We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

Hybrid singularly perturbed systems (SPSs) with time delay are considered and exponential stability of these systems is investigated. This work mainly covers switched and impulsive switched delay SPSs . Multiple Lyapunov functions technique as a tool is applied to these systems. Dwell and average dwell time approaches are used to organize the switching between subsystems (modes) so that the hybrid system is stable. Systems with all stable modes are first discussed and, after developing lemmas to ensure existence of growth rates of unstable modes, these systems are then extended to include, in addition, unstable modes. Sufficient conditions showing that impulses contribute to yield stability properties of impulsive switched systems that consist of all unstable subsystems are also established. A number of illustrative examples are presented to help motivate the study of these systems.

Designers use simulations to observe the behaviour of a system and to make design decisions to improve dynamic performance. However, for complex dynamic systems, these simulations are often time-consuming and, for robust design purposes, numerous simulations are required as a range of design variables is investigated. Furthermore, the optimum set is desired to meet specifications at particular instances in time. In this thesis, the dynamic response of a system is broken into discrete time instances and recorded into a matrix. Each column of this matrix corresponds to a discrete time instance and each row corresponds to the response at a particular design variable set. Singular Value Decomposition (SVD) is then used to separate this matrix into two matrices: one that consists of information in parameter-space and the other containing information in time-space. Metamodels are then used to efficiently and accurately calculate the response at some arbitrary set of design variables at any time. This efficiency is especially useful in Monte Carlo simulation where the responses are required at a very large sample of design variable sets. This work is then extended where the normalized sensitivities along with the first and second moments of the response are required at specific times. Later, the procedure of calculating the metamodel at specific times and how this metamodel is used in parameter design or integrated design for finding the optimum parameters given specifications at specific time steps is shown. In conclusion, this research shows that SVD and metamodelling can be used to apply probabilistic robust design tools where specifications at certain times are required for the optimum performance of a system.

Hybrid singularly perturbed systems (SPSs) with time delay are considered and exponential stability of these systems is investigated. This work mainly covers switched and impulsive switched delay SPSs . Multiple Lyapunov functions technique as a tool is applied to these systems. Dwell and average dwell time approaches are used to organize the switching between subsystems (modes) so that the hybrid system is stable. Systems with all stable modes are first discussed and, after developing lemmas to ensure existence of growth rates of unstable modes, these systems are then extended to include, in addition, unstable modes. Sufficient conditions showing that impulses contribute to yield stability properties of impulsive switched systems that consist of all unstable subsystems are also established. A number of illustrative examples are presented to help motivate the study of these systems.

Designers use simulations to observe the behaviour of a system and to make design decisions to improve dynamic performance. However, for complex dynamic systems, these simulations are often time-consuming and, for robust design purposes, numerous simulations are required as a range of design variables is investigated. Furthermore, the optimum set is desired to meet specifications at particular instances in time. In this thesis, the dynamic response of a system is broken into discrete time instances and recorded into a matrix. Each column of this matrix corresponds to a discrete time instance and each row corresponds to the response at a particular design variable set. Singular Value Decomposition (SVD) is then used to separate this matrix into two matrices: one that consists of information in parameter-space and the other containing information in time-space. Metamodels are then used to efficiently and accurately calculate the response at some arbitrary set of design variables at any time. This efficiency is especially useful in Monte Carlo simulation where the responses are required at a very large sample of design variable sets. This work is then extended where the normalized sensitivities along with the first and second moments of the response are required at specific times. Later, the procedure of calculating the metamodel at specific times and how this metamodel is used in parameter design or integrated design for finding the optimum parameters given specifications at specific time steps is shown. In conclusion, this research shows that SVD and metamodelling can be used to apply probabilistic robust design tools where specifications at certain times are required for the optimum performance of a system.

We prove that if $1 > \alpha > 1/2$, then there exists a probability measure $\mu$ such that the Hausdorff dimension of its support is $\alpha$ and $\mu*\mu$ is a Lipschitz function of class $\alpha-1/2$.

We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. By analyzing the properties of the free boundary problem, we provide an explicit characterization of model parameters for which the value function is finite. Furthermore, we prove that the value function, as well as the slopes of the lines demarcating the no-trading region, can be expanded as a series of integer powers of [lambda superscript 1/3]. The coefficients of arbitrary order in this expansion can be computed. / text

In this paper we develop a computational procedure to investigate linear fracture of two-dimensional problems in isotropic linearly elastic media. A symmetric Galerkin boundary element method (SGBEM), based on a weakly singular, weak-form traction integral equation, is adopted to model these fractures. In particular we consider multiple interacting cracks in an unbounded domain subject to internal pressure and remote stress. The growth of the cracks is driven by either linearly dependent injection pressures or volumes in each crack. A variety of crack geometries are investigated. / text