DOCSLIB.ORG

Ruin Probability

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ruin Probability Ruin probability Dan Kucerovsky1 Department of Mathematics and Statistics University of New Brunswick Fredericton, New Brunswick, Canada Abstract We discuss studying ruin probability by Levy processes, and in particular the problem of approximat- ing them through compound sum of exponentials leads us to some interesting mathematical problems. Complex analysis and Weiner-Hopf factorization turn out to be useful tools. Email: [email protected] 1speaker 1 11 On the importance of data mining methods in actuarial science Shima Ara1, Saman Insurance Company, Tehran, Iran Abstract Nowadays, by rapid advancements in technology, the actuaries often encounter large amount of ob- servations from various insurance fields. Modeling and analyzing such big data raise several challenges in the application. In recent years, data mining methods attracted many interest in various fields, as well as insurance. Actuaries should be ready to combine their traditional knowledge with the data mining approach to deal with the new world of big data. In this talk, we will review the impact of these methods in actuarial science and present findings of employing such methods on life and nonlife insurance in Saman insurance company. 1speaker 1 12 Mortality modeling of skin cancer patients with actuarial applications Amin Hassan Zadeh1 Shahid Beheshti University, Tehran, Iran Raoufeh Asghari Shahid Beheshti University, Tehran, Iran Abstract In this article, the Markovian aging process is used to model mortality of patients with skin cancer. The time till death is assumed to have a phase-type distribution (which is defined in a Markov chain environment) with interpretable parameters. The underlying continuous-time Markov chain has one absorbing state (death) and nx +1 (x is the age when the patient is diagnosed with cancer) transient states. Each transient state represents a physiological age, and aging is transitions from one physiological age to the next one until the process reaches to its end. The transition can occur from any other state to the absorbing state. For patients with skin cancer in the United States, we estimate unknown parameters related to the aging process that can be useful for comparing the physiological age processes of patients with cancer and healthy people. For different age intervals, we estimate physiological age parameters for both males and females. The index of conditional expected physiological age of the patients with skin cancer at given ages are calculated and compared with the US total population. By using the bootstrap techniques, confidence bands and confidence intervals are constructed for the estimated survival curves and aging process parameters, respectively. The fitting results have been used for pricing the substandard annuities. Keywords: Aging process; mortality modeling; phase-type distribution; skin cancer; substandard annuities, boot- strap. 1speaker 1 13 A sample article title 1 Use of Decision Tree in Classifying Customers of Third-Party Liability Motor Insurance Farbod Khanizadeh1 Insurance Research Center, Tehran, Iran Mohammadreza Asghari Oskoei Allameh Tabataba'i University, Tehran, Iran Azadeh Bahador Insurance Research Center, Tehran, Iran Abstract The purpose of this study was to present a pattern through which insurance companies are enabled to classify the risk level of their customer and to predict the possibility of future claims. We have analyzed an insurance claim dataset provided by an Iranian insurance company with a sample size of 6000. According to the structure of the dataset, a supervised learning algorithm was used to describe the underlying relationships between variables. The proposed algorithm, decision tree, was implemented using Python programming language. Based on the results, age, vehicle type and marital status were the main three factors contributing in prediction of claims. Keywords: Decision Tree, Supervised Learning, Machine Learning, Classification, TPL. 1 Introduction Motor third party liability insurance (TPL) is a type of policy financially protecting third parties against both physical damage and bodily injury caused by car accidents. Under third party insurance Act , TPL is a compulsory insurance product in Iran. This is a sound reason for insurers to make an attempt to have their customers' risks assessed as accurately as possible. Furthermore, Iran insurance industry is going to experience the transition from auto-based TPL to the driver-based one [1]. Currently premiums are calculated solely according to auto features. By driver-based policies we mean the situation in which both drivers' and autos' characteristics would affect premium rates. The new regulations are going into effect by the end of 2021. Therefore, insurance companies need to review their risk assessment methods and make necessary amendments. In fact, in near future drivers’ traits will also play a crucial role in TLP ratemaking. Hence in this article we worked with a dataset of features describing both drivers and vehicles. Basically