Option pricing in market models driven by telegraph processes with jumps

Esta tesis está dividida en dos partes: en la primera parte se presentan y estudian los procesos telegráficos, los procesos de Poisson con compensador telegráfico y los procesos telegráficos con saltos. El estudio presentado en esta primera parte incluye el cálculo de las distribuciones de cada proceso, las medias y varianzas, así como las funciones generadoras de momentos entre otras propiedades. Utilizando estas propiedades en la segunda parte se estudian los modelos de valoración de opciones basados en procesos telegráficos con saltos. En esta parte se da una descripción de cómo calcular las medidas neutrales al riesgo, se encuentra la condición de no arbitraje en este tipo de modelos y por último se calcula el precio de las opciones Europeas de compra y venta.

An approximation to the standard of living index: the Colombian case

El documento busca mostrar las principales propiedades de un indicador de estándar de vida dentro del enfoque teórico de Amartya Sen. Establecemos un puente entre conceptos tales como: bienestar económico, bienestar, logro de agencia y estándar de vida. La metodología del Optimal scaling fue usada para probar las propiedades de Monotonicidad, No independencia de alternativas irrelevantes, Concavidad, informatividad y sustituibilidad.

Input usage, output mix and industry deregulation: an analysis of the Australian dairy manufacturing industry

In this paper we estimate a Translog output distance function for a balanced panel of state level data for the Australian dairy processing sector. We estimate a fixed effects specification employing Bayesian methods, with and without the imposition of monotonicity and curvature restrictions. Our results indicate that Tasmania and Victoria are the most technically efficient states with New South Wales being the least efficient. The imposition of theoretical restrictions marginally affects the results especially with respect to estimates of technical change and industry deregulation. Importantly, our bias estimates show changes in both input use and output mix that result from deregulation. Specifically, we find that deregulation has positively biased the production of butter, cheese and powders.

Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays

In this paper, we study the periodic oscillatory behavior of a class of bidirectional associative memory (BAM) networks with finite distributed delays. A set of criteria are proposed for determining global exponential periodicity of the proposed BAM networks, which assume neither differentiability nor monotonicity of the activation function of each neuron. In addition, our criteria are easily checkable. (c) 2005 Elsevier Inc. All rights reserved.

Global exponential periodicity of a class of neural networks with recent-history distributed delays

In this paper, we propose to study a class of neural networks with recent-history distributed delays. A sufficient condition is derived for the global exponential periodicity of the proposed neural networks, which has the advantage that it assumes neither the differentiability nor monotonicity of the activation function of each neuron nor the symmetry of the feedback matrix or delayed feedback matrix. Our criterion is shown to be valid by applying it to an illustrative system. (c) 2005 Elsevier Ltd. All rights reserved.

A geometric approach to generalized Stokes conjectures

We consider the Stokes conjecture concerning the shape of extreme two-dimensional water waves. By new geometric methods including a nonlinear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.

The Stokes conjecture for waves with vorticity

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity. We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity. In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far. Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

Extended factorizations of exponential functionals of Lévy processes

Pardo, Patie, and Savov derived, under mild conditions, a Wiener-Hopf type factorization for the exponential functional of proper Lévy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by considering the exponential functional for killed Lévy processes. As a by-product, we derive some interesting fine distributional properties enjoyed by a large class of this random variable, such as the absolute continuity of its distribution and the smoothness, boundedness or complete monotonicity of its density. This type of results is then used to derive similar properties for the law of maxima and first passage time of some stable Lévy processes. Thus, for example, we show that for any stable process with $\rho\in(0,\frac{1}{\alpha}-1]$, where $\rho\in[0,1]$ is the positivity parameter and $\alpha$ is the stable index, then the first passage time has a bounded and non-increasing density on $\mathbb{R}_+$. We also generate many instances of integral or power series representations for the law of the exponential functional of Lévy processes with one or two-sided jumps. The proof of our main results requires different devices from the one developed by Pardo, Patie, Savov. It relies in particular on a generalization of a transform recently introduced by Chazal et al together with some extensions to killed Lévy process of Wiener-Hopf techniques. The factorizations developed here also allow for further applications which we only indicate here also allow for further applications which we only indicate here.

