In this thesis, we address the problem of minimizing the average delay of data pack-ets served by a transmitter on a static, point-to-point link. The transmitter dynamically chooses state-dependent admission and transmission rates, while adhering to average throughput and transmission power constraints. The transmitter is modelled as an infinite buffer Markov queue with adjustable arrival and service rates. Data packets arrive at the system according to a Poisson process with rate, Λ, and are admitted at a rate, λnwith 0≤ λn ≤ Λ, depending on the number, n, of data packets present in the system. The packet size is assumed to be exponentially distributed, and the controller chooses a transmission rate, µn, at most equal to a maximum value, M, depending on the current backlog, n, in the system. The objective is to minimize the average delay of data packets in the system subject to a throughput lower bound constraint, while satisfying an upper bound on the average transmission power. This constrained MDP problem is solved using a Lagrange relaxation approach and analysed for the cases with throughput and power constraints that are achievable with equality by appropri-ate values of the Lagrange multipliers. A procedure is developed, based on explicit formulae, using which optimal admission and service rates as a function of the packet queue length are obtained.

Continuous-time Markov chains (CTMC's) form a convenient mathematical framework for analyzing random systems across many different disciplines. A specific research problem that is often of interest is to try to predict maximum probability sequences of state transitions given initial or boundary conditions. This work shows how to solve this problem exactly through an efficient dynamic programming algorithm. We demonstrate our approach through two different applications - ranking mutational pathways of HIV virus based on their probabilities, and determining the most probable failure sequences in complex fault-tolerant engineering systems. Even though CTMC's have been used extensively to realistically model many types of complex processes, it is often a standard practice to eventually simplify the model in order to perform the state evolution analysis. As we show here, simplifying approaches can lead to inaccurate and often misleading solutions. Therefore we expect our algorithm to find a wide range of applications across different domains.

The pricing of options is one of the key problems in mathematical finance. In recent years, pricing models that are based on the continuous time random walk (CTRW), an anomalous diffusive random walk model widely used in physics, have been introduced. In this thesis, we investigate the pricing of European call options with CTRW and generalized CTRW models within the Black-Scholes framework. Here, the non-Markovian character of the underlying pricing model is manifest in Black-Scholes PDEs with fractional time derivatives containing memory terms. The inclusion of non-zero interest rates leads to a distinction between different types of \forward" and \backward" options, which are easily mapped onto each other in the standard Markovian framework, but exhibit significant dfferences in the non-Markovian case. The backward-type options require us in particular to include the multi-point statistics of the non-Markovian pricing model. Using a representation of the CTRW in terms of a subordination (time change) of a normal diffusive process with an inverse L evy-stable process, analytical results can be obtained. The extension of the formalism to arbitrary waiting time distributions and general payoff functions is discussed. The pricing of path-dependent Asian options leads to further distinctions between different variants of the subordination. We obtain analytical results that relate the option price to the solution of generalized Feynman-Kac equations containing non-local time derivatives such as the fractional substantial derivative. Results for L evy-stable and tempered L evy-stable subordinators, power options, arithmetic and geometric Asian options are presented.

Continuous-time (CT) data conversion and continuous-time digital signal processing (DSP) are an interesting alternative to conventional methods of signal conversion and processing. This alternative proposes time-based encoding that may not suffer from aliasing; shows superior spectral properties (e.g. no quantization noise floor); and enables time-based, event-driven, flexible signal processing using digital circuits, thus scaling well with technology. Despite these interesting features, this approach has so far been limited by the CT encoder, due to both its relatively poor energy efficiency and the constraints it imposes on the subsequent CT DSP. In this thesis, we present three principles that address these limitations and help improve the CT ADC/DSP system. First, an adaptive-resolution encoding scheme that achieves first-order reconstruction with simple circuitry is proposed. It is shown that for certain signals, the scheme can significantly reduce the number of samples generated per unit of time for a given accuracy compared to schemes based on zero-order-hold reconstruction, thus promising to lead to low dynamic power dissipation at the system level. Presented next is a novel time-based CT ADC architecture, and associated encoding scheme, that allows a compact, energy-efficient circuit implementation, and achieves first-order quantization error spectral shaping. The design of a test chip, implemented in a 0.65-V 28-nm FDSOI process, that includes this CT ADC and a 10-tap programmable FIR CT DSP to process its output is described. The system achieves 32 dB – 42 dB SNDR over a 10 MHz – 50 MHz bandwidth, occupies 0.093 mm2, and dissipates 15 µW–163 µW as the input amplitude goes from zero to full scale. Finally, an investigation into the possibility of CT encoding using voltage-controlled oscillators is undertaken, and it leads to a CT ADC/DSP system architecture composed primarily of asynchronous digital delays. The latter makes the system highly digital and technology-scaling-friendly and, hence, is particularly attractive from the point of view of technology migration. The design of a test chip, where this delay-based CT ADC/DSP system architecture is used to implement a 16-tap programmable FIR filter, in a 1.2-V 28-nm FDSOI process, is described. Simulations show that the system will achieve a 33 dB – 40 dB SNDR over a 600 MHz bandwidth, while dissipating 4 mW.

