<p>The presence of vibration is an important problem in many engineering applications. Various passive techniques have traditionally been used in order to reduce waves and vibrations, and their harmful effects. Passive techniques are, however, difficult to apply in the low frequency region. In addition, the use of passive techniques often involve adding mass to the system, which is undesirable in many applications.</p><p>As an alternative, active techniques can be used to manipulate system dynamics and to control the propagation of waves and vibrations. This thesis deals with modeling, estimation and active control of systems that have resonant dynamics. The systems are exposed to stochastic disturbances. Some of them excite the system and generate vibrational responses and other corrupt measured signals. </p><p>Feedback control of a beam with attached piezoelectrical elements is studied. A detailed modeling approach is described and system identification techniques are employed for model order reduction. Disturbance attenuation of a non-measured variable shows to be difficult. This issue is further analyzed and the problems are shown to depend on fundamental design limitations.</p><p>Feedforward control of traveling waves is also considered. A device with properties analogous to those of an electrical diode is introduced. An `ideal´ feedforward controller based on the mechanical properties of the system is derived. It has, however, poor noise rejection properties and it therefore needs to be modified. A number of feedforward controllers that treat the measurement noise in a statistically sound way are derived.</p><p>Separation of overlapping traveling waves is another topic under investigation. This operation also is sensitive to measurement noise. The problem is thoroughly analyzed and Kalman filtering techniques are employed to derive wave estimators with high statistical performance. </p><p>Finally, a nonlinear regression problem with close connections to unbalance estimation of rotating machinery is treated. Different estimation techniques are derived and analyzed with respect to their statistical accuracy. The estimators are evaluated using the example of separator balancing. </p>

L'objectif de ma thèse est d'étudier mathématiquement un indicateur de rupture de volatilité très utilisé par les praticiens en salle de marché. L'indicateur bandes de Bollinger appartient à la famille des méthodes dites d'analyse technique et donc repose exclusivement sur l'historique récente du cours considéré et un principe déduit des observations passées des marchés, indépendamment de tout modèle mathématique. Mon travail consiste à étudier les performances de cet indicateur dans un univers qui serait gouverné par des équations différentielles stochastiques (Black -Scholes) dont le coefficient de diffusion change sa valeur à un temps aléatoire inconnu et inobservable, pour un praticien désirant maximiser une fonction objectif (par exemple, une certaine utilité espérée de la valeur du portefeuille à une certaine maturité). Dans le cadre du modèle, l'indicateur de Bollinger peut s'interpréter comme un estimateur de l'instant de la prochaine rupture. On montre dans le cas des petites volatilités, que le comportement de la densité de l'indicateur dépend de la volatilité, ce qui permet pour un ratio de volatilité assez grand, de détecter via l'estimation de la distribution de l'indicateur dans quel régime de volatilité on se situe. Aussi, dans le cas des grandes volatilités, on montre par une approche via la transformée de Laplace, que le comportement asymptotique des queues de distribution de l'indicateur dépend de la volatilité. Ce qui permet de détecter le changement des grandes volatilités. Ensuite, on s'intéresse à une étude comparative entre l'indicateur de Bollinger et l'estimateur classique de la variation quadratique pour la détection de changement de la volatilité. Enfin, on étudie la gestion optimale de portefeuille qui est décrite par un problème stochastique non standard en ce sens que les contrôles admissibles sont contraints à être des fonctionnelles des prix observés. On résout ce problème de contrôle en s'inspirant de travaux de Pham and Jiao pour décomposer le problème initial d'allocation de portefeuille en un problème de gestion après la rupture et un problème avant la rupture, et chacun de ces problèmes est résolu par la méthode de la programmation dynamique . Ainsi, un théorème de verification est prouvé pour ce problème de contrôle stochastique. / The aim of my thesis is to study mathematically an indicator widely used by the practitioners in the trading market, and designed to detect changes in the volatility term . The Bollinger Bands indicator belongs to the family of methods known as technical analysis which consist in looking t the past price movement in order to predict its future price movements independently of any mathematical model. We study the performance of this indicator in a universe that is governed by a stochastic differential equations (Black-Scholes) such that the volatility changes at an unknown and unobservable random time, for a practitioner seeking to maximize an objective function (for instance, the expected utility of the wealth at a certain maturity). Within the framework of the model, Bollinger indicator can be interpreted as an estimator of the time at which the volatility changes its value. We show that in the case of small volatilities, the density behavior of the indicator depends on the value of the volatility, which allows that for large ratio of volatility, to detect via the distribution estimation in which regime of volatility we are. Also , for the case of large volatilities, we show by an approach via the Laplace transform that the asymptotic tails behavior of the indictor depends on the volatility value. This allows to detect a change for large volatilities. Next, we compare two indicators designed to detect a volatility change: the Bollinger bands and the quadratic variation indicators. Finally, we study the optimal portfolio allocation which is described by a non-standard stochastic problem in view of that the admissible controls need to be adapted to the filtration generated by the prices. We resolve this control problem by an approach used by Pham and Jiao to separate the initial allocation problem into an allocation problem after the rupture and an problem before the rupture, and each one of these problems is resolved by the dynamic programming method. Also, a verification theorem is proved for this stochastic control problem.

