# Probability and Stochastic Modeling : The Mathematics of Insurance

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## Description

Ideal for students preparing for level 300 actuarial exams in the US, Actuarial Models: The Mathematics of Insurance provides a comprehensive exposition of insurance process models and presents mathematical setups and methods used in Actuarial Modeling. Divided into three self-contained and explicitly designated parts of different levels of difficulty, this book examines standard as well as advanced topics such as modern utility theory, martingale technique, models with payments of dividends, reinsurance models, and classification of distributions. It provides practical skills in analysis of insurance processes. This text discusses a number of topics not commonly found in existing Actuarial Mathematics textbooks, including achievements of the modern Risk Evaluation theory, premium principles, accuracy of normal and Poisson approximation, and a reinsurance market model. The main text is preceded by introductory chapters containing basic facts from Probability Theory, Calculus, and the Theory of Interest. The reader will not have to refer to outside sources; everything is under one cover and in the same unified notation and style.
The book includes many examples, practice problems, and exercises on numerical calculations using Excel(R). It includes preliminary examination material for the Society of Actuaries and the Casualty Actuarial Society (CAS), providing, in particular, real problems from past CAS exams.show more

## Product details

- Mixed media product | 656 pages
- 177.8 x 251.5 x 40.6mm | 1,156.67g
- 30 Sep 2006
- Taylor & Francis Inc
- Chapman & Hall/CRC
- Boca Raton, FL, United States
- English
- 110 black & white illustrations, 13 black & white tables
- 1584885866
- 9781584885863