we built a model to classify customers based on their risk factors divided into demographic information and some significant properties of cars ([2], [3], [4], [5]). Decision tree is one of the common supervised learning algorithms used for both classification and regression. The method can be applied across a broad range of disciplines ([6], [7], [8], [9], [10], [11] ). Due to the nature of insurance industry, risk classification, decision tree has received considerable attention within this field ([12], [13], 1 speaker 14 Farbod Khanizadeh, Mohammadreza Asghari Oskoei, Azadeh Bahador [14], [15] ). The focus of this article was to develop a tree classifier to help insurers predict their high and low risk policyholders ([16], [17], [18]). Having gathered the data set, four steps were followed to build up the model: 1. Data preprocessing (replacing missing values, encoding categorical variables, split data into training and test set) 2. Build the tree classifier 3. Cross-validation and model evaluation 4. Avoid over-fitting (pre-pruning) 5. Model visualization Regarding step two, we used the "Gini Index" as a measure of node impurity. Basically in building the binary tree, Gini index is the splitting criterion at each node and would determine the importance of each feature. Gini index is defined as: 2 Gini= 1- ∑(Pi) Where Pi corresponds to the probability of each class. The next section would present key findings and outcomes of the research. Note that the learning algorithm was evaluated on a dataset containing information regarding TLP claims. In the given dataset there were 6000 observations for 5 features namely; policy holder age, type of vehicle, marital status, gender and the dependent variable claim with two levels (Yes/No). 2 Main results To obtain a reliable model we evaluated the accuracy of our classifier over both training and test sets. This was conducted through specifying the optimum value of a hyper parameter maximum depth of the tree. With the help of k-fold cross validation (for k=10) we tried thirty values for the depth and following figure was derived: As shown in the figure, accuracy reaches lower score as we increase the maximum depth of tree. The highest accuracy belongs to the first five values. Therefore, we pick the highest depth corresponds to the best accuracy score (i.e. max_depth=5). According to the above figure we have prevented a model from over-fitting via pre-pruning method. In fact the hyper parameter max_depth is used to set the maximum depth of our classifier. The following diagram depicts the decision tree of height 5: 15 Use of Decision Tree in Classifying Customers of Third-Party Liability Motor Insurance The model consists of 19 leaves which are terminal nodes and our tree starts with the most important features of the data set, the policyholder age. Furthermore the root consists of 4200 samples obtained based on the ratio between a training and validation sets (i.e. 70:30). As one can see the model is easy to interpret and enjoys high accuracy. Acknowledgment Authors are grateful for supports received from Iran Insurance Research Center (IRC). References [1] Compulsory Third Party Liability Insurance Damage due to Vehicle Accidents (2015) . [2] A.C. Yeo, K.A. Smith, R.J Willis, M. Brooks. Clustering technique for risk classification and prediction of claim costs in the automobile insurance industry. Intelligent Systems in Accounting, Finance & Management. 2001 Mar;10(1):39-50. [3] C.K Fan, W.Y. Wang. A comparison of underwriting decision making between telematics-enabled UBI and traditional auto insurance. Advances in Management and Applied Economics. 2017;7(1):17. [4] J. Son, M. Park, B.B. Park. The effect of age, gender and roadway environment on the acceptance and effectiveness of Advanced Driver Assistance Systems. Transportation research part F: traffic psychology and behaviour. 2015 May 1;31:12-24. [5] Y. Bian, C. Yang, J.L. Zhao, L. Liang. Good drivers pay less: A study of usage-based vehicle insurance models. Transportation research part A: policy and practice. 2018 Jan 1;107:20-34. [6] W.Y. Loh. Classification and regression trees. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery. 2011 Jan;1(1):14-23. 16 Farbod Khanizadeh, Mohammadreza Asghari Oskoei, Azadeh Bahador [7] C. Machuca, M.V. Vettore, M. Krasuska,S.R. Baker, P.G. Robinson. Using classification and regression tree modelling to investigate response shift patterns in dentine hypersensitivity. BMC medical research methodology. 2017 Dec;17(1):120. [8] Y.Y. Song, L.U. Ying. Decision tree methods: applications for classification and prediction. Shanghai archives of psychiatry. 2015 Apr 25;27(2):130. [9] V. Podgorelec, P. Kokol, B. Stiglic, I. Rozman. Decision trees: an overview and their use in medicine. Journal of medical systems. 2002 Oct 1;26(5):445-63. [10] L. E. Brandão, J.S. Dyer, W.J. Hahn. Using binomial decision trees to solve real-option valuation problems. Decision Analysis. 2005 Jun;2(2):69-88. [11] M. Zięba, S.K Tomczak, J.M. Tomczak. Ensemble boosted trees with synthetic features generation in application to bankruptcy prediction.
