One of the classical problems in finance is that of an economic unit who aims at maximizing his expected life-time utility from consumption and/or terminal wealth by an effective asset-liability management. The purpose of this thesis is to determine the optimal investment strategies , from the point of view of their economic utility, for individual and institutional investors such pension funds.

Unsere alltägliche Erfahrung ist mit verschiedenen akustischen Einfluessen wie Lärm, aber auch Musik verbunden. Jeder weiss, wie Lärm stören kann und Kommunikation behindert oder gar unterbindet. Ähnliche optische Effekte sind bekannt: starkes Schneetreiben oder Regengüsse verschlechtern die Sicht und lassen uns Umrisse nur noch schemenhaft erkennen. Jedoch koennen ähnliche Stimuli auch sehr positive Auswirkungen haben: Autofahrer fahren bei leiser Musik konzentrierter -- die Behauptung von Schulkindern, nur bei dröhnenden Bässen die Mathehausaufgaben richtig rechnen zu können, ist allerdings nicht wissenschaftlich erwiesen. Außerordentlich interessant aus dieser Sicht sind auch Reizleitungsprozesse: Reize werden nur weitergleitet, wenn die strukturlosen Signale der Neuronen mit ausreichend starker Intensität erfolgen, also ein Schwellwert überschritten ist. <br /> <br /> Der Physiker Dr. Alexei Zaikin von der Universität Potsdam beschäftigt sich mit sogenannten rauschinduzierten Phänomenen aus theorischer Sicht. Sein Forschungsgebiet sind Prozesse, bei denen Rauschen mehrfach das Systemverhalten beeinflusst: ist es ausreichend gross, d.h. größer als ein kritischer Wert, wird eine reguläre Struktur gebildet, die durch das immernoch vorhandene Rauschen mit der Struktur des Nachbarsystems synchronisiert. Um ein solches System mit kritischem Wert zu erhalten, bedarf es einer weiteren Rauschquelle. Herr Zaikin analysierte noch weitere Beispiele solcher doppelt stochastischen Effekte. Die Ausarbeitung derartiger theoretischer Grundlagen ist wichtig, da diese Prozesse in der Neurophysik, in technischen Kommunikationssystemen und in den Lebenswissenschaften eine Rolle spielen. / Our every-day experience is connected with different acoustical noise or music. Usually noise plays the role of nuisance in any communication and destroys any order in a system. Similar optical effects are known: strong snowing or raining decreases quality of a vision. In contrast to these situations noisy stimuli can also play a positive constructive role, e.g. a driver can be more concentrated in a presence of quiet music. Transmission processes in neural systems are of especial interest from this point of view: excitation or information will be transmitted only in the case if a signal overcomes a threshold.<br /> <br /> Dr. Alexei Zaikin from the Potsdam University studies noise-induced phenomena in nonlinear systems from a theoretical point of view. Especially he is interested in the processes, in which noise influences the behaviour of a system twice: if the intensity of noise is over a threshold, it induces some regular structure that will be synchronized with the behaviour of neighbour elements. To obtain such a system with a threshold one needs one more noise source. Dr. Zaikin has analyzed further examples of such doubly stochastic effects and developed a concept of these new phenomena. These theoretical findings are important, because such processes can play a crucial role in neurophysics, technical communication devices and living sciences.

The stochastic Galerkin and collocation methods are used to solve an advection-diusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection-diusion equation onto the stochastic basis functions. High-order summationby- parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steadystate

