Upper and Lower Bounds for Ruin Probability

In this note we discuss upper and lower bound for the ruin probability in an insurance model with very heavy-tailed claims and interarrival times.

Explicit ruin formulas for models with dependence among risks

We show that a simple mixing idea allows one to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.

Ruin Theory with Excess of Loss Reinsurance and Reinstatements

The present paper studies the probability of ruin of an insurer, if excess of loss reinsurance with reinstatements is applied. In the setting of the classical Cramer-Lundberg risk model, piecewise deterministic Markov processes are used to describe the free surplus process in this more general situation. It is shown that the finite-time ruin probability is both the solution of a partial integro-differential equation and the fixed point of a contractive integral operator. We exploit the latter representation to develop and implement a recursive algorithm for numerical approximation of the ruin probability that involves high-dimensional integration. Furthermore we study the behavior of the finite-time ruin probability under various levels of initial surplus and security loadings and compare the efficiency of the numerical algorithm with the computational alternative of stochastic simulation of the risk process. (C) 2011 Elsevier Inc. All rights reserved.

Properties of a risk measure derived from ruin theory

This paper studies a risk measure inherited from ruin theory and investigates some of its properties. Specifically, we consider a value-at-risk (VaR)-type risk measure defined as the smallest initial capital needed to ensure that the ultimate ruin probability is less than a given level. This VaR-type risk measure turns out to be equivalent to the VaR of the maximal deficit of the ruin process in infinite time. A related Tail-VaR-type risk measure is also discussed.

Ruin problems under IBNR Dynamics

In this paper, we consider a discrete-time risk process allowing for delay in claim settlement, which introduces a certain type of dependence in the process. From martingale theory, an expression for the ultimate ruin probability is obtained, and Lundberg-type inequalities are derived. The impact of delay in claim settlement is then investigated. To this end, a convex order comparison of the aggregate claim amounts is performed with the corresponding non-delayed risk model, and numerical simulations are carried out with Belgian market data.

A class of bivariate Erlang distributions and ruin probabilities in multivariate risk models

Nous y introduisons une nouvelle classe de distributions bivariées de type Marshall-Olkin, la distribution Erlang bivariée. La transformée de Laplace, les moments et les densités conditionnelles y sont obtenus. Les applications potentielles en assurance-vie et en finance sont prises en considération. Les estimateurs du maximum de vraisemblance des paramètres sont calculés par l'algorithme Espérance-Maximisation. Ensuite, notre projet de recherche est consacré à l'étude des processus de risque multivariés, qui peuvent être utiles dans l'étude des problèmes de la ruine des compagnies d'assurance avec des classes dépendantes. Nous appliquons les résultats de la théorie des processus de Markov déterministes par morceaux afin d'obtenir les martingales exponentielles, nécessaires pour établir des bornes supérieures calculables pour la probabilité de ruine, dont les expressions sont intraitables.

A teoria da ruína aplicada em um modelo de empresa financeira com risco de crédito

In this work we study a new risk model for a firm which is sensitive to its credit quality, proposed by Yang(2003): Are obtained recursive equations for finite time ruin probability and distribution of ruin time and Volterra type integral equation systems for ultimate ruin probability, severity of ruin and distribution of surplus before and after ruin

Teoria da Ruína em um Modelo de Markov com dois Estados

In this work, we present a risk theory application in the following scenario: In each period of time we have a change in the capital of the ensurance company and the outcome of a two-state Markov chain stabilishs if the company pays a benece it heat to one of its policyholders or it receives a Hightimes c > 0 paid by someone buying a new policy. At the end we will determine once again by the recursive equation for expectation the time ruin for this company

Modelo de risco controlado por resseguro e desigualdades para a probabilidade de ruína

In the work reported here we present theoretical and numerical results about a Risk Model with Interest Rate and Proportional Reinsurance based on the article Inequalities for the ruin probability in a controlled discrete-time risk process by Ros ario Romera and Maikol Diasparra (see [5]). Recursive and integral equations as well as upper bounds for the Ruin Probability are given considering three di erent approaches, namely, classical Lundberg inequality, Inductive approach and Martingale approach. Density estimation techniques (non-parametrics) are used to derive upper bounds for the Ruin Probability and the algorithms used in the simulation are presented

Aplicação da teoria da confiabilidade em estacas escavadas de grande diâmetro para obtenção das superfícies resistentes de projeto

In this work, the author looks forward to develop a new method capable of incorporate the concepts of the Reliability Theory and Ruin Probability in Deep Foundations, in order to do a better quantification of the uncertainties, which is intrinsic in all geotechnical projects, meanly because we don't know all the properties of the materials that we work with. Using the methodologies of Decourt Quaresma and David Cabral, resistance surfaces have been developed utilizing the data achieved from the Standard Penetration Tests performed in the field of study, in conjecture with the loads defined in the executive project of the piles. The construction of resistance surfaces shows to be a very useful tool for decision making, no matter in which phase it is current on, projecting or execution. The surfaces were developed by Kriging (using the software Surfer® 12), making it easier to visualize the geotechnical profile of the field of study. Comparing the results, the conclusion was that a high safety factor doesn't mean higher security. It is fundamental to consider the loads and resistance of the piles in the whole field, carefully choosing the project methodology responsible to define the diameter and length of the piles

Aplicação da teoria da confiabilidade em estacas escavadas de grande diâmetro para obtenção das superfícies resistentes de projeto

In this work, the author looks forward to develop a new method capable of incorporate the concepts of the Reliability Theory and Ruin Probability in Deep Foundations, in order to do a better quantification of the uncertainties, which is intrinsic in all geotechnical projects, meanly because we don't know all the properties of the materials that we work with. Using the methodologies of Decourt Quaresma and David Cabral, resistance surfaces have been developed utilizing the data achieved from the Standard Penetration Tests performed in the field of study, in conjecture with the loads defined in the executive project of the piles. The construction of resistance surfaces shows to be a very useful tool for decision making, no matter in which phase it is current on, projecting or execution. The surfaces were developed by Kriging (using the software Surfer® 12), making it easier to visualize the geotechnical profile of the field of study. Comparing the results, the conclusion was that a high safety factor doesn't mean higher security. It is fundamental to consider the loads and resistance of the piles in the whole field, carefully choosing the project methodology responsible to define the diameter and length of the piles

Risk Measures for Classical and Perturbed Risk Processes - a Survey

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

A Modified Model of Risk Business

2000 Mathematics Subject Classification: 60K10, 62P05

On the Poisson Process of Order k Space

2010 Mathematics Subject Classification: 60E05, 62P05.