In this thesis, we propose two continuous time stochastic volatility models with long memory that generalize two existing models. More importantly, we provide analytical formulae that allow us to study option prices numerically, rather than by means of simulation. We are not aware about analytical results in continuous time long memory case. In both models, we allow for the non-zero correlation between the stochastic volatility and stock price processes. We numerically study the effects of long memory on the option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter in short memory models. We also find that long memory models have the potential to accommodate the short term options and the decay of volatility skew better than the corresponding short memory stochastic volatility models.

Most of the existing research on mathematical morphology is restricted to the deterministic case. This thesis addresses the void in the results on the stochastic properties of morphological filters. The primary results include analysis of the stochastic properties of morphological operations, such as dilation, erosion, closing and opening. Two unbiased morphological filters are introduced and a quantitative description of the probability distribution function of morphological operations on independent, identically distributed random signals is obtained. Design of an optimal morphological filter in the sense of a criterion proposed here is also discussed. A brief, but systematic description of the definitions and properties of deterministic morphological operations on sets is presented to establish the necessary background for the analysis of the filter stochastic properties. / Graduation date: 1992

High prices in any industry, agricultural especially, tend to spur new investment opportunities. Recent prices for pecans have been high relative to their historical pattern, suggesting investment opportunities for pecans. Prior to any investment, the investor needs to know what products consumers are demanding and how profitable it is to grow those products. This study assessed Texas consumers' preferences for pecan products and the profitability of growing pecans in the central Texas region. A choice experiment was conducted amongst Texas consumers to reveal consumers' preferences and determine their willingness-to-pay for the attributes comprising pecan products. A stochastic production model was formulated to determine the profitability of three different types of pecan orchards: a native orchard with no irrigation, an improved varieties orchard with irrigation, and an improved varieties orchard without irrigation. Results from the choice experiment indicated that consumers preferred large size pecans, native variety pecans, pecan halves, United States-grown pecans, and Texas-grown pecans. The choice experiment also found that consumers were heterogeneous in their preferences for all attributes except pecan variety and U.S. origin. Results from the stochastic production model indicated that the most profitable pecan orchard in central Texas was the irrigated improved orchard.

Integrated robust planning systems that cover all levels of SC hierarchy have become increasingly important. Strategic, tactical, and operational SC plans should not be generated in isolation to avoid infeasible and conflicting decisions. On the other hand, enterprise planning systems contain over millions of records that are processed in each planning iteration. In such enterprises, the ability to generate robust plans is vital to their success because such plans can save the enterprise resources that may otherwise have to be reserved for likely SC plan changes. A robust SC plan is valid in all circumstances and does not need many corrections in the case of interruption, error, or disturbance. Such a reliable plan is proactive as well as reactive. Proactivity can be achieved by forecasting the future events and taking them into account in planning. Reactivity is a matter of agility, the capability of keeping track of system behaviour and capturing alarming signals from its environment, and the ability to respond quickly to the occurrence of an unforeseen event. Modeling such a system behaviour and providing solutions after such an event is extremely important for a SC. This study focuses on integrated supply chain planning with stochastic dynamic considerations. An integrated tactical-operational model is developed and then segregated into two sub-models which are solved iteratively. A SC is a stochastic dynamic system whose state changes over time often in an unpredictable manner. As a result, the customer demand is treated as an uncertain parameter and is handled by exploiting scenario-based stochastic programming. The increase in the number of scenarios makes it difficult to obtain quick and good solutions. As such, a Branch and Fix algorithm is developed to segregate the stochastic model into isolated islands so as to make the computationally intractable problem solvable. However not all the practitioners, planners, and managers are risk neutral. Some of them may be concerned about the risky extreme scenarios. In view of this, the robust optimization approach is also adopted in this thesis. Both the solution robustness and model robustness are taken into account in the tactical model. Futhermore, the dynamic behaviour of a SC system is handled with the concept of Model Predictive Control (MPC).