This thesis has applied the theory of real options to study forestry investment decision-making under stochastic timber prices. Suitable models have been developed for the stochastic timber prices, after addressing major issues in characterisation of the price process. First, the assumption of stochastic timber price process was based on detailed unit root tests, incorporating structural breaks in time-series analysis. The series was found to be stationary around shifting mean, justifying the assumption of mean reversion model. Due to shift in the mean, long-run mean to which the prices tended to revert could not be assumed constant. Accordingly, it was varied in discreet steps as per the breaks identified in the tests. The timber price series failed the normality test implying fat tails in the data. To account for these fat tails, ‘jumps’ were incorporated in the mean reversion model. The results showed that the option values for the jump model were higher than the mean reversion model and threshold levels for investment implied different optimal paths. Ignoring jumps could provide sub-optimal results leading to erroneous decisions. Second, the long-run mean to which prices reverted was assumed to shift continuously in a random manner. This was modeled through the incorporation of stochastic level and slope in the trend of the prices. Since the stochastic level and slope were not observable in reality, a Kalman-filter approach was used for the estimation of model parameters. The price forecasts from the model were used to estimate option values for the harvest investment decisions. Third, investment in a carbon sequestration project from managed forests was evaluated using real options, under timber price stochasticity. The option values and threshold levels for investment were estimated, under baseline and mitigation scenarios. Results indicated that carbon sequestration from managed forests might not be a viable investment alternative due to existing bottlenecks. Overall, the research stressed upon the need for market information and adaptive management, with a pro-active approach, for efficient investment decisions in forestry.

This thesis has applied the theory of real options to study forestry investment decision-making under stochastic timber prices. Suitable models have been developed for the stochastic timber prices, after addressing major issues in characterisation of the price process. First, the assumption of stochastic timber price process was based on detailed unit root tests, incorporating structural breaks in time-series analysis. The series was found to be stationary around shifting mean, justifying the assumption of mean reversion model. Due to shift in the mean, long-run mean to which the prices tended to revert could not be assumed constant. Accordingly, it was varied in discreet steps as per the breaks identified in the tests. The timber price series failed the normality test implying fat tails in the data. To account for these fat tails, ‘jumps’ were incorporated in the mean reversion model. The results showed that the option values for the jump model were higher than the mean reversion model and threshold levels for investment implied different optimal paths. Ignoring jumps could provide sub-optimal results leading to erroneous decisions. Second, the long-run mean to which prices reverted was assumed to shift continuously in a random manner. This was modeled through the incorporation of stochastic level and slope in the trend of the prices. Since the stochastic level and slope were not observable in reality, a Kalman-filter approach was used for the estimation of model parameters. The price forecasts from the model were used to estimate option values for the harvest investment decisions. Third, investment in a carbon sequestration project from managed forests was evaluated using real options, under timber price stochasticity. The option values and threshold levels for investment were estimated, under baseline and mitigation scenarios. Results indicated that carbon sequestration from managed forests might not be a viable investment alternative due to existing bottlenecks. Overall, the research stressed upon the need for market information and adaptive management, with a pro-active approach, for efficient investment decisions in forestry.

Runway systems at airports have been identified as a major source of delay in the aviation system and efficient runway operations are, therefore, important to maintain and/or increase the capacity of the entire aviation system. The goal of the airport runway scheduling problem is to schedule a set of aircraft and minimize a given objective while maintaining separation requirements and enforcing other operational constraints. Uncertain factors such as weather, surrounding traffic and pilot behavior affect when aircraft can be scheduled, and these factors need to be considered in planning models. In this thesis we propose two stochastic programs to address the stochastic airport runway scheduling problem and similarly structured machine scheduling problems. In the first part, we develop a two-stage stochastic integer programming model and analyze it by developing alternative formulations and solution methods. As part of our analysis, we first show that a restricted version of the stochastic runway scheduling problem is equivalent to a machine scheduling problem on a single machine with sequence dependent setup times and stochastic due dates. We then extend this restricted model by considering characteristics specific to the runway scheduling problem and present two different stochastic integer programming models. We derive some tight valid inequalities for these formulations, and we propose a solution methodology based on sample average approximation and Lagrangian based scenario decomposition. Realistic data sets are then used to perform a detailed computational study involving implementations and analyses of several different configurations of the models. The results from the computational tests indicate that practically implementable truncated versions of the proposed solution algorithm almost always produce very high quality solutions. In the second part, we propose a sampling based stochastic program for a general machine scheduling problem with similar characteristics as the airport runway scheduling problem. The sampling based approach allows us to capture more detailed aspects of the problem, such as taxiway operations crossing active runways. The model is based on the stochastic branch and bound algorithm with several enhancements to improve the computational performance. More specifically, we incorporate a method to dynamically update the sample sizes in various parts of the branching tree, effectively decreasing the runtime without worsening the solution quality. When applied to runway scheduling, the algorithm is able to produce schedules with makespans that are 5% to 7% shorter than those obtained by optimal deterministic methods. Additional contributions in this thesis include the development of a global cost function, capturing all relevant costs in airport runway scheduling and trading off different, sometimes conflicting, objectives. We also analyze the impact of including environmental factors in the scheduling process.