MASTER THESIS ABSTRACT TITLE: Trading Strategy in Incomplete Market AUTHOR: Tomáš Bunčák DEPARTMENT: Department of Probability and Mathematical Statistics, Charles University in Prague SUPERVISOR: Andrea Karlová We focus on the problem of finding optimal trading strategies (in a meaning corresponding to hedging of a contingent claim) in the realm of incomplete markets mainly. Although various ways of hedging and pricing of contingent claims are outlined, main subject of our study is the so-called mean-variance hedging (MVH). Sundry techniques used to treat this problem can be categorized into two approaches, namely a projection approach (PA) and a stochastic control approach (SCA). We review the methodologies used within PA in diversely general market models. In our research concerning SCA, we examine the possibility of using the methods of optimal stochastic control in MVH, and we study the problem of our interest in several settings of market models; involving cases of pure diffusion models and a jump- diffusion case. In order to reach an exemplary comparison, we provide solutions of the MVH problem in the setting of the Heston model via techniques of both of the approaches. Some parts of the thesis are accompanied with numerical illustrations.

This thesis deals with discrete-time Markov jump linear systems (MJLS) with Markov chain in a general Borel space S. Several control issues have been addressed for this class of dynamic systems, including stochastic stability (SS), linear quadratic (LQ) optimal control synthesis, fllter design and a separation principle. Necessary and sffcient conditions for SS have been derived. It was shown that SS is equivalent to the spectral radius of an operator being less than 1 or to the existence of a solution to a \\Lyapunov-like\" equation. Based on the SS concept, the finite- and infinite-horizon LQ optimal control problems were tackled. The solution to the finite- (infinite-)horizon LQ optimal control problem was derived from the associated control S-coupled Riccati difference (algebraic) equations. By S-coupled it is meant that the equations are coupled via an integral over a transition probability kernel having a density with respect to a in-finite measure on the Borel space S. The design of linear Markov jump filters was analyzed and a solution to the finite- (infinite-)horizon filtering problem was obtained based on the associated filtering S-coupled Riccati difference (algebraic) equations. Conditions for the existence and uniqueness of a stabilizing positive semi-definite solution to the control and filtering S-coupled algebraic Riccati equations have also been derived. Finally a separation principle for discrete-time MJLS with Markov chain in a general state space was obtained. It was shown that the optimal controller for a partial information optimal control problem separates the partial information control problem into two problems, one associated with a filtering problem and the other associated with an optimal control problem with complete information. It is expected that the results obtained in this thesis may motivate further research on discrete-time MJLS with Markov chain in a general state space. / Esta tese trata de sistemas lineares com saltos markovianos (MJLS) a tempo discreto com cadeia de Markov em um espaço geral de Borel S. Vários problemas de controle foram abordados para esta classe de sistemas dinâmicos, incluindo estabilidade estocástica (SS), síntese de controle ótimo linear quadrático (LQ), projeto de filtros e um princípio da separação. Condições necessárias e suficientes para a SS foram obtidas. Foi demonstrado que SS é equivalente ao raio espectral de um operador ser menor que 1 ou à existência de uma solução para uma equação de Lyapunov. Os problemas de controle ótimo a horizonte finito e infinito foram abordados com base no conceito de SS. A solução para o problema de controle ótimo LQ a horizonte finito (infinito) foi obtida a partir das associadas equações a diferenças (algébricas) de Riccati S-acopladas de controle. Por S-acopladas entende-se que as equações são acopladas por uma integral sobre o kernel estocástico com densidade de transição em relação a uma medida in-finita no espaço de Borel S. O projeto de filtros lineares markovianos foi analisado e uma solução para o problema da filtragem a horizonte finito (infinito) foi obtida com base nas associadas equações a diferenças (algébricas) de Riccati S-acopladas de filtragem. Condições para a existência e unicidade de uma solução positiva semi-definida e estabilizável para as equações algébricas de Riccati S-acopladas associadas aos problemas de controle e filtragem também foram obtidas. Por último, foi estabelecido um princípio da separação para MJLS a tempo discreto com cadeia de Markov em um espaço de estados geral. Foi demonstrado que o controlador ótimo para um problema de controle ótimo com informação parcial separa o problema de controle com informação parcial em dois problemas, um deles associado a um problema de filtragem e o outro associado a um problema de controle ótimo com informação completa. Espera-se que os resultados obtidos nesta tese possam motivar futuras pesquisas sobre MJLS a tempo discreto com cadeia de Markov em um espaço de estados geral.