Transreal arithmetic as a consistent basis for paraconsistent logics

Paraconsistent logics are non-classical logics which allow non-trivial and consistent reasoning about inconsistent axioms. They have been pro- posed as a formal basis for handling inconsistent data, as commonly arise in human enterprises, and as methods for fuzzy reasoning, with applica- tions in Artificial Intelligence and the control of complex systems. Formalisations of paraconsistent logics usually require heroic mathe- matical efforts to provide a consistent axiomatisation of an inconsistent system. Here we use transreal arithmetic, which is known to be consis- tent, to arithmetise a paraconsistent logic. This is theoretically simple and should lead to efficient computer implementations. We introduce the metalogical principle of monotonicity which is a very simple way of making logics paraconsistent. Our logic has dialetheaic truth values which are both False and True. It allows contradictory propositions, allows variable contradictions, but blocks literal contradictions. Thus literal reasoning, in this logic, forms an on-the- y, syntactic partition of the propositions into internally consistent sets. We show how the set of all paraconsistent, possible worlds can be represented in a transreal space. During the development of our logic we discuss how other paraconsistent logics could be arithmetised in transreal arithmetic.

Proximal methods for nonlinear programming: double regularization and inexact subproblems

This paper describes the first phase of a project attempting to construct an efficient general-purpose nonlinear optimizer using an augmented Lagrangian outer loop with a relative error criterion, and an inner loop employing a state-of-the art conjugate gradient solver. The outer loop can also employ double regularized proximal kernels, a fairly recent theoretical development that leads to fully smooth subproblems. We first enhance the existing theory to show that our approach is globally convergent in both the primal and dual spaces when applied to convex problems. We then present an extensive computational evaluation using the CUTE test set, showing that some aspects of our approach are promising, but some are not. These conclusions in turn lead to additional computational experiments suggesting where to next focus our theoretical and computational efforts.

Monotone approximation of aggregation operators using least squares splines

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.

Input usage, output mix and industry deregulation: an analysis of the Australian dairy manufacturing industry

In this paper we estimate a Translog output distance function for a balanced panel of state level data for the Australian dairy processing sector. We estimate a fixed effects specification employing Bayesian methods, with and without the imposition of monotonicity and curvature restrictions. Our results indicate that Tasmania and Victoria are the most technically efficient states with New South Wales being the least efficient. The imposition of theoretical restrictions marginally affects the results especially with respect to estimates of technical change and industry deregulation. Importantly, our bias estimates show changes in both input use and output mix that result from deregulation. Specifically, we find that deregulation has positively biased the production of butter, cheese and powders.

Shape preserving approximation using least squares splines

Least squares polynomial splines are an effective tool for data fitting, but they may fail to preserve essential properties of the underlying function, such as monotonicity or convexity. The shape restrictions are translated into linear inequality conditions on spline coefficients. The basis functions are selected in such a way that these conditions take a simple form, and the problem becomes non-negative least squares problem, for which effecitive and robust methods of solution exist. Multidimensional monotone approximation is achieved by using tensor-product splines with the appropriate restrictions. Additional inter polation conditions can also be introduced. The conversion formulas to traditional B-spline representation are provided.

Application of real and functional analysis to solve boundary value problems

This thesis is about using appropriate tools in functional analysis arid classical analysis to tackle the problem of existence and uniqueness of nonlinear partial differential equations. There being no unified strategy to deal with these equations, one approaches each equation with an appropriate method, depending on the characteristics of the equation. The correct setting of the problem in appropriate function spaces is the first important part on the road to the solution. Here, we choose the setting of Sobolev spaces. The second essential part is to choose the correct tool for each equation. In the first part of this thesis (Chapters 3 and 4) we consider a variety of nonlinear hyperbolic partial differential equations with mixed boundary and initial conditions. The methods of compactness and monotonicity are used to prove existence and uniqueness of the solution (Chapter 3). Finding a priori estimates is the main task in this analysis. For some types of nonlinearity, these estimates cannot be easily obtained, arid so these two methods cannot be applied directly. In this case, we first linearise the equation, using linear recurrence (Chapter 4). In the second part of the thesis (Chapter 5), by using an appropriate tool in functional analysis (the Sobolev Imbedding Theorem), we are able to improve previous results on a posteriori error estimates for the finite element method of lines applied to nonlinear parabolic equations. These estimates are crucial in the design of adaptive algorithms for the method, and previous analysis relies on, what we show to be, unnecessary assumptions which limit the application of the algorithms. Our analysis does not require these assumptions. In the last part of the thesis (Chapter 6), staying with the theme of choosing the most suitable tools, we show that using classical analysis in a proper way is in some cases sufficient to obtain considerable results. We study in this chapter nonexistence of positive solutions to Laplace's equation with nonlinear Neumann boundary condition. This problem arises when one wants to study the blow-up at finite time of the solution of the corresponding parabolic problem, which models the heating of a substance by radiation. We generalise known results which were obtained by using more abstract methods.

The nash solution and relevant expansions

In this paper we propose a new axiom, Relevant Balanced Expansion Monotonicity (RBEM), to replace Nash’s original Independence of Irrelevant Alternatives (HA) axiom. Using the RBEM axiom and the other three original axioms suggested by Nash we provide a characterization of the Nash Solution.