In this thesis I discuss the method of time-change and its applications in quantitative finance. I mainly consider the time change by writing a continuous diffusion process as a Brownian motion subordinated by a subordinator process. I divide the time change method into two cases: deterministic time change and stochastic time change. The difference lies in whether the subordinator process is a deterministic function of time or a stochastic process of time. Time-changed Brownian motion with deterministic time change provides a new viewpoint to deal with option pricing under stochastic interest rates and I utilize this idea in pricing various exotic options under stochastic interest rates. Time-changed Brownian motion with stochastic time change is more complicated and I give the equivalence in law relation governing the ``original time" and the ``new stochastic time" under different clocks. This is readily applicable in pricing a new product called ``timer option". It can also be used in pricing barrier options under the Heston stochastic volatility model. Conclusion and further research directions in exploring the ideas of time change method in other areas of quantitative finance are in the last chapter.

In this thesis I discuss the method of time-change and its applications in quantitative finance. I mainly consider the time change by writing a continuous diffusion process as a Brownian motion subordinated by a subordinator process. I divide the time change method into two cases: deterministic time change and stochastic time change. The difference lies in whether the subordinator process is a deterministic function of time or a stochastic process of time. Time-changed Brownian motion with deterministic time change provides a new viewpoint to deal with option pricing under stochastic interest rates and I utilize this idea in pricing various exotic options under stochastic interest rates. Time-changed Brownian motion with stochastic time change is more complicated and I give the equivalence in law relation governing the ``original time" and the ``new stochastic time" under different clocks. This is readily applicable in pricing a new product called ``timer option". It can also be used in pricing barrier options under the Heston stochastic volatility model. Conclusion and further research directions in exploring the ideas of time change method in other areas of quantitative finance are in the last chapter.

