Generalized Poisson Models and their Applications in Insurance and Finance

Modern Probability and Statistics

Contents:

Foreword

Preface

BASIC NOTIONS OF PROBABILITY THEORY

Random variables, their distributions and moments

Generating and characteristic functions

Random vectors. Stochastic independence

Weak convergence of random variables and distribution functions

Poisson theorem

Law of large numbers. Central limit theorem. Stable laws

The Berry-Esseen inequality

Asymptotic expansions in the central limit theorem

Elementary properties of random sums

Stochastic processes

POISSON PROCESS

The definition and elementary properties of a Poisson process

Poisson process as a model of chaotic displacement of points in time

The asymptotic normality of a Poisson process

Elementary rarefaction of renewal processes

CONVERGENCE OF SUPERPOSITIONS OF INDEPENDENT STOCHASTIC PROCESSES

Characteristic features of the problem

Approximation of distributions of randomly indexed random sequences by special mixtures

The transfer theorem. Relations between the limit laws for random sequences with random and non-random indices

Convergence of distributions of randomly indexed sequences to identifiable location or scale mixtures. The asymptotic behavior of extremal random sumsNecessary and sufficient conditions for the convergence of distributions of random sequences with independent random indices

Convergence of distributions of random sums. The central limit theorem and the law of large numbers for random sums

A general theorem on the asymptotic behavior of superpositions of independent stochastic processes

The transfer theorem for random sums of independent identically distributed random variables in the double array limit scheme

COMPOUND POISSON DISTRIBUTIONS

Mixed and compound Poisson distributions

Discrete compound Poisson distributions

The asymptotic normality of compound Poisson distributions. The Berry-Esseen inequality for Poisson random sums. Non-central Lyapunov fractions

Asymptotic expansions for compound Poisson distributions

The asymptotic expansions for the quantiles of compound Poisson distributions

Exponential inequalities for the probabilities of large deviations of Poisson random sums. An analog of Bernshtein-Kolmogorov inequality

The application of Esscher transforms to the approximation of the tails of compound Poisson distributions

Estimates of convergence rate in local limit theorems for Poisson random sums

CLASSICAL RISK PROCESSES

The definition of the classical risk process. Its asymptotic normality

The Pollaczek-Khinchin-Beekman formula for the ruin probability in the classical risk process

Approximations for the ruin probability with small safety loading

Asymptotic expansions for the ruin probability with small safety loading

Approximations for the ruin probability

Asymptotic approximations for the distribution of the surplus in general risk processes

A problem of inventory control

A non-classical problem of optimization of the initial capital

DOUBLY STOCHASTIC POISSON PROCESSES (COX PROCESSES)

The asymptotic behavior of random sums of random indicators

Mixed Poisson processes

The modified Pollaczek-Khinchin-Beekman formula

The definition and elementary properties of doubly stochastic Poisson processes

The asymptotic behavior of Cox processes

COMPOUND COX PROCESSES WITH ZERO MEAN

Definition. Examples

Conditions of convergence of the distributions of compound Cox processes with zero mean. Limit laws

Convergence rate estimates

Asymptotic expansions for the distributions of compound Cox processes with zero mean

Asymptotic expansions for the quantiles of compound Cox processes with zero mean

Exponential inequalities for the probabilities of large deviations of compound Cox processes with zero mean

Limit theorems for extrema of compound Cox processes with zero mean

Estimates of the rate of convergence of extrema of compound Cox processes with zero mean

MODELING EVOLUTION OF STOCK PRICES BY COMPOUND COX PROCESSES

Introduction

Normal and stable models

Heterogeneity of operational time and normal mixtures

Inhomogeneous discrete chaos and Cox processes

Restriction of the class of mixing distributions

Heavy-tailedness of scale mixtures of normals

The case of elementary increments with non-zero means

Models within the double array limit scheme

Quantiles of the distributions of stock prices

COMPOUND COX PROCESSES WITH NONZERO MEAN

Definition. Examples

Conditions of convergence of compound Cox processes with nonzero mean. Limit laws

Convergence rate estimates for compound Cox processes with nonzero mean

Asymptotic expansions for the distributions of compound Cox processes with nonzero mean

Asymptotic expansions for the quantiles of compound Cox processes with nonzero mean

Exponential inequalities for the negative values of the surplus in collective risk models with stochastic intensity of insurance payments

Limit theorems with extrema of compound Cox processes with nonzero mean

Convergence rate estimates for extrema of compound Cox processes with nonzero mean

Minimum admissible reserve of an insurance company with stochastic intensity of insurance payments

Optimization of the initial capital of an insurance company in a static insurance model with random portfolio size

FUNCTIONAL LIMIT THEOREMS FOR COMPOUND COX PROCESSES

Functional limit theorems for non-centered compound Cox processes

Functional limit theorems for nonrandomly centered compound Cox processes

GENERALIZED RISK PROCESSES

The definition of generalized risk processes

Conditions of convergence of the distributions of generalized risk processes

Convergence rate estimates for generalized risk processes

Asymptotic expansions for the distributions of generalized risk processes

Asymptotic expansions for the quantiles of generalized risk processes

Exponential inequalities for the probabilities of negative values of generalized risk processes

STATISTICAL INFERENCE CONCERNING THE PARAMETERS OF RISK PROCESSES

Statistical estimation of the ruin probability in classical risk processes

Specific features of statistical estimation of ruin probability for a generalized risk process

A nonparametric estimator of the ruin probability for a generalized risk process

Interval estimator of the ruin probability for a generalized risk process

Computational aspects of the construction of confidence intervals for the ruin probability in generalized risk processes

Bibliography

Index

2002; xx pages
ISBN 90-6764-366-1
Price: EUR 229/US$ 266