Investiga-se, em tempo discreto, o problema multi-período de otimização de carteiras generalizado em média-variância cujos coeficientes de mercado são modulados por uma cadeia de Markov finita. O problema multi-período generalizado de média-variância com saltos Markovianos (PGMV ) é um problema de controle estocástico sem restrição cuja função objetivo consiste na maximização da soma ponderada ao longo do tempo da combinação linear de três elementos: o valor esperado da riqueza do investidor, o quadrado da esperança desta riqueza e a esperança do quadrado deste patrimônio. A principal contribuição deste trabalho é a derivação analítica de condições necessárias e suficientes para a determinação de uma estratégia ótima de investimento para o problema PGMV . A partir deste modelo são derivadas várias formulações de médiavariância, como o modelo tradicional cujo objetivo é maximizar o valor esperado da riqueza final do investidor, dado um nível de risco (variância) do portfólio no horizonte de investimento, bem como o modelo mais complexo que busca maximizar a soma ponderada das esperanças da riqueza ao longo do tempo, limitando a perda deste patrimônio em qualquer momento. Adicionalmente, derivam-se formas fechadas para a solução dos problemas citados quando as restrições incidem somente no instante final. Outra contribuição deste trabalho é a extensão do modelo PGMV para a solução do problema de seleção de carteiras em média-variância com o objetivo de superar um benchmark estocástico, com restrições sobre o valor esperado ou sobre a variância do tracking error do portfólio. Por fim, aplicam-se os resultados obtidos em exemplos numéricos cujo universo de investimento são todas as ações do IBOVESPA. / In this work we deal with a discrete-time multi-period mean-variance portfolio selection model with the market parameters subject to Markov regime switching. The multi-period generalized mean-variance portfolio selection model with regime switching (PGMV ) is an unrestricted stochastic control problem, in which the objective function involves the maximization of the weighted sum of a linear combination of three parts: the expected wealth, the square of the expected wealth and the expected value of the wealth squared. The main contribution of this work is the analytical derivation of necessary and sufficient conditions for the existence of an optimal control strategy to this PGMV model. We show that several mean-variance models are derived from the PGMV model, as the traditional formulation in which the objective is to maximize the expected terminal wealth for a given final risk (variance), or the complex one in which the objective function is to maximize the weighted sum of the wealth throughout its investment horizon, with control over maximum wealth lost. Additionally, we derive closed forms solutions for the above models when the restrictions are just in the final time. Another contribution of this work is to extend the PGMV model to solve the multi-period portfolio selection problem of beating a stochastic benchmark with control over the tracking error variance or its expected value. Finally, we run numerical examples in which the investment universe is formed by all the stocks belonging to the IBOVESPA.

In dieser Dissertation lösen wir eine Klasse stochastischer Kontrollprobleme und konstruieren optimale Handelsstrategien in illiquiden Märkten. In Kapitel 1 betrachten wir einen Investor, der sein Portfolio nahe an einer stochastischen Zielfunktion halten möchte. Gesucht ist eine Strategie (aus aktiven und passiven Orders), die die Abweichung vom Zielportfolio und die Handelskosten minimiert. Wir zeigen Existenz und Eindeutigkeit einer optimalen Strategie. Wir beweisen eine Version des stochastischen Maximumprinzips und leiten damit ein Kriterium für Optimalität mittels einer gekoppelten FBSDE her. Wir beweisen eine zweite Charakterisierung mittels Kauf- und Verkaufregionen. Das Portfolioliquidierungsproblem wird explizit gelöst. In Kapitel 2 verallgemeinern wir die Klasse der zulässigen Strategien auf singuläre Marktorders. Wie zuvor zeigen wir Existenz und Eindeutigkeit einer optimalen Strategie. Im zweiten Schritt beweisen wir eine Version des Maximumprinzips im singulären Fall, die eine notwendige und hinreichende Optimalitätsbedingung liefert. Daraus leiten wir eine weitere Charakterisierung mittels Kauf-, Verkaufs- und Nichthandelsregionen ab. Wir zeigen, dass Marktorders nur benutzt werden, wenn der Spread klein genug ist. Wir schließen dieses Kapitel mit einer Fallstudie über Portfolioliquidierung ab. Das dritte Kapitel thematisiert Marktmanipulation in illiquiden Märkten. Wenn Transaktionen einen Einfluß auf den Aktienpreis haben, dann können Optionsbesitzer damit den Wert ihres Portfolios beeinflussen. Wir betrachten mehrere Agenten, die europäische Derivate halten und den Preis des zugrundeliegenden Wertpapiers beeinflussen. Wir beschränken uns auf risikoneutrale und CARA-Investoren und zeigen die Existenz eines eindeutigen Gleichgewichts, das wir mittels eines gekoppelten Systems nichtlinearer PDEs charakterisieren. Abschließend geben wir Bedingungen an, wie diese Art von Marktmanipulation verhindert werden kann. / In this thesis we study a class of stochastic control problems and analyse optimal trading strategies in limit order markets. The first chapter addresses the problem of curve following. We consider an investor who wants to keep his stock holdings close to a stochastic target function. We construct the optimal strategy (comprising market and passive orders) which balances the penalty for deviating and the cost of trading. We first prove existence and uniqueness of an optimal control. The optimal trading strategy is then characterised in terms of the solution to a coupled FBSDE involving jumps via a stochastic maximum principle. We give a second characterisation in terms of buy and sell regions. The application of portfolio liquidation is studied in detail. In the second chapter, we extend our results to singular market orders using techniques of singular stochastic control. We first show existence and uniqueness of an optimal control. We then derive a version of the stochastic maximum principle which yields a characterisation of the optimal trading strategy in terms of a nonstandard coupled FBSDE. We show that the optimal control can be characterised via buy, sell and no-trade regions. We describe precisely when it is optimal to cross the bid ask spread. We also show that the controlled system can be described in terms of a reflected BSDE. As an application, we solve the portfolio liquidation problem with passive orders. When markets are illiquid, option holders may have an incentive to increase their portfolio value by using their impact on the dynamics of the underlying. In Chapter 3, we consider a model with competing players that hold European options and whose trading has an impact on the price of the underlying. We establish existence and uniqueness of equilibrium results and show that the equilibrium dynamics can be characterised in terms of a coupled system of non-linear PDEs. Finally, we show how market manipulation can be reduced.

