Credit risk & forward price models

Doutoramento em Gestão

On multimodality of obnoxious faclity location models

Obnoxious single facility location models are models that have the aim to find the best location for an undesired facility. Undesired is usually expressed in relation to the so-called demand points that represent locations hindered by the facility. Because obnoxious facility location models as a rule are multimodal, the standard techniques of convex analysis used for locating desirable facilities in the plane may be trapped in local optima instead of the desired global optimum. It is assumed that having more optima coincides with being harder to solve. In this thesis the multimodality of obnoxious single facility location models is investigated in order to know which models are challenging problems in facility location problems and which are suitable for site selection. Selected for this are the obnoxious facility models that appear to be most important in literature. These are the maximin model, that maximizes the minimum distance from demand point to the obnoxious facility, the maxisum model, that maximizes the sum of distance from the demand points to the facility and the minisum model, that minimizes the sum of damage of the facility to the demand points. All models are measured with the Euclidean distances and some models also with the rectilinear distance metric. Furthermore a suitable algorithm is selected for testing multimodality. Of the tested algorithms in this thesis, Multistart is most appropriate. A small numerical experiment shows that Maximin models have on average the most optima, of which the model locating an obnoxious linesegment has the most. Maximin models have few optima and are thus not very hard to solve. From the Minisum models, the models that have the most optima are models that take wind into account. In general can be said that the generic models have less optima than the weighted versions. Models that are measured with the rectilinear norm do have more solutions than the same models measured with the Euclidean norm. This can be explained for the maximin models in the numerical example because the shape of the norm coincides with a bound of the feasible area, so not all solutions are different optima. The difference found in number of optima of the Maxisum and Minisum can not be explained by this phenomenon.

Hilbert Space Valued Signal Plus Noise Models: Analysis of Structural Breaks under High Dimensionality and Temporal Dependence

This thesis is concerned with change point analysis for time series, i.e. with detection of structural breaks in time-ordered, random data. This long-standing research field regained popularity over the last few years and is still undergoing, as statistical analysis in general, a transformation to high-dimensional problems. We focus on the fundamental »change in the mean« problem and provide extensions of the classical non-parametric Darling-Erdős-type cumulative sum (CUSUM) testing and estimation theory within highdimensional Hilbert space settings. In the first part we contribute to (long run) principal component based testing methods for Hilbert space valued time series under a rather broad (abrupt, epidemic, gradual, multiple) change setting and under dependence. For the dependence structure we consider either traditional m-dependence assumptions or more recently developed m-approximability conditions which cover, e.g., MA, AR and ARCH models. We derive Gumbel and Brownian bridge type approximations of the distribution of the test statistic under the null hypothesis of no change and consistency conditions under the alternative. A new formulation of the test statistic using projections on subspaces allows us to simplify the standard proof techniques and to weaken common assumptions on the covariance structure. Furthermore, we propose to adjust the principal components by an implicit estimation of a (possible) change direction. This approach adds flexibility to projection based methods, weakens typical technical conditions and provides better consistency properties under the alternative. In the second part we contribute to estimation methods for common changes in the means of panels of Hilbert space valued time series. We analyze weighted CUSUM estimates within a recently proposed »high-dimensional low sample size (HDLSS)« framework, where the sample size is fixed but the number of panels increases. We derive sharp conditions on »pointwise asymptotic accuracy« or »uniform asymptotic accuracy« of those estimates in terms of the weighting function. Particularly, we prove that a covariance-based correction of Darling-Erdős-type CUSUM estimates is required to guarantee uniform asymptotic accuracy under moderate dependence conditions within panels and that these conditions are fulfilled, e.g., by any MA(1) time series. As a counterexample we show that for AR(1) time series, close to the non-stationary case, the dependence is too strong and uniform asymptotic accuracy cannot be ensured. Finally, we conduct simulations to demonstrate that our results are practically applicable and that our methodological suggestions are advantageous.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.