Ruin and related quantities in some advanced insurance risk models

Risk theory has been a very active research area over the last decades. The main objectives of the theory are to find adequate stochastic processes which can model the surplus of a (non-life) insurance company and to analyze the risk related quantities such as ruin time, ruin probability, expected discounted penalty function and expected discounted dividend/tax payments. The study of these ruin related quantities provides crucial information for actuaries and decision makers. This thesis consists of the study of four different insurance risk models which are essentially related. The ruin and related quantities are investigated by using different techniques, resulting in explicit or asymptotic expressions for the ruin time, the ruin probability, the expected discounted penalty function and the expected discounted tax payments. - La recherche en théorie du risque a été très dynamique au cours des dernières décennies. D'un point de vue théorique, les principaux objectifs sont de trouver des processus stochastiques adéquats permettant de modéliser le surplus d'une compagnie d'assurance non vie et d'analyser les mesures de risque, notamment le temps de ruine, la probabilité de ruine, l'espérance de la valeur actuelle de la fonction de pénalité et l'espérance de la valeur actuelle des dividendes et taxes. L'étude de ces mesures associées à la ruine fournit des informations cruciales pour les actuaires et les décideurs. Cette thèse consiste en l'étude des quatre différents modèles de risque d'assurance qui sont essentiellement liés. La ruine et les mesures qui y sont associées sont examinées à l'aide de différentes techniques, ce qui permet d'induire des expressions explicites ou asymptotiques du temps de ruine, de la probabilité de ruine, de l'espérance de la valeur actuelle de la fonction de pénalité et l'espérance de la valeur actuelle des dividendes et taxes.

Some optimization and decision problems in proportional reinsurance

Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function (Gerber & Shiu, 1998) is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs.

Étude empirique de distributions associées à la Fonction de Pénalité Escomptée

On présente une nouvelle approche de simulation pour la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine, pour des modèles de risque déterminés par des subordinateurs de Lévy. Cette approche s'inspire de la décomposition "Ladder height" pour la probabilité de ruine dans le Modèle Classique. Ce modèle, déterminé par un processus de Poisson composé, est un cas particulier du modèle plus général déterminé par un subordinateur, pour lequel la décomposition "Ladder height" de la probabilité de ruine s'applique aussi. La Fonction de Pénalité Escomptée, encore appelée Fonction Gerber-Shiu (Fonction GS), a apporté une approche unificatrice dans l'étude des quantités liées à l'événement de la ruine été introduite. La probabilité de ruine et la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine sont des cas particuliers de la Fonction GS. On retrouve, dans la littérature, des expressions pour exprimer ces deux quantités, mais elles sont difficilement exploitables de par leurs formes de séries infinies de convolutions sans formes analytiques fermées. Cependant, puisqu'elles sont dérivées de la Fonction GS, les expressions pour les deux quantités partagent une certaine ressemblance qui nous permet de nous inspirer de la décomposition "Ladder height" de la probabilité de ruine pour dériver une approche de simulation pour cette fonction de densité conjointe. On présente une introduction détaillée des modèles de risque que nous étudions dans ce mémoire et pour lesquels il est possible de réaliser la simulation. Afin de motiver ce travail, on introduit brièvement le vaste domaine des mesures de risque, afin d'en calculer quelques unes pour ces modèles de risque. Ce travail contribue à une meilleure compréhension du comportement des modèles de risques déterminés par des subordinateurs face à l'éventualité de la ruine, puisqu'il apporte un point de vue numérique absent de la littérature.

Some Applications of Markov Additive Processes as Models in Insurance and Financial Mathematics

Cette thèse est principalement constituée de trois articles traitant des processus markoviens additifs, des processus de Lévy et d'applications en finance et en assurance. Le premier chapitre est une introduction aux processus markoviens additifs (PMA), et une présentation du problème de ruine et de notions fondamentales des mathématiques financières. Le deuxième chapitre est essentiellement l'article "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" écrit en collaboration avec Manuel Morales et publié dans la revue European Actuarial Journal. Cet article étudie le problème de ruine pour un processus de risque markovien additif. Une identification de systèmes de Lévy est obtenue et utilisée pour donner une expression de l'espérance de la fonction de pénalité actualisée lorsque le PMA est un processus de Lévy avec changement de régimes. Celle-ci est une généralisation des résultats existant dans la littérature pour les processus de risque de Lévy et les processus de risque markoviens additifs avec sauts "phase-type". Le troisième chapitre contient l'article "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" qui est soumis pour publication. Cet article présente une extension de l'espérance de la fonction de pénalité actualisée pour un processus subordinateur de risque perturbé par un mouvement brownien. Cette extension contient une série de fonctions escomptée éspérée des minima successives dus aux sauts du processus de risque après la ruine. Celle-ci a des applications importantes en gestion de risque et est utilisée pour déterminer la valeur espérée du capital d'injection actualisé. Finallement, le quatrième chapitre contient l'article "The Minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model" écrit en collaboration avec Romuald Hervé Momeya et publié dans la revue Asia-Pacific Financial Market. Cet article présente de nouveaux résultats en lien avec le problème de l'incomplétude dans un marché financier où le processus de prix de l'actif risqué est décrit par un modèle exponentiel markovien additif. Ces résultats consistent à charactériser la mesure martingale satisfaisant le critère de l'entropie. Cette mesure est utilisée pour calculer le prix d'une option, ainsi que des portefeuilles de couverture dans un modèle exponentiel de Lévy avec changement de régimes.

