Stochastic control is a broad tool with applications in several areas of academic interest. The financial literature is full of examples of decisions made under uncertainty and stochastic control is a natural framework to deal with these problems. Problems such as optimal trading, option pricing and economic policy all fall under the purview of stochastic control. These problems often face nonlinearities that make analytical solutions infeasible and thus numerical methods must be employed to find approximate solutions. In this dissertation three types of stochastic control formulations are used to model applications in finance and numerical methods are developed to solve the resulting nonlinear problems. To begin with, optimal stopping is applied to option pricing. Next, impulse control is used to study the problem of interest rate control faced by a nation's central bank, and finally a new type of hybrid control is developed and applied to an investment decision faced by money managers. / text

University of Macau / Faculty of Science and Technology / Department of Mathematics

Cette thèse porte sur l'étude de quelques problèmes de contrôle stochastique dans un contexte de risque de liquidité et d'impact sur le prix des actifs. La thèse se compose de quatre chapitres.Dans le deuxième chapitre, on propose une modélisation d'un problème d'animation de marché dans un contexte de risque de liquidité en présence de contraintes d'inventaire et de changements de régime. Cette formulation peut être considérée comme étant une extension de précédentes études sur ce sujet. Le résultat principal de cette partie est la caractérisation de la fonction valeur comme solution unique, au sens de la viscosité, d'un système d'équations d'Hamilton-Jacobi-Bellman . On enrichit notre étude par la donnée de quelques résultats numériques.Dans le troisième chapitre, on propose un schéma d'approximation numérique pour résoudre un problème d'optimisation de portefeuille dans un contexte de risque de liquidité et d'impact sur le prix des actifs. On montre que la fonction valeur peut être obtenue comme limite d'une procédure itérative dont chaqueitération représente un problème d'arrêt optimal et on utilise un algorithme numérique, basé sur la quantification optimale, pour calculer la fonction valeur ainsi que la politique de contrôle. La convergence du schéma numérique est obtenue via des critères de monotonicité, stabilité et consistance.Dans le quatrième chapitre, on s'intéresse à un problème couplé de contrôle singulier et de contrôle impulsionnel dans un contexte d'illiquidité. On propose une formulation mathématique pour modéliser la distribution de dividendes et la politique d'investissement d'une entreprise sujette à des contraintes de liquidité. On montre que, sous des coûts de transaction et un impact sur le prix des actifs illiquides, la fonction valeur de l'entreprise est l'unique solution de viscosité d'une équation d'Hamilton-Jacobi-Bellman. On propose aussi une méthode numérique itérative pour calculer la stratégie optimale d'achat, de vente et de distribution de dividendes. / The purpose of this thesis is to study some stochastic control problems with liquidity risk and price impact. The thesis contains four chapters.The second chapter is devoted to the modeling aspects of a market making problem in a liquidity risk framework under inventory constraints and switching regimes. This formulation can be seen as an extension of previous studies on this subject. The main result is the characterization of the value functions as the unique viscosity solutions to the associated Hamilton-Jacobi-Bellman system. We further enrich our study with some numerical results.In the third section, we introduce a numerical scheme to solve an impulse control problem under state constraints arising from optimal portfolio selection under liquidity risk and price impact. We show that the value function could be obtained as the limit of an iterative procedure where each step is an optimal stopping problem and we use a numerical approximation algorithm based on quantization procedure to compute the value function and the optimal policy. The main result is to prove the convergence of our numerical scheme using monotonicity, stability and consistency properties.In the fourth section, we study a mixed singular and impulse control problem with liquidity risks and constraints. We propose a mathematical modeling to the dividend and investment policy of a firm operating under uncertain environment and liquidity risks. Our main contribution is to show that, under transaction costs and impact on the illiquid asset price, the firm's value function is the unique viscosity solution of a certain Hamilton-Jacobi-Bellman equation. We also formulated an iterative numerical procedure to compute the optimal dividend and investment policy.

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Pairs trading has been a popular statistical arbitrage strategy among hedge funds. One important research field in pairs trading is to maximize the return under differential constraints and assumptions. In this thesis, we develop two models to optimize the performance of pairs trading. In the static model, we find the analytic solution of optimal thresholds for pairs trading to maximize the long run profit per unit time. Comparison is made between the optimal rules we developed and the common practice. To overcome limitations of the static model, we extend our research to dynamic pairs trading, where a continuous time Markov chain is used to model the change of parameters. Our objective is to maximize the expected return in the finite horizon under the Constant Relative Risk Aversion (CRRA) utility. Numerical examples are presented to illustrate the impact of price limits, risk aversion rate and regime switching on consumers' investment decision.

Thesis (Ph.D.)--University of Florida, 2005. / Title from title page of source document. Document formatted into pages; contains 158 pages. Includes vita. Includes bibliographical references.