Load more

Recommended publications
  • Official Journal of the Bernoulli Society for Mathematical Statistics And
    Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty One Number Four November 2015 ISSN: 1350-7265 CONTENTS Papers BEN ALAYA, M., HAJJI, K. and KEBAIER, A. 1947 Importance sampling and statistical Romberg method REBELLES, G. 1984 Pointwise adaptive estimation of a multivariate density under independence hypothesis OGIHARA, T. 2024 Local asymptotic mixed normality property for nonsynchronously observed diffusion processes ADAMCZAK, R. and BEDNORZ, W. 2073 Some remarks on MCMC estimation of spectra of integral operators AKASHI, F., LIU, Y. and TANIGUCHI, M. 2093 An empirical likelihood approach for symmetric α-stable processes VOS, P. and WU, Q. 2120 Maximum likelihood estimators uniformly minimize distribution variance among distribution unbiased estimators in exponential families BOURGUIN, S. 2139 Poisson convergence on the free Poisson algebra FANG, X. and RÖLLIN, A. 2157 Rates of convergence for multivariate normal approximation with applications to dense graphs and doubly indexed permutation statistics CHONG, C. and KLÜPPELBERG, C. 2190 Integrability conditions for space–time stochastic integrals: Theory and applications CHEN, F. and NKURUNZIZA, S. 2217 Optimal method in multiple regression with structural changes GIACOMIN, G. and MERLE, M. 2242 Weak noise and non-hyperbolic unstable fixed points: Sharp estimates on transit and exit times BULL, A.D. 2289 Adaptive-treed bandits (continued) The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Mathematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), and Zentralblatt für Mathematik (also avalaible on the MATH via STN database and Compact MATH CD-ROM). A list of forthcoming papers can be found online at http://www.
    [Show full text]
  • Convoluted Fractional Poisson Process
    ALEA, Lat. Am. J. Probab. Math. Stat. 18, 1241–1265 (2021) DOI: 10.30757/ALEA.v18-46 Convoluted Fractional Poisson Process Kuldeep Kumar Kataria and Mostafizar Khandakar Department of Mathematics Indian Institute of Technology Bhilai Raipur-492015, India. E-mail address: [email protected], [email protected] Abstract. In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). It has integer valued jumps of size j ≥ 1. The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. A special case of the CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a Lévy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property. 1. Introduction The Poisson process is a renewal process with exponentially distributed waiting times. This Lévy process is often used to model the counting phenomenon. Empirically, it is observed that the process with heavy-tailed distributed waiting times offers a better model than the ones with light-tailed distributed waiting times. For this purpose several fractional generalizations of the homogeneous Poisson process are introduced and studied by researchers in the past two decades.
    [Show full text]
  • Text Mining and Ruin Theory: a Case Study of Research on Risk Models with Dependence ∗
    Text mining and ruin theory: A case study of research on risk models with dependence ∗ Authors: Renata G. Alcoforado { ISEG & CEMAPRE, Universidade de Lisboa; Department of Accounting and Actuarial Sciences, Universidade Federal dePernambuco, Portugal & Brazil ([email protected]) Alfredo D. Eg´ıdio dos Reis { ISEG & CEMAPRE, Universidade de Lisboa Portugal ([email protected]) Abstract: • This paper aims to analyze unstructured data using a text mining approach. The work was motivated in order to organize and structure research in Risk Theory. In our study, the subject to be analyzed is composed by 27 published papers of the risk and ruin theory topic, area of actuarial science. They were coded into 32 categories. For the purpose, all data was analyzed and figures were produced using the software NVivo 11 plus. Software NVivo is a specialized tool in analyzing unstructured data, although it is commonly used just for qualitative research. We used it for Quali-Quant analysis. Key-Words: • Big data; Unstructured data; Text mining; Risk theory; Ruin theory; Dependence modeling. AMS Subject Classification: • 49A05, 78B26. ∗The opinions expressed in this text are those of the authors and do not necessarily reflect the views of any organization. 2 Renata G. Alcoforado and Alfredo D. Eg´ıdiodos Reis Using the \revstat.sty" Package 3 1. INTRODUCTION As widely known, Big Data is an area of great development in statistics. We can define Big Data as \a phenomenon defined by the rapid acceleration in the expanding volume of high velocity, complex, and diverse types of data. Big Data is often defined along three dimensions { volume, velocity, and variety" (TechAmerica Foundation's Federal Big Data Commission, 2012).
