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 Index