Projected gradient approach to the numerical solution of the SCoTLASS

The SCoTLASS problem-principal component analysis modified so that the components satisfy the Least Absolute Shrinkage and Selection Operator (LASSO) constraint-is reformulated as a dynamical system on the unit sphere. The LASSO inequality constraint is tackled by exterior penalty function. A globally convergent algorithm is developed based on the projected gradient approach. The algorithm is illustrated numerically and discussed on a well-known data set. (c) 2004 Elsevier B.V. All rights reserved.

A modified Primal-Dual Logarithmic-Barrier Method for solving the Optimal Power Flow problem with discrete and continuous control variables

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A modified Primal-Dual Logarithmic-Barrier Method for solving the Optimal Power Flow problem with discrete and continuous control variables

The aim of solving the Optimal Power Flow problem is to determine the optimal state of an electric power transmission system, that is, the voltage magnitude and phase angles and the tap ratios of the transformers that optimize the performance of a given system, while satisfying its physical and operating constraints. The Optimal Power Flow problem is modeled as a large-scale mixed-discrete nonlinear programming problem. This paper proposes a method for handling the discrete variables of the Optimal Power Flow problem. A penalty function is presented. Due to the inclusion of the penalty function into the objective function, a sequence of nonlinear programming problems with only continuous variables is obtained and the solutions of these problems converge to a solution of the mixed problem. The obtained nonlinear programming problems are solved by a Primal-Dual Logarithmic-Barrier Method. Numerical tests using the IEEE 14, 30, 118 and 300-Bus test systems indicate that the method is efficient. (C) 2012 Elsevier B.V. All rights reserved.

El problema del rozamiento en el análisis de estructuras de fábrica mediante modelos de sólidos rígidos