    [Show full text]
  • On the Profitability of Selfish Blockchain Mining Under Consideration of Ruin Hansjörg Albrecher, Pierre-Olivier Goffard
    On the profitability of selfish blockchain mining under consideration of ruin Hansjörg Albrecher, Pierre-Olivier Goffard To cite this version: Hansjörg Albrecher, Pierre-Olivier Goffard. On the profitability of selfish blockchain mining under consideration of ruin. 2021. hal-02649025v3 HAL Id: hal-02649025 https://hal.archives-ouvertes.fr/hal-02649025v3 Preprint submitted on 26 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the profitability of selfish blockchain mining under consideration of ruin Hansjörg Albrecher∗ and Pierre-O. Goffard† Abstract Mining blocks on a blockchain equipped with a proof of work consensus protocol is well-known to be resource-consuming. A miner bears the operational cost, mainly electricity consumption and IT gear, of mining, and is compensated by a capital gain when a block is discovered. This paper aims at quantifying the profitability of min- ing when the possible event of ruin is also considered. This is done by formulating a tractable stochastic model and using tools from applied probability and analysis, including the explicit solution of a certain type of advanced functional differential equation. The expected profit at a future time point is determined for the situation when the miner follows the protocol as well as when he/she withholds blocks.
    [Show full text]
  • Duality in Ruin Problems for Ordered Risk Models Pierre-Olivier Goffard, Claude Lefèvre
    Duality in ruin problems for ordered risk models Pierre-Olivier Goffard, Claude Lefèvre To cite this version: Pierre-Olivier Goffard, Claude Lefèvre. Duality in ruin problems for ordered risk models. 2016. hal-01398910v2 HAL Id: hal-01398910 https://hal.archives-ouvertes.fr/hal-01398910v2 Preprint submitted on 9 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Duality in ruin problems for ordered risk models Pierre-Olivier Goffard∗, Claude Lef`evrey September 18, 2017 Abstract On one hand, an ordered dual risk model is considered where the profit arrivals are governed by an order statistic point process (OSPP). First, the ruin time distribution is obtained in terms of Abel-Gontcharov polynomials. Then, by duality, the ruin time distribution is deduced for an insurance model where the claim amounts correspond to the inter-arrival times in an OSPP. On the other hand, an ordered insurance model is considered with an OSPP as claim arrival process. Lef`evreand Picard [28] determined the finite-time ruin probability in terms of Appell polynomials. Duality is used to derive the ruin probability in a dual model where the profit sizes correspond to the inter-arrival times of an OSPP.
    [Show full text]
  • Text Mining and Ruin Theory: a Case Study of Research on Risk Models with Dependence
    REVSTAT – Statistical Journal Volume 18, Number 4, October 2020, 483–499 TEXT MINING AND RUIN THEORY: A CASE STUDY OF RESEARCH ON RISK MODELS WITH DEPENDENCE Authors: Renata G. Alcoforado – ISEG & CEMAPRE, Universidade de Lisboa, Portugal and Department of Accounting and Actuarial Sciences, Universidade Federal de Pernambuco, Brazil [email protected] Alfredo D. Eg´ıdio dos Reis – ISEG & CEMAPRE, Universidade de Lisboa Portugal [email protected] Received: December 2017 Revised: June 2018 Accepted: June 2018 Abstract: • This paper aims to analyze unstructured data using a text mining approach. The work was motivated in order to organize and structure research in Risk Theory. In our study, the subject to be analyzed is composed by 27 published papers of the risk and ruin theory topic, area of actuarial science. They were coded into 32 categories. For the purpose, all data was analyzed and figures were produced using the software NVivo 11 plus. Software NVivo is a specialized tool in analyzing unstructured data, although it is commonly used just for qualitative research. We used it for Quali-Quant analysis. Keywords: • big data; unstructured data; text mining; risk theory; ruin theory; dependence modeling. AMS Subject Classification: • 49A05, 78B26. 484 Renata G. Alcoforado and Alfredo D. Eg´ıdio dos Reis 1. INTRODUCTION As widely known, Big Data is an area of great development in statistics. We can define Big Data as “a phenomenon defined by the rapid acceleration in the expanding volume of high velocity, complex, and diverse types of data. Big Data is often defined along three dimensions — volume, velocity, and variety” (TechAmerica Foundation’s Federal Big Data Commission [44], 2012).