Delta-sigma modulators are currently a very popular technique for making high-resolution analog-to-digital and digital-to-analog converters. These oversampled data converters have several advantages over conventional Nyquist-rate converters, including an insensitivity to many analog component imperfections, a simpler antialiasing filter and reduced accuracy requirements in the sample and hold. Though the initial uses of delta-sigma modulators were in the audio field, the development of bandpass modulators opened up the application range to radar systems, digital communication systems and instruments which convert IF, or even RF, analog signals directly to digital form. This thesis presents a method used to analyze and synthesize continuous-time delta-sigma modulators for given specifications. A fourth-order prototype continuous-time bandpass delta-sigma modulator employing g[subscript m]-LC resonator structure is demonstrated on a PCB board and measurement results corroborate the theory. To allow the construction of very high performance delta-sigma modulators, this thesis presents an architecture for a multibit DAC constructed from unit elements which shapes element mismatches. Theoretical analysis and simulation shows that this architecture greatly increases the noise attenuation in the band-of-interest and facilitates the use of multibit quantization in delta-sigma modulators. The methods presented in this thesis will allow high-frequency wideband bandpass delta-sigma modulators to be constructed. / Graduation date: 1996

We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent.<br> However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time. / Nous étudions les notions de monotonie et de monotonie complète pour les processus de Markov (ou chaînes de Markov à temps continu) prenant leurs valeurs dans un espace partiellement ordonné. Ces deux notions ne sont pas équivalentes, comme c'est le cas lorsque le temps est discret. Cependant, nous établissons que pour certains ensembles partiellement ordonnés, l'équivalence a lieu en temps continu bien que n'étant pas vraie en temps discret.

Continuous-time Markov chains (CTMC's) form a convenient mathematical framework for analyzing random systems across many different disciplines. A specific research problem that is often of interest is to try to predict maximum probability sequences of state transitions given initial or boundary conditions. This work shows how to solve this problem exactly through an efficient dynamic programming algorithm. We demonstrate our approach through two different applications - ranking mutational pathways of HIV virus based on their probabilities, and determining the most probable failure sequences in complex fault-tolerant engineering systems. Even though CTMC's have been used extensively to realistically model many types of complex processes, it is often a standard practice to eventually simplify the model in order to perform the state evolution analysis. As we show here, simplifying approaches can lead to inaccurate and often misleading solutions. Therefore we expect our algorithm to find a wide range of applications across different domains.