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## Table of contents

Preface Acknowledgments Introduction SOME PRELIMINARY NOTATIONS AND FACTS FROM PROBABILITY THEORY, THE THEORY OF INTEREST, AND CALCULUS Probability and Random Variables Sample space, events, probability measure Independence and conditional probabilities Random variables, random vectors, and their distributions Expectation Definitions Integration by parts and a formula for expectation A general definition of expectation Can we encounter an infinite expected value in models of real phenomena? Moments of r.v.'s. Correlation Inequalities for deviations Linear transformations of r.v.'s. Normalization Some Basic Distributions Discrete distributions Continuous distributions Moment Generating Functions Laplace transform An example when a m.g.f. does not exist The m.g.f.'s of basic distributions The moment generating function and moments Expansions for m.g.f.'s Convergence of Random Variables and Distributions Some Facts and Formulas from the Theory of Interest Compound interest Nominal rate Discount and annuities Accumulated value Effective and nominal discount rates Appendix. Some Notations and Facts from Calculus The "small o and big O" notation Taylor expansions Concavity COMPARISON OF RANDOM VARIABLES. PREFERENCES OF INDIVIDUALS Comparison of Random Variables. Some Particular Criteria Preference order Several simple criteria On coherent measures of risk Comparison of R.V.'S and Limit Theorems of Probability Theory A diversion to Probability Theory: two limit theorems A simple model of insurance with many clients St. Petersburg's paradox Expected Utility Expected utility maximization (EUM) Utility and insurance How we may determine the utility function in particular cases Risk aversion A new view: EUM as a linear criterion Non-Linear Criteria Allais' paradox Weighted utility Implicit or comparative utility Rank Dependent Expected Utility Remarks Optimal Payment from the Standpoint of the Insured Arrow's theorem A generalization Exercises AN INDIVIDUAL RISK MODEL FOR A SHORT PERIOD The Distribution of an Individual Payment The distribution of the loss given that it has occurred The distribution of the loss X The distribution of the payment and types of insurance The Aggregate Payment Convolutions Moment generating functions Normal and Other Approximations Normal approximation How to take into account the asymmetry of S. The G-approximation Asymptotic expansions and Normal Power (NP) approximation Exercises CONDITIONAL EXPECTATIONS How to Compute Conditional Expectations. The Conditioning Procedure Conditional expectation given a r.v Properties of conditional expectations Conditioning and some useful formulas Conditional expectation given a r.vec. Formula for Total Expectation and Conditional Expectation Given a Partition Conditional expectation given an event The formula for total expectation Expectation given a partition Conditional Expectations Given Random Variables or Vectors The discrete case The general case One More Important Property of Conditional Expectations Conditioning on partitions Conditioning on r.v.'s or r.vec.'s A General Approach to Conditional Expectations Conditional expectation relative to a s-algebra Conditional expectation given a r.v. or a r.vec Properties of conditional expectations Some Proofs Exercises A COLLECTIVE RISK MODEL FOR A SHORT PERIOD Three Basic Propositions Counting or Frequency Distributions The Poisson distribution and Poisson's theorem Some other "counting" distributions The Distribution of the Aggregate Claim The case of a homogeneous group The case of several homogeneous groups Normal Approximation of the Distribution of the Aggregate Claim A limit theorem Estimation of premiums The accuracy of normal approximation Proof of Theorem10 Exercises RANDOM PROCESSES. I. COUNTING AND COMPOUND PROCESSES. MARKOV CHAINS. MODELING CLAIM AND CASH FLOWS A General Framework and Typical Situations Preliminaries Processes with independent increments Markov processes Poisson and Other Counting Processes The homogeneous Poisson process The non-homogeneous Poisson process The Cox process Compound Processes Markov Chains. Cash Flows in the Markov Environment Preliminaries Variables defined on a Markov chain. Cash flows The first step analysis. An infinite horizon Limiting probabilities and stationary distributions The ergodicity property and classification of states Exercises RANDOM PROCESSES. II. BROWNIAN MOTION AND MARTINGALES. HITTING TIMES Brownian Motions and its Generalization Further properties of the standard Brownian motion The Brownian motion with drift Geometric Brownian motion Martingales General properties and examples Martingale transform Optional stopping time and some applications Generalizations Exercises GLOBAL CHARACTERISTICS OF THE SURPLUS PROCESS. RUIN MODELS. MODELS WITH PAYING DIVIDENDS. Introduction Ruin Models Adjustment coefficients and ruin probabilities Computing adjustment coefficients Trade-off between the premium and the initial surplus Three cases when the ruin probability may be computed precisely The martingale approach. The renewal approach Some recurrent relations and computational aspects Criteria Connected with Paying Dividends A general model The case of the simple random walk Finding an optimal strategy Exercises SURVIVAL DISTRIBUTIONS The Distribution of the Lifetime Survival functions and force of mortality The time-until-death for a person of a given age Curtate-future-lifetime Survivorship groups Life tables and interpolation Some analytical laws of mortality A Multiple Decrement Model A single life Another view: net probabilities of decrement A survivorship group Proof of Proposition 1 . Multiple Life Models The joint distribution The lifetime of statuses A model of dependency: conditional independence Exercises LIFE INSURANCE MODELS A General Model The present value of a future payment The present value of payments to many clients Some Particular Types of Contracts Whole life insurance Deferred whole life insurance Term insurance Endowments Varying Benefits Certain payments Random payments Multiple Decrement and Multiple Life Models Multiple decrements Multiple life insurance On the Actuarial Notation Exercises ANNUITY MODELS Introduction. Two Approaches to Computing Annuities Continuous annuities Discrete annuities Level Annuities. A Connection with Insurance Certain annuities. Some notation Random annuities Some Particular Types of Level Annuities. Examples Whole life annuities Temporary annuities Deferred annuities Certain and life annuity More on Varying Payment Annuities with m-thly Payments Multiple Decrements and Multiple Life Models Multiple decrement Multiple life annuities Exercises PREMIUMS AND RESERVES Some General Premium Principles Premium Annuities Preliminaries. General principles Benefit premiums. The case of a single risk Accumulated values Percentile premium Exponential premiums Reserves Definitions and preliminary remarks Examples of direct calculations Formulas for some standard types of insurance Recursive relations Exercises RISK EXCHANGES: REINSURANCE AND COINSURANCE Reinsurance from the Standpoint of a Cedent Some optimization considerations Proportional reinsurance. Adding a new contract to an existing portfolio Long-term insurance. Ruin probability as a criterion Risk Exchange and Reciprocity of Companies A general framework and some examples Two more examples with expected utility maximization The case of the mean-variance criterion Reinsurance Market A model of the exchange market of random assets An example concerning reinsurance Exercises Tables References Answers to Exercises Subject Indexshow more