    [Show full text]
  • Mathematical Tripos Part III Lecture Courses in 2013-2014
    Mathematical Tripos Part III Lecture Courses in 2013-2014 Department of Pure Mathematics & Mathematical Statistics Department of Applied Mathematics & Theoretical Physics Notes and Disclaimers. • Students may take any combination of lectures that is allowed by the timetable. The examination timetable corresponds to the lecture timetable and it is therefore not possible to take two courses for examination that are lectured in the same timetable slot. There is no requirement that students study only courses offered by one Department. • The code in parentheses after each course name indicates the term of the course (M: Michaelmas; L: Lent; E: Easter), and the number of lectures in the course. Unless indicated otherwise, a 16 lecture course is equivalent to 2 credit units, while a 24 lecture course is equivalent to 3 credit units. Please note that certain courses are non-examinable, and are indicated as such after the title. Some of these courses may be the basis for Part III essays. • At the start of some sections there is a paragraph indicating the desirable previous knowledge for courses in that section. On one hand, such paragraphs are not exhaustive, whilst on the other, not all courses require all the pre-requisite material indicated. However you are strongly recommended to read up on the material with which you are unfamiliar if you intend to take a significant number of courses from a particular section. • The courses described in this document apply only for the academic year 2013-14. Details for subsequent years are often broadly similar, but not necessarily identical. The courses evolve from year to year.
    [Show full text]
  • Ruin Models with Investment Income 417 the 20Th Century the Theory of Stochastic Processes Was Far Less Developed, and Also Far Less Known, Than It Is Today
    Probability Surveys Vol. 5 (2008) 416–434 ISSN: 1549-5787 DOI: 10.1214/08-PS134 Ruin models with investment income Jostein Paulsen∗ Department of Mathematics University of Bergen Johs. Brunsgt. 12 5008 Bergen, Norway Abstract: This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro- differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of pa- pers published after 1998. For more references to papers published before that, the reader can consult [47]. AMS 2000 subject classifications: Primary 60G99; secondary 60G40, 60G44, 60J25, 60J75. Keywords and phrases: Ruin probability, Risk theory, Compounding assets. Received June 2008. Contents 1 Introduction................................. 416 2 SomegeneralresultsintheMarkovmodel . 419 3 Adiscretetimemodel ........................... 420 4 Analytical and numerical solutions . 421 5 Asymptoticresults ............................. 422 6 Inequalities ................................. 425 7 Stochasticinterestrates . 425 8 Minimization of ruin probabilities . 426 References.................................... 429 arXiv:0806.4125v2 [q-fin.RM] 18 Dec 2008 1. Introduction The problem of ruin has a long history in risk theory, going back to Lundberg [40]. In Lundberg’s model, the company did not earn any investment on its capital. An obvious reason for this assumption, although there may be other reasons as well, is that the mathematics is easier, and back in the first half of ∗This survey was written while the author was a visiting professor at the departments of mathematics and statistics at The University of Chicago 416 J.
    [Show full text]
  • A Risk Surplus Model Using Hawkes Point Processes
    U.U.D.M. Project Report 2015:11 A Risk Surplus Model using Hawkes Point Processes Hanna Magnusson Examensarbete i matematik, 15 hp Handledare och examinator: Ingemar Kaj Maj 2015 Department of Mathematics Uppsala University A Risk Surplus Model using Hawkes Point Processes Hanna Magnusson Abstract In this thesis we define a Hawkes process with exponential decay that later on is used in an application to insurance. We have also applied a simulation algorithm for the Hawkes process that are able to model cluster arrival of claims. In the classical risk model one uses a homogeneous Poisson process to model the arrival of claims which is not realistic. What is most crucial for an insurance company is to price their premiums such that the probability of bankruptcy is small. Hence, we discuss the effect of modelling claim arrivals with a Hawkes process and different choices of premium principles with respect to the probability of ruin. Contents 1 Introduction 1 2 Hawkes Process 1 3 Simulation Algorithm 4 3.1 An Illustration of a Hawkes Process . 5 4 Application of a Hawkes Process in Insurance 5 4.1 The Cram´er-Lundberg Model . 5 4.2 Net Profit Condition . 8 4.3 The Expected Value Principle . 8 4.4 Hawkes Point Processes in a Risk Model . 9 4.5 The Variance Principle . 16 5 Summary and Conclusion 18 1 Introduction Insurance companies offer their customers insurance against a risk in exchange for a premium. The main challenge for actuaries is to find an optimal premium, meaning that the probability of bankruptcy for the company should be small and the price of the pre- mium not too high.