Aunque la transmisión del virus de la lengua azul (VLA) aparentemente se interrumpe durante el invierno, los brotes a menudo reaparecen en la siguiente temporada. Para evaluar la probabilidad de que el VLA sobreviva al invierno a través de la persistencia en vectores adultos, hospedadores o una combinación de ambos, se desarrolló un modelo de evaluación del riesgo estocástico. Además, el modelo permite evaluar el papel que juega el número residual de vectores presentes durante el invierno, y el papel desempeñado por los Culicoides con comportamiento endofílico. El modelo se aplicó a un escenario real, la persistencia del VLA en Alemania entre 2006 y 2007. Los resultados mostraron que la presencia de vectores durante el invierno permitiría la transmisión durante este periodo, y que mientras que la transmisión se vería favorecida por el comportamiento endofílico, su efecto era limitado. A pesar de que la transmisión por los mecanismos estudiados era posible, la probabilidad parecía demasiado baja para explicar la reaparición de la enfermedad observada en Alemania. Las redes de transporte y comercio continúan expandiéndose, y una consecuencia importante es la importación de patógenos transmitidos por vectores. Un aspecto importante de la epidemiología de la lengua azul (LA) es la posibilidad de que Culicoides infectados sean introducidos a través de redes de transporte y comerciales. Por tanto, se desarrolló un modelo estocástico de evaluación de riesgos para calcular la probabilidad de aparición de un brote de lengua azul, por introducción de Culicoides a través de estas redes. El modelo se aplicó al riesgo de una epidemia del serotipo 8 en España en 2007 por el transporte de Culicoides desde los países afectados del norte de Europa, independientemente del mecanismo de introducción del vector. El riesgo anual ponderado parecía ser bajo (entre 3.2x10-7 y 6.4x10-12), aunque había grandes diferencias dependiendo del país de origen, con las probabilidades más elevadas desde Bélgica, Holanda, Alemania y Francia. Para que este mecanismo supusiera un riesgo significativo para los países libres, el número de vectores transportados tendría que ser elevado. La hipótesis de que los toros infectados pueden excretar virus a través de su esperma dio lugar a restricciones en el comercio internacional de semen y al establecimiento de medidas preventivas. Sin embargo, ni el riesgo de transmisión de LA por el semen, ni la eficacia de estas medidas se han evaluado cuantitativamente. El objetivo del estudio fue calcular por medio de un modelo estocástico, en caso de introducción de la LA en un centro de recogida de semen bovino (CRS), tanto el riesgo de transmisión del VLA a través de semen de bovino, como la reducción de dicho riesgo por aplicación de medidas preventivas. El modelo se aplicó a diferentes escenarios, dependiendo por ejemplo del tipo de prueba diagnóstica empleada, el intervalo entre los controles de los toros donantes, o la tasa de dispersión del VLA dentro del CRS. Los controles cada 60 días por ensayo por inmunoabsorción ligado a enzimas (ELISA) parece ser un método poco eficaz para reducir el riesgo de transmisión por semen en comparación con la reacción en cadena de polimerasa (PCR) cada 28 días. Un aumento en la tasa de propagación del VLA dentro del CRS reduce el riesgo de transmisión por el semen. El almacenamiento de semen por 30 días antes de su distribución parece ser eficaz para reducir el riesgo de transmisión. El análisis de sensibilidad identificó la probabilidad de excreción del VLA en semen como un parámetro crucial. Sin embargo, existe un alto grado de incertidumbre asociado a este parámetro, con diferencias significativas en función del serotipo del virus. / Even though bluetongue virus (BTV) transmission is apparently interrupted during winter, bluetongue outbreaks often reappear in the next season (overwintering). In order to assess the probability of BTV overwintering by persistence in adult vectors, ruminants (through prolonged viraemia) or a combination of both, a quantitative risk assessment model was developed. Furthermore, the model allowed the role played by the residual number of vectors present during winter to be examined, and the effect of a proportion of Culicoides living inside buildings (endophilic behaviour) to be explored. The model was then applied to a real scenario: overwintering in Germany between 2006 and 2007. The results showed that the limited number of vectors active during winter seemed to allow the transmission of BTV during this period, and that while transmission was favoured by the endophilic behaviour of some Culicoides, its effect was limited. Even though transmission was possible, the likelihood of BTV overwintering by the mechanisms studied seemed too low to explain the observed re-emergence of the disease. Transport and trade networks continue to expand, and one important consequences of this expansion is vector-borne pathogen importation. One important aspect of BT epidemiology is the potential for movement of infected adults Culicoides via local and global transportation networks. Therefore, a stochastic risk assessment model was constructed to assess the probability of development of a BTV outbreak as a consequence of the introduction of infected Culicoides via transport and trade networks. The model was applied to calculate the risk of a BTV-8 epidemic in Spain in 2007 as the consequence of the transport of a Culicoides from the affected Northern European countries, regardless of the mechanism by which the midge was introduced. The weighted annual risk by transportation of a single Culicoides from the affected Northern European countries seemed to be low (between 3.2x10-7 and 6.4x10-12) although there were major differences among countries, with the highest risks by Culicoides imported from Belgium, the Netherlands, Germany and France. For this mechanism to pose a significant risk to BTV-free countries, large number of vectors would have to be transported. The hypothesis that infected bulls could excrete BTV in their semen led to restrictions on international trade of ruminant semen and the establishment of measures to prevent BTV transmission by semen. However, neither the risk of BTV transmission by semen nor the effectiveness of these measures was estimated quantitatively. The objective of the study was to assess, in case of introduction of BTV into a bovine semen collection centre (SCC), both the risk of BTV transmission by bovine semen and the risk reduction achieved by the preventive measures available, by means of a stochastic risk assessment model. The model was applied to different scenarios, depending on for example the type of diagnostic test and the interval between the controls of donor bulls, or the rate of BTV spread within the SCC. Enzyme-linked immunosorbant assay (ELISA) controls of donor bulls every 60 days seemed to be an ineffective method for reducing the risk of BTV transmission in contrast to polymerase chain reaction (PCR) tests every 28 days. An increase in the rate of spread within the SCC resulted in a reduced risk of BTV transmission by semen. The storage of semen for 30 days prior to dispatch seemed to be an efficient way of reducing the risk of transmission by semen. The sensitivity analysis identified the probability of BTV shedding in semen as a crucial parameter in the probability of BTV transmission by semen. However, there is a great degree of uncertainty associated with this parameter, with significant differences depending on the BTV serotype.