## Review quote

"This book covers practically all essential topics from modern probability and stochastic processes, which in one or another way are included in various university programs. So, is there something making this book different or better than many others available in the market? The answer is yes! The author presents the material systematically and rigorously. Of primary importance, however, is the emphasis on the modeling aspects. The large number of well-chosen and carefully described practical cases is a tremendous advantage of the book. strongly recommended as a basic source for a variety of courses." G. Schroder, Zentralblatt MATH 1268 "The author s mode of presentation is well chosen. New concepts are introduced with illuminating discussion and illustrations. The book is physically nicely presented. this text has something new to offer, mainly for business-oriented students, and I recommend that instructors give it serious consideration when choosing a text for their courses. Anthony G. Pakes, Mathematical Reviews, September 2013 "This is a superbly written text on probability and stochastic processes for students who have had two semesters of calculus and an introductory course in linear algebra. This includes upper division students in science and engineering including statistics and mathematics, as well as students in fields such as economics and finance. In addition, it will be a wonderful book for self study for many others. Important and well-chosen examples illustrate the theory throughout, and a large body of exercises supplements the text. It gives a lucid presentation of basic probability theory, including Markov chains and martingales. A special feature of this book is a marvelous exposition of many interesting aspects of financial mathematics that are generally considered rather intricate and inaccessible at this level. This book carries the imprint of a distinguished mathematician and teacher with expertise in probability theory and many of its special applications to mathematical economics and finance. It is an outstanding addition to the field requiring only a modest background in mathematics." Rabi Bhattacharya, Department of Mathematics, University of Arizona, Tucson, USA "Written in a lively and stimulating manner, the book makes a very good impression. The author, having extensive teaching experience and an undoubted literary talent, has managed to create an original introduction to modern probability theory. The successful combination of a variety of examples, exercises and applications with deep and nontrivial ideas makes the book interesting not only for beginning students, but also for professionals working with probabilistic problems. I believe that the book can serve as an ideal textbook for anyone interested in probability theory and its applications. The book will take a worthy place in the literature on probabilistic issues." Youri Davydov, Laboratoire Paul Painleve, Universite des Sciences et Technologies de Lille, France "The author has produced a comprehensive introduction to probability theory and stochastic processes, including martingales and Brownian motion. The text is suitable for students with a standard background in calculus and linear algebra. The approach is rigorous without being pedantic, and the text is liberally sprinkled with examples. Throughout, there is a welcome emphasis on stochastic modeling. Of note is the fairly early introduction and use of conditional expectations.The main text is complemented by a large collection of exercises with a wide range of difficulty. The book is in fact two-in-one, as a series of roadside markers guides the reader through two possible courses of study, one consisting of material suitable for a one-semester course, and the other a more in-depth journey suitable for a two-semester course. This book is a welcome and attractive addition to the list of textbooks available for an upper division probability course and would even be suitable for a graduate-level introduction to non-measure-theoretic probability and stochastic processes." Patrick J. Fitzsimmons, Department of Mathematics, University of California-San Diego, La Jolla, USA"show more

## About Vladimir I. Rotar

Vladimir I. Rotar is a professor in the Department of Mathematics and Statistics at San Diego State University. Dr. Rotar has authored four books and more than 100 scientific papers on probability theory and its applications in leading mathematical journals.show more