La evaluación de la seguridad de estructuras antiguas de fábrica es un problema abierto.El material es heterogéneo y anisótropo, el estado previo de tensiones difícil de conocer y las condiciones de contorno inciertas. A comienzos de los años 50 se demostró que el análisis límite era aplicable a este tipo de estructuras, considerándose desde entonces como una herramienta adecuada. En los casos en los que no se produce deslizamiento la aplicación de los teoremas del análisis límite estándar constituye una herramienta formidable por su simplicidad y robustez. No es necesario conocer el estado real de tensiones. Basta con encontrar cualquier solución de equilibrio, y que satisfaga las condiciones de límite del material, en la seguridad de que su carga será igual o inferior a la carga real de inicio de colapso. Además esta carga de inicio de colapso es única (teorema de la unicidad) y se puede obtener como el óptimo de uno cualquiera entre un par de programas matemáticos convexos duales. Sin embargo, cuando puedan existir mecanismos de inicio de colapso que impliquen deslizamientos, cualquier solución debe satisfacer tanto las restricciones estáticas como las cinemáticas, así como un tipo especial de restricciones disyuntivas que ligan las anteriores y que pueden plantearse como de complementariedad. En este último caso no está asegurada la existencia de una solución única, por lo que es necesaria la búsqueda de otros métodos para tratar la incertidumbre asociada a su multiplicidad. En los últimos años, la investigación se ha centrado en la búsqueda de un mínimo absoluto por debajo del cual el colapso sea imposible. Este método es fácil de plantear desde el punto de vista matemático, pero intratable computacionalmente, debido a las restricciones de complementariedad 0 y z 0 que no son ni convexas ni suaves. El problema de decisión resultante es de complejidad computacional No determinista Polinomial (NP)- completo y el problema de optimización global NP-difícil. A pesar de ello, obtener una solución (sin garantía de exito) es un problema asequible. La presente tesis propone resolver el problema mediante Programación Lineal Secuencial, aprovechando las especiales características de las restricciones de complementariedad, que escritas en forma bilineal son del tipo y z = 0; y 0; z 0 , y aprovechando que el error de complementariedad (en forma bilineal) es una función de penalización exacta. Pero cuando se trata de encontrar la peor solución, el problema de optimización global equivalente es intratable (NP-difícil). Además, en tanto no se demuestre la existencia de un principio de máximo o mínimo, existe la duda de que el esfuerzo empleado en aproximar este mínimo esté justificado. En el capítulo 5, se propone hallar la distribución de frecuencias del factor de carga, para todas las soluciones de inicio de colapso posibles, sobre un sencillo ejemplo. Para ello, se realiza un muestreo de soluciones mediante el método de Monte Carlo, utilizando como contraste un método exacto de computación de politopos. El objetivo final es plantear hasta que punto está justificada la busqueda del mínimo absoluto y proponer un método alternativo de evaluación de la seguridad basado en probabilidades. Las distribuciones de frecuencias, de los factores de carga correspondientes a las soluciones de inicio de colapso obtenidas para el caso estudiado, muestran que tanto el valor máximo como el mínimo de los factores de carga son muy infrecuentes, y tanto más, cuanto más perfecto y contínuo es el contacto. Los resultados obtenidos confirman el interés de desarrollar nuevos métodos probabilistas. En el capítulo 6, se propone un método de este tipo basado en la obtención de múltiples soluciones, desde puntos de partida aleatorios y calificando los resultados mediante la Estadística de Orden. El propósito es determinar la probabilidad de inicio de colapso para cada solución.El método se aplica (de acuerdo a la reducción de expectativas propuesta por la Optimización Ordinal) para obtener una solución que se encuentre en un porcentaje determinado de las peores. Finalmente, en el capítulo 7, se proponen métodos híbridos, incorporando metaheurísticas, para los casos en que la búsqueda del mínimo global esté justificada. Abstract Safety assessment of the historic masonry structures is an open problem. The material is heterogeneous and anisotropic, the previous state of stress is hard to know and the boundary conditions are uncertain. In the early 50's it was proven that limit analysis was applicable to this kind of structures, being considered a suitable tool since then. In cases where no slip occurs, the application of the standard limit analysis theorems constitutes an excellent tool due to its simplicity and robustness. It is enough find any equilibrium solution which satisfy the limit constraints of the material. As we are certain that this load will be equal to or less than the actual load of the onset of collapse, it is not necessary to know the actual stresses state. Furthermore this load for the onset of collapse is unique (uniqueness theorem), and it can be obtained as the optimal from any of two mathematical convex duals programs However, if the mechanisms of the onset of collapse involve sliding, any solution must satisfy both static and kinematic constraints, and also a special kind of disjunctive constraints linking the previous ones, which can be formulated as complementarity constraints. In the latter case, it is not guaranted the existence of a single solution, so it is necessary to look for other ways to treat the uncertainty associated with its multiplicity. In recent years, research has been focused on finding an absolute minimum below which collapse is impossible. This method is easy to set from a mathematical point of view, but computationally intractable. This is due to the complementarity constraints 0 y z 0 , which are neither convex nor smooth. The computational complexity of the resulting decision problem is "Not-deterministic Polynomialcomplete" (NP-complete), and the corresponding global optimization problem is NP-hard. However, obtaining a solution (success is not guaranteed) is an affordable problem. This thesis proposes solve that problem through Successive Linear Programming: taking advantage of the special characteristics of complementarity constraints, which written in bilinear form are y z = 0; y 0; z 0 ; and taking advantage of the fact that the complementarity error (bilinear form) is an exact penalty function. But when it comes to finding the worst solution, the (equivalent) global optimization problem is intractable (NP-hard). Furthermore, until a minimum or maximum principle is not demonstrated, it is questionable that the effort expended in approximating this minimum is justified. XIV In chapter 5, it is proposed find the frequency distribution of the load factor, for all possible solutions of the onset of collapse, on a simple example. For this purpose, a Monte Carlo sampling of solutions is performed using a contrast method "exact computation of polytopes". The ultimate goal is to determine to which extent the search of the global minimum is justified, and to propose an alternative approach to safety assessment based on probabilities. The frequency distributions for the case study show that both the maximum and the minimum load factors are very infrequent, especially when the contact gets more perfect and more continuous. The results indicates the interest of developing new probabilistic methods. In Chapter 6, is proposed a method based on multiple solutions obtained from random starting points, and qualifying the results through Order Statistics. The purpose is to determine the probability for each solution of the onset of collapse. The method is applied (according to expectations reduction given by the Ordinal Optimization) to obtain a solution that is in a certain percentage of the worst. Finally, in Chapter 7, hybrid methods incorporating metaheuristics are proposed for cases in which the search for the global minimum is justified.