Columns subjected to time varying axial load may exhibit dynamic instability due to parametric resonance. This type of instability is inherent in structures; it is not due to material or geometrical imperfections, and can occur even in perfectly constructed structures. This characteristic makes parametric resonance a very difficult to predict and therefore dangerous phenomenon. In this thesis the stability of a structural column under bounded noise axial load is studied by use of Lyapunov exponents. Bounded noise is especially useful as a loading because it may be used to represent both wide and narrow band processes, making the stability equations developed general enough to handle a wide variety of real world probabilistic loadings. The equation of motion of the first mode of vibration for this system is a second-order nonlinear stochastic ordinary differential equation. The nonlinearity makes the system exhibit bifurcating behaviour where stability shifts from the trivial solution to a non-zero mean stationary solution. The stability of the trivial and non-trivial solutions is important in obtaining a complete picture of the dynamical behaviour of the system. The effect that damping, the amplitude of noise, and the level of nonlinearity have on the stability of a structural column is studied using both analytical and numerical approaches. The largest Lyapunov exponent of the trivial solution is determined analytically by using time averaged versions of the original equation of motion. The validity of the analytical time averaged equation of motion is also verified with Monte Carlo simulations. Due to the mathematical complexity the largest Lyapunov exponent of the non-trivial stationary solutions is obtained using Monte Carlo simulation only.

Although modern society is critically reliant on power grids, even modern power grids are subject to unavoidable outages due to storms, lightning strikes, and equipment failures. The situation in developing countries is even worse, with frequent load shedding lasting several hours a day due to unreliable generation. We study the use of battery storage to allow a set of homes in a single residential neighbour- hood to avoid power outages. Due to the high cost of storage, our goal is to choose the smallest battery size such that, with high target probability, there is no loss of power despite a grid out- age. Recognizing that the most common approach today for mitigating outages is to use a diesel generator (genset), we study the related problem of minimizing the carbon footprint of genset operation. Drawing on recent results, we model both problems as buffer sizing problems that can be ad- dressed using stochastic network calculus. We show that this approach greatly improves battery sizing in contrast to prior approaches. Specifically, a numerical study shows that, for a neigh- bourhood of 100 homes, our approach computes a battery size, which is less than 10% more than the minimum possible size necessary to satisfy a one day in ten years loss probability (2.7 ∗ 10^4 ). Moreover, we are able to estimate the carbon footprint reduction, compared to an exact numerical analysis, within a factor of 1.7. We also study the genset scheduling problem when the rate of genset fuel consumption is given by an affine function instead of a linear function of the current power. We give alternate scheduling, an online scheduling strategy that has a competitive ratio of (k1 G/C +k2)/(k1+k2) , where G is the genset capacity, C is the battery charging rate, and k1, k2 are the affine function constants. Numerically, we show that for a real industrial load alternate scheduling is very close to the offline optimal strategy.