In this work we study the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises. We consider the multiperiod and finite time horizon optimization of a mean-variance cost function under a new criterion. In this new problem, we apply a constraint on the total output variance weighted by its risk parameter while maximizing the expected output. The optimal control law is obtained from a set of interconnected Riccati difference equations, extending previous results in the literature. The application of our results is exemplified by numerical simulations of a portfolio of stocks and a risk-free asset. / Neste trabalho, estudamos o problema do controle ótimo estocástico de sistemas lineares em tempo discreto sujeitos a saltos Markovianos e ruídos multiplicativos. Consideramos a otimização multiperíodo, com horizonte de tempo finito, de um funcional da média-variância sob um novo critério. Neste novo problema, maximizamos o valor esperado da saída do sistema ao mesmo tempo em que limitamos a sua variância total ponderada pelo seu parâmetro de risco. A lei de controle ótima é obtida através de um conjunto de equações de diferenças de Riccati interconectadas, estendendo resultados anteriores da literatura. São apresentadas simulações numéricas para uma carteira de investimentos com ações e um ativo de risco para exemplificarmos a aplicação de nossos resultados.

Nesta tese, apresentamos uma metodologia concreta para calcular os controles -ótimos para sistemas estocásticos não-Markovianos. A análise trajetória a trajetória e o uso da estrutura de discretização proposta por Leão e Ohashi [36] conjuntamente com argumentos de seleção mensuráveis, nos forneceu uma estrutura para transformar um problema infinito dimensional para um finito dimensional. Desta forma, garantimos uma descrição concreta para uma classe bastante geral de problemas. / In this thesis, we present a concrete methodology to calculate the -optimal controls for non-Markovian stochastic systems. A pathwise analysis and the use of the discretization structure proposed by Leão and Ohashi [36] jointly with measurable selection arguments, allows us a structure to transform an infinite dimensional problem into a finite dimensional. In this way, we guarantee a concrete description for a rather general class of stochastic problems.

Stochastic dynamic programming is a recursive method for solving sequential or multistage decision problems. It helps economists and mathematicians construct and solve a huge variety of sequential decision making problems in stochastic cases. Research on stochastic dynamic programming is important and meaningful because stochastic dynamic programming reflects the behavior of the decision maker without risk aversion; i.e., decision making under uncertainty. In the solution process, it is extremely difficult to represent the existing or future state precisely since uncertainty is a state of having limited knowledge. Indeed, compared to the deterministic case, which is decision making under certainty, the stochastic case is more realistic and gives more accurate results because the majority of problems in reality inevitably have many unknown parameters. In addition, time scale calculus theory is applicable to any field in which a dynamic process can be described with discrete or continuous models. Many stochastic dynamic models are discrete or continuous, so the results of time scale calculus are directly applicable to them as well. The aim of this thesis is to introduce a general form of a stochastic dynamic sequence problem on complex discrete time domains and to find the optimal sequence which maximizes the sequence problem.

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.