Thermal food processing optimization: Algorithms and software

The algorithms and graphic user interface software package ?OPT-PROx? are developed to meet food engineering needs related to canned food thermal processing simulation and optimization. The adaptive random search algorithm and its modification coupled with penalty function?s approach, and the finite difference methods with cubic spline approximation are utilized by ?OPT-PROx? package (http://tomakechoice. com/optprox/index.html). The diversity of thermal food processing optimization problems with different objectives and required constraints are solvable by developed software. The geometries supported by the ?OPT-PROx? are the following: (1) cylinder, (2) rectangle, (3) sphere. The mean square error minimization principle is utilized in order to estimate the heat transfer coefficient of food to be heated under optimal condition. The developed user friendly dialogue and used numerical procedures makes the ?OPT-PROx? software useful to food scientists in research and education, as well as to engineers involved in optimization of thermal food processing.

3D Frangi-based lung vessel enhancement filter penalizing airways

This paper describes a fully automatic simultaneous lung vessel and airway enhancement filter. The approach consists of a Frangi-based multiscale vessel enhancement filtering specifically designed for lung vessel and airway detection, where arteries and veins have high contrast with respect to the lung parenchyma, and airway walls are hollow tubular structures with a non negative response using the classical Frangi's filter. The features extracted from the Hessian matrix are used to detect centerlines and approximate walls of airways, decreasing the filter response in those areas by applying a penalty function to the vesselness measure. We validate the segmentation method in 20 CT scans with different pathological states within the VESSEL12 challenge framework. Results indicate that our approach obtains good results, decreasing the number of false positives in airway walls.

Coupling the folding of homologous proteins

The empirical observation that homologous proteins fold to similar structures is used to enhance the capabilities of an ab initio algorithm to predict protein conformations. A penalty function that forces homologous proteins to look alike is added to the potential and is employed in the coupled energy optimization of several homologous proteins. Significant improvement in the quality of the computed structures (as compared with the computational folding of a single protein) is demonstrated and discussed.

Particle Swarm Optimization Based Reactive Power Dispatch for Power Networks with Distributed Generation

Reactive power is critical to the operation of the power networks on both safety aspects and economic aspects. Unreasonable distribution of the reactive power would severely affect the power quality of the power networks and increases the transmission loss. Currently, the most economical and practical approach to minimizing the real power loss remains using reactive power dispatch method. Reactive power dispatch problem is nonlinear and has both equality constraints and inequality constraints. In this thesis, PSO algorithm and MATPOWER 5.1 toolbox are applied to solve the reactive power dispatch problem. PSO is a global optimization technique that is equipped with excellent searching capability. The biggest advantage of PSO is that the efficiency of PSO is less sensitive to the complexity of the objective function. MATPOWER 5.1 is an open source MATLAB toolbox focusing on solving the power flow problems. The benefit of MATPOWER is that its code can be easily used and modified. The proposed method in this thesis minimizes the real power loss in a practical power system and determines the optimal placement of a new installed DG. IEEE 14 bus system is used to evaluate the performance. Test results show the effectiveness of the proposed method.

The cross-entropy method for continuous multi-extremal optimization

In recent years, the cross-entropy method has been successfully applied to a wide range of discrete optimization tasks. In this paper we consider the cross-entropy method in the context of continuous optimization. We demonstrate the effectiveness of the cross-entropy method for solving difficult continuous multi-extremal optimization problems, including those with non-linear constraints.

Finite element analysis of the statics of and vibrations in axisymmetric shells

The finite element process is now used almost routinely as a tool of engineering analysis. From early days, a significant effort has been devoted to developing simple, cost effective elements which adequately fulfill accuracy requirements. In this thesis we describe the development and application of one of the simplest elements available for the statics and dynamics of axisymmetric shells . A semi analytic truncated cone stiffness element has been formulated and implemented in a computer code: it has two nodes with five degrees of freedom at each node, circumferential variations in displacement field are described in terms of trigonometric series, transverse shear is accommodated by means of a penalty function and rotary inertia is allowed for. The element has been tested in a variety of applications in the statics and dynamics of axisymmetric shells subjected to a variety of boundary conditions. Good results have been obtained for thin and thick shell cases .

Risk Measures for Classical and Perturbed Risk Processes - a Survey

2000 Mathematics Subject Classification: 60B10, 60G17, 60G51, 62P05.