Columns subjected to time varying axial load may exhibit dynamic instability due to parametric resonance. This type of instability is inherent in structures; it is not due to material or geometrical imperfections, and can occur even in perfectly constructed structures. This characteristic makes parametric resonance a very difficult to predict and therefore dangerous phenomenon. In this thesis the stability of a structural column under bounded noise axial load is studied by use of Lyapunov exponents. Bounded noise is especially useful as a loading because it may be used to represent both wide and narrow band processes, making the stability equations developed general enough to handle a wide variety of real world probabilistic loadings. The equation of motion of the first mode of vibration for this system is a second-order nonlinear stochastic ordinary differential equation. The nonlinearity makes the system exhibit bifurcating behaviour where stability shifts from the trivial solution to a non-zero mean stationary solution. The stability of the trivial and non-trivial solutions is important in obtaining a complete picture of the dynamical behaviour of the system. The effect that damping, the amplitude of noise, and the level of nonlinearity have on the stability of a structural column is studied using both analytical and numerical approaches. The largest Lyapunov exponent of the trivial solution is determined analytically by using time averaged versions of the original equation of motion. The validity of the analytical time averaged equation of motion is also verified with Monte Carlo simulations. Due to the mathematical complexity the largest Lyapunov exponent of the non-trivial stationary solutions is obtained using Monte Carlo simulation only.

This thesis focuses on task sequencing and manpower scheduling to develop robust schedules for an aircraft manufacturer. The production of an aircraft goes through a series of multiple workstations, each consisting of a large number of interactive tasks and a limited number of working zones. The duration of each task varies from operator to operator, because most operations are performed manually. These factors limit the ability of managers to balance, optimize, and change the statement of work in each workstation. In addition, engineers spend considerable amount of time to manually develop schedules that may be incompatible with the changes in the production rate. To address the above problems, the current state of work centers are first analyzed. Then, several deterministic mathematical programming models are developed to minimize the total production labour cost for a target cycle time. The mathematical models seek to find optimal schedules by eliminating and/or considering the effect of overtime on the production cost. The resulting schedules decrease the required number of operators by 16% and reduce production cycle time of work centers by 53% to 67%. Using these models, the time needed to develop a schedule is reduced from 36 days to less than a day. To handle the stochasticity of the task durations, a two-stage stochastic programming model is developed to minimize the total production labour cost and to find the number of operators that are able to work under every scenario. The solution of the two-stage stochastic programming model finds the same number of operators as that of the deterministic models, but reduces the time to adjust production schedules by 88%.

There are two sources of information available in empirical research in finance: one corresponding to historical data and the other to prices currently observed in the markets. When proposing a model, it is desirable to use information from both sources. However in modern finance, where stochastic differential equations have been one of the main modeling tools, the common models are typically different for historical data and for current market data. The former are usually assumed to be time homogeneous, while the latter are typically time in-homogeneous. This practice can be explained by the fact that a time-homogeneous model is stationary and easier to estimate, while time-inhomogeneous model are required in order to replicate market data sufficiently well without creating arbitrage opportunities. In this thesis, we study methods of statistical inference, both parametric and non-parametric, for stochastic differential equations with time-dependent parameters. In the first part, we propose a new class of stochastic differential equation with time-dependent drift and diffusion terms, where some of the parameters change according to a hidden Markov process. We show that under some technical conditions this innovative way of modeling switching times renders the resulting model stationary. We also explore different approaches to estimate parameters in our proposed model. Our simulation studies demonstrate that the parameters of the model can be efficiently estimated by using a version of the filtering method proposed in the literature. We illustrate our model and the proposed estimation method by applying them to interest rate data, and we detect significant time variations in early 1980s, when targets of the monetary policy in the United States were changed. One of the known drawbacks of parametric models is the risk of model misspecification. In the second part of the thesis, we allow the drift to be time-dependent and nonparametric, and our objective is to estimate it using a single trajectory of the process. The main idea underlying this method is to approximate the time-dependent function with a sequence of polynomials. Since we can estimate efficiently only a finite number of parameters for any finite length of data, in our method we propose to relate the number of parameters to the length of the observed trajectory. This idea is similar to the method of sieves proposed by Grenander (Abstract Inference, 1981). The asymptotic analysis that we present is based on the assumption that the length of available data $T$ increases to infinity. We investigate two cases, one is a Brownian motion with time-dependent drift and the other corresponds to a class of mean-reverting stochastic differential equations with time-dependent mean-reversion level. In both cases we prove asymptotic consistency and normality of a modified maximum likelihood estimator of the projected time-dependent component. The main challenge in proving our results in the second case stems from two features of the problem: one is due to the fact that coefficients of projections change with $T$ and the other is related to the confounding effect between the mean-reversion speed and the level function. By applying our method to the same interest rate data we use in the first part, we find another evidence of time-variation in the drift term.