We consider the problem of staffing large-scale service centers with multiple customer classes and agent types operating under quality-of-service (QoS) constraints. We introduce formulations for a class of staffing problems, minimizing the cost of staffing while requiring that the long-run average QoS achieves a certain pre-specified level. The queueing models we use to define such service center staffing problems have random inter-arrival times and random service times. The models we study differ with respect to whether the arrival rates are deterministic or stochastic. In the deterministic version of the service center staffing problem, we assume that the customer arrival rates are known deterministically. It is computationally challenging to solve our service center staffing problem with deterministic arrival rates. Thus, we provide an approximation and prove that the solution of our approximation is asymptotically optimal in the sense that the gap between the optimal value of the exact model and the objective function value of the approximate solution shrinks to zero as the size of the system grows large. In our work, we also focus on doubly stochastic service center systems; that is, we focus on solving large-scale service center staffing problems when the arrival rates are uncertain in addition to the inherent randomness of the system's inter-arrival times and service times. This brings the modeling closer to reality. In solving the service center staffing problems with deterministic arrival rates, we provide a solution procedure for solving staffing problems for doubly stochastic service center systems. We consider a decision making scheme in which we must select staffing levels before observing the arrival rates. We assume that the decision maker has distributional information about the arrival rates at the time of decision making. In the presence of arrival-rate uncertainty, the decision maker's goal is to minimize the staffing cost, while ensuring the QoS achieves a given level. We show that as the system scales large in size, there is at most one key scenario under which the probability of waiting converges to a non-trivial value, i.e., a value strictly between 0 and 1. That is, the system is either over- or under-loaded in any other scenario as the size of the system grows to infinity. Exploiting this result, we propose a two-step solution procedure for the staffing problem with random arrival rates. In the first step, we use the desired QoS level to identify the key scenario corresponding to the optimal staffing level. After finding the key scenario, the random arrival-rate model reduces to a deterministic arrival-rate model. In the second step, we solve the resulting model, with deterministic arrival rate, by using the approximation model we point to above. The approximate optimal staffing level obtained in this procedure asymptotically converges to the true optimal staffing level for the random arrival-rate problem. The decision making scheme we sketch above, assumes that the distribution of the random arrival rates is known at the time of decision making. In reality this distribution must be estimated based on historical data and experience, and needs to be updated as new observations arrive. Another important issue that arises in service center management is that in the daily operation in service centers, the daily operational period is split into small decision time periods, for example, hourly periods, and then the staffing decisions need to be made for all such time periods. Thus, to achieve an overall optimal daily staffing policy, one must deal with the interaction among staffing decisions over adjacent time periods. In our work, we also build a model that handles the above two issues. We build a two-stage stochastic model with recourse that provides the staffing decisions over two adjacent decision time periods, i.e., two adjacent decision stages. The model minimizes the first stage staffing cost and the expected second stage staffing cost while satisfying a service quality constraint on the second stage operation. A Bayesian update is used to obtain the second-stage arrival-rate distribution based on the first-stage arrival-rate distribution and the arrival observations in the first stage. The second-stage distribution is used in the constraint on the second stage service quality. After reformulation, we show that our two-stage model can be expressed as a newsvendor model, albeit with a demand that is derived from the first stage decision. We provide an algorithm that can solve the two-stage staffing problem under the most commonly used QoS constraints. This work uses stochastic programming methods to solve problems arising in queueing networks. We hope that the ideas that we put forward in this dissertation lead to other attempts to deal with decision making under uncertainty for queueing systems that combine techniques from stochastic programming and analysis tools from queueing theory. / text

Fluid networks are useful tools for analyzing complex manufacturing environments especially in semiconductor wafer fabrication. The makespan of a fluid network is defined as the time to drain the system, when there is fluid present in the buffers initially. Based on this definition, the question of determining the allocation of resources so as to minimize the makespan of a fluid network is known as the makespan problem. In the deterministic version of the makespan problem, it is assumed that the parameters of the system, such as incoming rates, service rates and initial inventory, are known deterministically. The deterministic version of the makespan problem for reentrant lines and multiclass fluid networks has been investigated in the literature and an analytical solution for the problem is well-known. In this work, we provide another formulation for the deterministic makespan problem and prove that the problem can be solved for each station separately. Optimal solutions for the deterministic makespan problem have been used as a guide to develop heuristics methods to solve makespan scheduling problem in the job-shop context in the literature. This provides one motivation for further investigation of the fluid makespan problem. In this work our main focus is solving the makespan problem when the problem parameters are uncertain. This uncertainty may be caused by various factors such as the unpredictability of the arrival process or randomness in machine availability due to failures. In the presence of parameter uncertainty, the decision maker's goal is to optimally allocate the capacity in order to minimize the expected value of the makespan. We assume that the decision maker has distributional information about the parameters at the time of decision making. We consider two decision making schemes. In the first scheme, the controller sets the allocations before observing the parameters. After the initial allocations are set, they cannot be changed. In the second scheme, the controller is allowed a recourse action after a data collection process. It is shown that in terms of obtaining the optimal control, both schemes differ considerably from the deterministic version of the problem. We formulate both schemes using stochastic programming techniques. The first scheme is easier to analyze since the resulting model is convex. Unfortunately, under the second decision scheme, the objective function is non-convex. We develop a branch-and-bound methodology to solve the resulting stochastic non-convex program. Finally, we identify some special cases where the stochastic problem is analytically solvable. This work uses stochastic programming techniques to formulate and solve a problem arising in queueing networks. Stochastic programming and queueing systems are two major areas of Operations Research that deal with decision making under uncertainty. To the best of our knowledge, this dissertation is one of the first works that brings these two major areas together.