El objetivo de esta tesis es modelar y predecir la volatilidad de las series financieras con modelos de volatilidad en tiempo discreto y continuo.En mi primer capítulo, intento modelar las principales características de las series financieras, como a persistencia y curtosis. Los modelos de volatilidad estocástica estimados son extensiones directas de los modelos de Gallant y Tauchen (2001), donde incluyo un elemento de retro-alimentación. Este elemento es de extrema importancia porque permite captar el hecho de que períodos de alta volatilidad están, en general, seguidos de periodos de gran volatilidad y viceversa. En este capítulo, como en toda la tesis, uso el método de estimación eficiente de momentos de Gallant y Tauchen (1996). De la estimación surgen dos modelos posibles de describir los datos, el modelo logarítmico con factor de volatilidad y retroalimentación y el modelo logarítmico con dos factores de volatilidad. Como no es posible elegir entre ellos basados en los tests efectuados en la fase de la estimación, tendremos que usar el método de reprogección para obtener mas herramientas de comparación. El modelo con un factor de volatilidad se comporta muy bien y es capaz de captar la "quiebra" de los mercados financieros de 1987.En el segundo capítulo, hago la evaluación del modelo con dos factores de volatilidad en términos de predicción y comparo esa predicción con las obtenidas con los modelos GARCH y ARFIMA. La evaluación de la predicción para los tres modelos es hecha con la ayuda del R2 de las regresiones individuales de la volatilidad "realizada" en una constante y en las predicciones. Los resultados empíricos indican un mejor comportamiento del modelo en tiempo continuo. Es más, los modelos GARCH y ARFIMA parecen tener problemas en seguir la marcha de la volatilidad "realizada". Finalmente, en el tercer capítulo hago una extensión del modelo de volatilidad estocástica de memoria larga de Harvey (2003). O sea, introduzco un factor de volatilidad de corto plazo. Este factor extra aumenta la curtosis y ayuda a captar la persistencia (que es captada con un proceso integrado fraccional, como en Harvey (1993)). Los resultados son probados y el modelo implementado empíricamente. / The purpose of my thesis is to model and forecast the volatility of the financial series of returns by using both continuous and discrete time stochastic volatility models.In my first chapter I try to fit the main characteristics of the financial series of returns such as: volatility persistence, volatility clustering and fat tails of the distribution of the returns.The estimated logarithmic stochastic volatility models are direct extensions of the Gallant and Tauchen's (2001) by including the feedback feature. This feature is of extreme importance because it allows to capture the low variability of the volatility factor when the factor is itself low (volatility clustering) and it also captures the increase in volatility persistence that occurs when there is an apparent change in the pattern of volatility at the very end of the sample. In this chapter, as well as in all the thesis, I use Efficient Method of Moments of Gallant and Tauchen (1996) as an estimation method. From the estimation step, two models come out, the logarithmic model with one factor of volatility and feedback (L1F) and the logarithmic model with two factors of volatility (L2). Since it is not possible to choose between them based on the diagnostics computed at the estimation step, I use the reprojection step to obtain more tools for comparing models. The L1F is able to reproject volatility quite well without even missing the crash of 1987.In the second chapter I fit the continuous time model with two factors of volatility of Gallant and Tauchen (2001) for the return of a Microsoft share. The aim of this chapter is to evaluate the volatility forecasting performance of the continuous time stochastic volatility model comparatively to the ones obtained with the traditional GARCH and ARFIMA models. In order to inquire into this, I estimate using the Efficient Method of Moments (EMM) of Gallant and Tauchen (1996) a continuous time stochastic volatility model for the logarithm of asset price and I filter the underlying volatility using the reprojection technique of Gallant and Tauchen (1998). Under the assumption that the model is correctly specified, I obtain a consistent estimator of the integrated volatility by fitting a continuous time stochastic volatility model to the data. The forecasting evaluation for the three estimated models is going to be done with the help of the R2 of the individual regressions of realized volatility on the volatility forecasts obtained from the estimated models. The empirical results indicate the better performance of the continuous time model in the out-of-sample periods compared to the ones of the traditional GARCH and ARFIMA models. Further, these two last models show difficulties in tracking the growth pattern of the realized volatility. This probably is due to the change of pattern in volatility in this last part of the sample. Finally, in the third chapter I come back to the model specification and I extend the long memory stochastic volatility model of Harvey (1993) by introducing a short run volatility factor. This extra factor increases kurtosis and helps the model capturing volatility persistence (that it is captured by a fractionally integrated process as in Harvey (1993) ). Futhermore, considering some restrictions of the parameters it is possible to fit the empirical fact of small first order autocorrelation of squared returns. All these results are proved theoretically and the model is implemented empirically using the S&P 500 composite index returns. The empirical results show the superiority of the model in fitting the main empirical facts of the financial series of returns.

The thesis proposes a third order continuous-time sigma delta modulator used in GSM. We used a special 1.5bit quantizer, and to use its three different states to reach a differential feedback path. That can improve the resolution of our circuit. Oversampling and noise shaping are two keys of sigma delta modulator. In structure, the continuous-time features can reduce power consumption. The proposed sigma delta modulator uses TSMC 0.35 m CMOS process and its sampling frequency is 10.8MHz, bandwidth is200KHz and oversampling ratio is 32.

The emergence of bandwidth-intensive services has created a need for high speed and high resolution data converters. Towards this end, system level design of a continuous-time sigma-delta modulator achieving 11 bits resolution over 100 MHz signal bandwidth by using a feed-forward topology is presented. The system is first built in the Simulink environment in MATLAB. The building blocks in the loop filter are modeled with non-idealities, and specifications for these blocks are obtained by simulations. An operational transconductor amplifier (OTA) with 100 mS transconductance, 70 dB linearity, and 34.2 mW power dissipation is designed to be used in the loop filter. Simulation results indicate that the 5th order loop filter implemented in the feed-forward architecture in transistor level shows lower power consumption, 105 mW, compared to the loop filter implemented by feedback architecture, 152 mW.