DISTRIBUTION OF THE TIME TO RUIN IN SOME SPARRE ANDERSEN RISK MODELS

2013 ◽  
Vol 43 (1) ◽  
pp. 39-59 ◽  
Author(s):  
Tianxiang Shi ◽  
David Landriault
Keyword(s):  
Busy Period ◽  
Risk Model ◽  
Research Problem ◽  
General Distribution ◽  
Risk Models ◽  
Expansion Theorem ◽  
The Laplace Transform ◽  
Lagrange Expansion ◽  
Passage Times ◽  
Distribution Assumption

AbstractThe finite-time ruin problem, which implicitly involves the inversion of the Laplace transform of the time to ruin, has been a long-standing research problem in risk theory. Existing results in the Sparre Andersen risk models are mainly based on an exponential assumption either on the interclaim times or on the claim sizes. In this paper, we utilize the multivariate version of Lagrange expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely the combination of n exponentials. A remark is further made to emphasize that this technique can also be applied to other areas of applied probability. For instance, the proposed methodology can be used to obtain the distribution of some first passage times for particular stochastic processes. As an illustration, the duration of a busy period in a queueing risk model will be examined.

scholarly journals Queues and Risk Models with Simultaneous Arrivals

2014 ◽  
Vol 46 (03) ◽  
pp. 812-831 ◽  
Author(s):  
E. S. Badila ◽  
O. J. Boxma ◽  
J. A. C. Resing ◽  
E. M. M. Winands
Keyword(s):  
Risk Model ◽  
Stochastic Decomposition ◽  
Two Dimensional ◽  
Riemann Boundary Value Problem ◽  
Risk Models ◽  
Riemann Boundary ◽  
Parallel Queues ◽  
The Laplace Transform ◽  
Claim Size Distribution ◽  
Stationary Workload

We focus on a particular connection between queueing and risk models in a multidimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue), we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline of how the two-dimensional risk model with a general two-dimensional claim size distribution (hence, without ordering of claim sizes) is related to a known Riemann boundary-value problem.

scholarly journals Queues and Risk Models with Simultaneous Arrivals

2014 ◽  
Vol 46 (3) ◽  
pp. 812-831 ◽  
Author(s):  
E. S. Badila ◽  
O. J. Boxma ◽  
J. A. C. Resing ◽  
E. M. M. Winands
Keyword(s):  
Risk Model ◽  
Stochastic Decomposition ◽  
Two Dimensional ◽  
Riemann Boundary Value Problem ◽  
Risk Models ◽  
Riemann Boundary ◽  
Parallel Queues ◽  
The Laplace Transform ◽  
Claim Size Distribution ◽  
Stationary Workload

We focus on a particular connection between queueing and risk models in a multidimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue), we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline of how the two-dimensional risk model with a general two-dimensional claim size distribution (hence, without ordering of claim sizes) is related to a known Riemann boundary-value problem.

scholarly journals A Two-Dimensional Risk Model with Proportional Reinsurance

2011 ◽  
Vol 48 (3) ◽  
pp. 749-765 ◽  
Author(s):  
Andrei L. Badescu ◽  
Eric C. K. Cheung ◽  
Landy Rabehasaina
Keyword(s):  
Laplace Transform ◽  
Risk Model ◽  
Poisson Processes ◽  
General Distribution ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Open Problems ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
The Laplace Transform

In this paper we consider an extension of the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008a). To this end, we assume that there are two insurers. The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer, which can be viewed as a different line of business of the same insurer or as a reinsurer, covers a proportion of the claims arising from one of these two compound Poisson processes. We derive the Laplace transform of the time until ruin of at least one insurer when the claim sizes follow a general distribution. The surplus level of the first insurer when the second insurer is ruined first is discussed at the end in connection with some open problems.

scholarly journals A Two-Dimensional Risk Model with Proportional Reinsurance

2011 ◽  
Vol 48 (03) ◽  
pp. 749-765 ◽  
Author(s):  
Andrei L. Badescu ◽  
Eric C. K. Cheung ◽  
Landy Rabehasaina
Keyword(s):  
Laplace Transform ◽  
Risk Model ◽  
Poisson Processes ◽  
General Distribution ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Open Problems ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
The Laplace Transform

In this paper we consider an extension of the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008a). To this end, we assume that there are two insurers. The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer, which can be viewed as a different line of business of the same insurer or as a reinsurer, covers a proportion of the claims arising from one of these two compound Poisson processes. We derive the Laplace transform of the time until ruin of at least one insurer when the claim sizes follow a general distribution. The surplus level of the first insurer when the second insurer is ruined first is discussed at the end in connection with some open problems.