We investigate the stabilization of unstable multidimensional partially observed single-station, multi-sensor (single-controller) and multi-controller (single-sensor) linear systems controlled over discrete noiseless channels under fixed-rate information constraints. Stability is achieved under communication requirements that are asymptotically tight in the limit of large sampling periods. Through the use of similarity transforms, sampling and random-time drift conditions we obtain a coding and control policy leading to the existence of a unique invariant distribution and finite second moment for the sampled state. We use a vector stabilization scheme in which all modes of the linear system visit a compact set together infinitely often. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2012-12-06 15:06:37.449

A thesis submitted for the degree of Doctor of Philosophy 2013 / This thesis presents a range of related pricing kernel models that are driven by incomplete information about a series of future unknowns. These unknowns may, for instance, represent fundamental macroeconomic, political or social random variables that are revealed at future times. They may also represent latent or hidden factors that are revealed asymptotically. We adopt the information-based approach of Brody, Hughston and Macrina (BHM) to model the information processes associated with the random variables. The market filtration is generated collectively by these information processes. By directly modelling the pricing kernel, we generate information-sensitive arbitrage-free models for the term structure of interest rates, the excess rate of return required by investors, and security prices. The pricing kernel is modelled by a supermartingale to ensure that nominal interest rates remain non-negative. To begin with, we primarily investigate finite-time pricing kernel models that are sensitive to Brownian bridge information. The BHM framework for the pricing of credit-risky instruments is extended to a stochastic interest rate setting. In addition, we construct recovery models, which take into consideration information about, for example, the state of the economy at the time of default. We examine various explicit examples of analytically tractable information-driven pricing kernel models. We develop a model that shares many of the features of the rational lognormal model, and investigate examples of heat kernel models. It is shown that these models may result in discount bonds and interest rates being bounded by deterministic functions. In certain situations, incoming information about random variables may exhibit jumps. To this end, we construct a more general class of nite-time pricing kernel models that are driven by Levy random bridges. Finally, we model the aggregate impact of uncertainties on a nancial market by randomised mixtures of Levy and Markov processes respectively. It is assumed that market participants have incomplete information about the underlying random mixture. We apply results from non-linear ltering theory and construct Flesaker-Hughston models and in nite-time heat kernel models based on these randomised mixtures.

The first essay considers a dynamic non-stationary inventory problem in which replenishment is made in fixed lot sizes (e.g., in full truckloads or full containers). We consider two separate cases: one with exogenous pricing and the other with endogenous pricing. In the first case (exogenous pricing), we show that when the ordering cost contains only a variable component, the reorder-point lot-size policy or (r, Q) policy is optimal for both single-stage and multi-echelon inventory systems. In the presence of a fixed cost, we establish the optimality of batch based (s, S ) policies for the single-stage inventory system. In the second case (endogenous pricing), we show that when the demand function has the additive form and there is only a variable ordering cost, the (r,Q) list-price policy is optimal for the single-stage system, where inventory replenishment follows an (r,Q) policy and the optimal price in each period depends on the order-up-to level. / The second essay analyzes a periodic-review, stochastic, inventory-control system in which the fixed order-cost is a step function of the order size. In particular, if the order size is within a specified limit, C, then the setup cost is K1; otherwise it is K2, where K2 ≥ K1. This cost structure is motivated from some industrial applications and transportation/production contracts used in practice. Under the condition that K1 ≤ K 2 ≤ K1, we introduce a new concept called C - (K1 ≤ K 2) convexity, which enables us to partially characterize the structure of an optimal ordering policy. For the general condition K 1 ≤ K2 , the analysis is facilitated with a different notion called strong K-convexity. Based on this analysis, we provide a partial characterization of the optimal policy and construct an easy-to-implement heuristic method that has near-optimal performance in random test instances. Our study extends or redevelops (with different techniques) several existing results in the literature. / The third essay studies a firm's periodic-review production/inventory ordering decisions when the next period's setup cost depends on the quantity produced/ ordered in the current period. In particular, if the current period's production/order quantity exceeds a specified threshold value, the system starts the next period in a "warm" state and no fixed setup cost is incurred; otherwise the state is considered "cold" and a positive setup cost is required for production/ ordering. We develop a dynamic programming formulation of the problem and provide a partial characterization of the optimal policy under the assumption that the demands follow a Polya or Uniform distribution. We use the structural results to develop fairly simple heuristic policies, which perform highly effectively in our computational experiments. / With increased globalization and competition in the current market, supply chain has become longer and more complicated than ever before. An effective and efficient supply chain is crucial and essential to a successful firm. In a supply chain, inventories are a very important component as the investment in inventories is enormous. This dissertation consists of three essays related to stochastic inventory management. / Yang, Yi. / Adviser: Youhua Chen. / Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 145-151). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

by Chen Xi. / "July 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 107-112). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.