    [Show full text]
  • Generalized Ornstein-Uhlenbeck Processes in Ruin Theory Jerôme Spielmann
    Generalized Ornstein-Uhlenbeck Processes in Ruin Theory Jerôme Spielmann To cite this version: Jerôme Spielmann. Generalized Ornstein-Uhlenbeck Processes in Ruin Theory. Probability [math.PR]. Université d’Angers, 2019. English. NNT : 2019ANGE0066. tel-03148887 HAL Id: tel-03148887 https://tel.archives-ouvertes.fr/tel-03148887 Submitted on 22 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE DE DOCTORAT DE L'UNIVERSITE D'ANGERS COMUE UNIVERSITE BRETAGNE LOIRE ECOLE DOCTORALE N° 601 Mathématiques et Sciences et Technologies de l'Information et de la Communication Spécialité : Mathématiques et leurs interactions Jérôme SPIELMANN Generalized Ornstein-Uhlenbeck Processes in Ruin Theory Thèse présentée et soutenue à Angers, le 13 décembre 2019 Unité de recherche : LAREMA UMR CNRS 6093 Thèse N° : 181271 Rapporteurs avant soutenance : Youri Kabanov Professeur, Université de Franche-Comté Jostein Paulsen Professor, Københavns Universitet Composition du Jury : Loïc Chaumont Professeur, Université d’Angers Alexander Lindner Professor, Universität
    [Show full text]
  • Fractional Poisson Process
    FRACTIONAL RISK PROCESS IN INSURANCE Arun KUMAR1 Nikolai LEONENKO2 Alois PICHLER3 Abstract The Poisson process suitably models the time of successive events and thus has numerous applications in statistics, in economics, it is also fun- damental in queueing theory. Economic applications include trading and nowadays particularly high frequency trading. Of outstanding importance are applications in insurance, where arrival times of successive claims are of vital importance. It turns out, however, that real data do not always support the genuine Poisson process. This has lead to variants and augmentations such as time dependent and varying intensities, for example. This paper investigates the fractional Poisson process. We introduce the process and elaborate its main characteristics. The exemplary ap- plication considered here is the Carm´er–Lundberg theory and the Sparre Andersen model. The fractional regime leads to initial economic stress. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the frac- tional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process. 1Department of Mathematics, Indian Institute of Technology Ropar, Rup- nagar, Punjab 140001, India [email protected] 2Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK LeonenkoN@Cardiff.ac.uk 3Chemnitz University of Technology, Faculty of Mathematics, Chemnitz, Germany [email protected] Key words: Fractional Poisson process, convex risk measures Mathematics Subject Classification (1991): arXiv:1808.07950v2 [math.ST] 13 Apr 2019 1 Introduction Companies, which are exposed to random orders or bookings, face the threat of ruin.
    [Show full text]
  • Working Paper Series
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Papers in Economics INTERNATIONAL CENTRE FOR ECONOMIC RESEARCH WORKING PAPER SERIES Ramsés H. Mena and Luis E. Nieto-Barajas EXCHANGEABLE CLAIM SIZES IN A COMPOUND POISSON TYPE PROCESS Working Paper no. 19/2007 October 2007 APPLIED MATHEMATICS AND QUANTITATIVE METHODS WORKING PAPER SERIES Exchangeable claim sizes in a compound Poisson type process Ramses´ H. Mena1∗and Luis E. Nieto-Barajas2 y 1Department of Probability and Statistics, IIMAS{UNAM, 01000 Mexico, D.F. 2Department of Statistics, ITAM, Rio Hondo 1, Progreso Tizapan, 01080 Mexico, D.F. ABSTRACT. When dealing with risk models the typical assumption of indepen- dence among claim size distributions is not always satisfied. Here we consider the case when the claim sizes are exchangeable and study the implications when constructing aggregated claims through compound Poisson type processes. In par- ticular, exchangeability is achieved through conditional independence and using parametric and nonparametric measures for the conditioning distribution. A full Bayesian analysis of the proposed model is carried out to illustrate. Key words: Bayes nonparametrics, compound Poisson process, exchangeable claim process, exchangeable sequence, risk model. 1. Introduction In insurance and risk modelling, the compound Poisson process has played a fundamental role. The compound Poisson process (CPP), denoted as fXt; t ≥ 0g, can be defined by N Xt Xt = Yi; (1) i=1 where fNt; t ≥ 0g is a homogeneous Poisson process with intensity λ > 0 and Y1;Y2;::: a se- quence of independent and identically distributed (nonnegative) random variables with a com- mon density f, independent of Nt.
    [Show full text]