scholarly journals Ruin problems in Markov-modulated risk models

2017 ◽  
Vol 12 (1) ◽  
pp. 23-48 ◽  
Author(s):  
David C.M. Dickson ◽  
Marjan Qazvini
Keyword(s):  
Continuous Time ◽  
Risk Model ◽  
Numerical Algorithms ◽  
Binomial Model ◽  
Risk Models ◽  
Time Of Ruin ◽  
Compound Binomial ◽  
Compound Binomial Model ◽  
Markov Modulated ◽  
The Time Of Ruin

AbstractChen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

scholarly journals On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment

1984 ◽  
Vol 14 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Jean-Marie Reinhard
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Risk Models ◽  
Markovian Environment ◽  
Necessary And Sufficient ◽  
Natural Way ◽  
Quantities Of Interest

AbstractWe consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

SOME ADVANCES ON THE ERLANG(n) DUAL RISK MODEL

2014 ◽  
Vol 45 (1) ◽  
pp. 127-150 ◽  
Author(s):  
Eugenio V. Rodríguez-Martínez ◽  
Rui M. R. Cardoso ◽  
Alfredo D. Egídio dos Reis
Keyword(s):  
Risk Model ◽  
Waiting Times ◽  
Dual Model ◽  
Time Of Ruin ◽  
Constant Rate ◽  
Dual Risk Model ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Renewal Risk ◽  
The Individual

AbstractThe dual risk model assumes that the surplus of a company decreases at a constant rate over time and grows by means of upward jumps, which occur at random times and sizes. It is said to have applications to companies with economical activities involved in research and development. This model is dual to the well-known Cramér-Lundberg risk model with applications to insurance. Most existing results on the study of the dual model assume that the random waiting times between consecutive gains follow an exponential distribution, as in the classical Cramér-Lundberg risk model. We generalize to other compound renewal risk models where such waiting times are Erlang(n) distributed. Using the roots of the fundamental and the generalized Lundberg's equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Furthermore, we compute expected discounted dividends, as well as higher moments, when the individual common gains follow a Phase-Type, PH(m), distribution. We also perform illustrations working some examples for some particular gain distributions and obtain numerical results.

An Evaluation of a Test-Driven Security Risk Analysis Approach Based on Two Industrial Case Studies

2019 ◽  
pp. 69-103 ◽  
Author(s):  
Gencer Erdogan ◽  
Phu H. Nguyen ◽  
Fredrik Seehusen ◽  
Ketil Stølen ◽  
Jon Hofstad ◽  
...  
Keyword(s):  
Risk Assessment ◽  
Case Studies ◽  
Risk Model ◽  
Case Study Research ◽  
Security Risk ◽  
Risk Models ◽  
First Case ◽  
Assessment Approach ◽  
Security Risk Assessment

Risk-driven testing and test-driven risk assessment are two strongly related approaches, though the latter is less explored. This chapter presents an evaluation of a test-driven security risk assessment approach to assess how useful testing is for validating and correcting security risk models. Based on the guidelines for case study research, two industrial case studies were analyzed: a multilingual financial web application and a mobile financial application. In both case studies, the testing yielded new information, which was not found in the risk assessment phase. In the first case study, new vulnerabilities were found that resulted in an update of the likelihood values of threat scenarios and risks in the risk model. New vulnerabilities were also identified and added to the risk model in the second case study. These updates led to more accurate risk models, which indicate that the testing was indeed useful for validating and correcting the risk models.

Assessing the Usefulness of Testing for Validating and Correcting Security Risk Models Based on Two Industrial Case Studies

2016 ◽  
pp. 1016-1037
Author(s):  
Gencer Erdogan ◽  
Fredrik Seehusen ◽  
Ketil Stølen ◽  
Jon Hofstad ◽  
Jan Øyvind Aagedal
Keyword(s):  
Risk Assessment ◽  
Case Studies ◽  
Web Application ◽  
Risk Model ◽  
Security Risk ◽  
Risk Models ◽  
First Case ◽  
New Information ◽  
Financial Application

The authors present the results of an evaluation in which the objective was to assess how useful testing is for validating and correcting security risk models. The evaluation is based on two industrial case studies. In the first case study the authors analyzed a multilingual financial Web application, while in the second case study they analyzed a mobile financial application. In both case studies, the testing yielded new information which was not found in the risk assessment phase. In particular, in the first case study, new vulnerabilities were found which resulted in an update of the likelihood values of threat scenarios and risks in the risk model. New vulnerabilities were also identified and added to the risk model in the second case study. These updates led to more accurate risk models, which indicate that the testing was indeed useful for validating and correcting the risk models.

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