might endanger the financial solvency of the plan. The appropriate and stable contributions become the goal of the plan manager in setting up his funding policy. Hence financial risk management in attaining the goal is especially vital to be examined. The study emphasizes on quantifying the mismanage risks in pension valuation. The stability and solvency issues are included in our financial risk management. Stochastic control is also reviewed and the methodology of the dynamic programming is explored. The performance measure proposed in Haberman and Sung (1994) and Chang (1999a) are employed to scrutinize the contribution rate risk and solvency risk. The risk measurement can provide extra information in disclosing the risk index for the plan sponsor. The results gain operative information besides the traditional accounting reporting. The optimal contribution based on managerial consideration can be implemented through dynamic mechanisms under the given demographic and economic parameters and the plan recursive constraints. Finally, Taiwan public employees retirement system (Tai-PERS) is illustrated to investigate the optimal results and funding levels through the proposed model. The optimal results can response the updated financial information and assist the plan manager in policy making under the multi-frameworks.

"January 2003" / Includes bibliographical references (leaves 120-129) / x, 129 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Presents performance analyses of dynamic, stochastic task scheduling policies for a real- time-communications system where tasks lose value as they are delayed in the system. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 2003

In this thesis, we develop partial differential equation (PDE) based numerical methods to solve certain optimal stochastic control problems in finance. The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. The HJB equation corresponds to the case when the controls are bounded while the HJB variational inequality corresponds to the unbounded control case. As a result, the solution to the stochastic control problem can be computed by solving the corresponding HJB equation/variational inequality as long as the convergence to the viscosity solution is guaranteed. We develop a unified numerical scheme based on a semi-Lagrangian timestepping for solving both the bounded and unbounded stochastic control problems as well as the discrete cases where the controls are allowed only at discrete times. Our scheme has the following useful properties: it is unconditionally stable; it can be shown rigorously to converge to the viscosity solution; it can easily handle various stochastic models such as jump diffusion and regime-switching models; it avoids Policy type iterations at each mesh node at each timestep which is required by the standard implicit finite difference methods. In this thesis, we demonstrate the properties of our scheme by valuing natural gas storage facilities---a bounded stochastic control problem, and pricing variable annuities with guaranteed minimum withdrawal benefits (GMWBs)---an unbounded stochastic control problem. In particular, we use an impulse control formulation for the unbounded stochastic control problem and show that the impulse control formulation is more general than the singular control formulation previously used to price GMWB contracts.

In this thesis, we develop partial differential equation (PDE) based numerical methods to solve certain optimal stochastic control problems in finance. The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. The HJB equation corresponds to the case when the controls are bounded while the HJB variational inequality corresponds to the unbounded control case. As a result, the solution to the stochastic control problem can be computed by solving the corresponding HJB equation/variational inequality as long as the convergence to the viscosity solution is guaranteed. We develop a unified numerical scheme based on a semi-Lagrangian timestepping for solving both the bounded and unbounded stochastic control problems as well as the discrete cases where the controls are allowed only at discrete times. Our scheme has the following useful properties: it is unconditionally stable; it can be shown rigorously to converge to the viscosity solution; it can easily handle various stochastic models such as jump diffusion and regime-switching models; it avoids Policy type iterations at each mesh node at each timestep which is required by the standard implicit finite difference methods. In this thesis, we demonstrate the properties of our scheme by valuing natural gas storage facilities---a bounded stochastic control problem, and pricing variable annuities with guaranteed minimum withdrawal benefits (GMWBs)---an unbounded stochastic control problem. In particular, we use an impulse control formulation for the unbounded stochastic control problem and show that the impulse control formulation is more general than the singular control formulation previously used to price GMWB contracts.

This thesis consists of a summary and five papers, dealing with financial applications of optimal stopping, optimal control and volatility. In Paper I, we present a method to recover a time-independent piecewise constant volatility from a finite set of perpetual American put option prices. In Paper II, we study the optimal liquidation problem under the assumption that the asset price follows a geometric Brownian motion with unknown drift, which takes one of two given values. The optimal strategy is to liquidate the first time the asset price falls below a monotonically increasing, continuous time-dependent boundary. In Paper III, we investigate the optimal liquidation problem under the assumption that the asset price follows a jump-diffusion with unknown intensity, which takes one of two given values. The best liquidation strategy is to sell the asset the first time the jump process falls below or goes above a monotone time-dependent boundary. Paper IV treats the optimal dividend problem in a model allowing for positive jumps of the underlying firm value. The optimal dividend strategy is of barrier type, i.e. to pay out all surplus above a certain level as dividends, and then pay nothing as long as the firm value is below this level. Finally, in Paper V it is shown that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Lévy models is the existence of jumps towards the strike price in the underlying process.