Generalized Poisson Models and their Applications in Insurance and Finance

Vladimir E. Bening, Victor Yu. Korolev

• Publisher: De Gruyter
• Year: 2012
• Print ISBN: 9789067643665
• eBook ISBN: 9783110936018
• Edition: 1
• Topic: Business
• Subtopic: Business Mathematics
• Index: Business

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Table of contents

• Foreword
• Preface
• 1 Basic notions of probability theory
• 1.1 Random variables, their distributions and moments
• 1.2 Generating and characteristic functions
• 1.3 Random vectors. Stochastic independence
• 1.4 Weak convergence of random variables and distribution functions
• 1.5 Poisson theorem
• 1.6 Law of large numbers. Central limit theorem. Stable laws
• 1.7 The Berry-Esseen inequality
• 1.8 Asymptotic expansions in the central limit theorem
• 1.9 Elementary properties of random sums
• 1.10 Stochastic processes
• 2 Poisson process
• 2.1 The definition and elementary properties of a Poisson process
• 2.2 Poisson process as a model of chaotic displacement of points in time
• 2.3 The asymptotic normality of a Poisson process
• 2.4 Elementary rarefaction of renewal processes
• 3 Convergence of superpositions of independent stochastic processes
• 3.1 Characteristic features of the problem
• 3.2 Approximation of distributions of randomly indexed random sequences by special mixtures
• 3.3 The transfer theorem. Relations between the limit laws for random sequences with random and non-random indices
• 3.4 Necessary and sufficient conditions for the convergence of distributions of random sequences with independent random indices
• 3.5 Convergence of distributions of randomly indexed sequences to identifiable location or scale mixtures. The asymptotic behavior of extremal random sums
• 3.6 Convergence of distributions of random sums. The central limit theorem and the law of large numbers for random sums
• 3.7 A general theorem on the asymptotic behavior of superpositions of independent stochastic processes
• 3.8 The transfer theorem for random sums of independent identically distributed random variables in the double array limit scheme
• 4 Compound Poisson distributions
• 4.1 Mixed and compound Poisson distributions
• 4.2 Discrete compound Poisson distributions
• 4.3 The asymptotic normality of compound Poisson distributions. The Berry-Esseen inequality for Poisson random sums. Non-central Lyapunov fractions
• 4.4 Asymptotic expansions for compound Poisson distributions
• 4.5 The asymptotic expansions for the quantiles of compound Poisson distributions
• 4.6 Exponential inequalities for the probabilities of large deviations of Poisson random sums. An analog of Bernshtein-Kolmogorov inequality
• 4.7 The application of Esscher transforms to the approximation of the tails of compound Poisson distributions
• 4.8 Estimates of convergence rate in local limit theorems for Poisson random sums
• 5 Classical risk processes
• 5.1 The definition of the classical risk process. Its asymptotic normality
• 5.2 The Pollaczek-Khinchin-Beekman formula for the ruin probability in the classical risk process
• 5.3 Approximations for the ruin probability with small safety loading
• 5.4 Asymptotic expansions for the ruin probability with small safety loading
• 5.5 Approximations for the ruin probability
• 5.6 Asymptotic approximations for the distribution of the surplus in general risk processes
• 5.7 A problem of inventory control
• 5.8 A non-classical problem of optimization of the initial capital
• 6 Doubly stochastic Poisson processes (Cox processes)
• 6.1 The asymptotic behavior of random sums of random indicators
• 6.2 Mixed Poisson processes
• 6.3 The modified Pollaczek-Khinchin-Beekman formula
• 6.4 The definition and elementary properties of doubly stochastic Poisson processes
• 6.5 The asymptotic behavior of Cox processes
• 7 Compound Cox processes with zero mean
• 7.1 Definition. Examples
• 7.2 Conditions of convergence of the distributions of compound Cox processes with zero mean. Limit laws
• 7.3 Convergence rate estimates
• 7.4 Asymptotic expansions for the distributions of compound Cox processes with zero mean
• 7.5 Asymptotic expansions for the quantiles of compound Cox processes with zero mean
• 7.6 Exponential inequalities for the probabilities of large deviations of compound Cox processes with zero mean
• 7.7 Limit theorems for extrema of compound Cox processes with zero mean
• 7.8 Estimates of the rate of convergence of extrema of compound Cox processes with zero mean
• 8 Modeling evolution of stock prices by compound Cox processes
• 8.1 Introduction
• 8.2 Normal and stable models
• 8.3 Heterogeneity of operational time and normal mixtures
• 8.4 Inhomogeneous discrete chaos and Cox processes
• 8.5 Restriction of the class of mixing distributions
• 8.6 Heavy-tailedness of scale mixtures of normals
• 8.7 The case of elementary increments with non-zero means
• 8.8 Models within the double array limit scheme
• 8.9 Quantiles of the distributions of stock prices
• 9 Compound Cox processes with nonzero mean
• 9.1 Definition. Examples
• 9.2 Conditions of convergence of compound Cox processes with nonzero mean. Limit laws
• 9.3 Convergence rate estimates for compound Cox processes with nonzero mean
• 9.4 Asymptotic expansions for the distributions of compound Cox processes with nonzero mean
• 9.5 Asymptotic expansions for the quantiles of compound Cox processes with nonzero mean
• 9.6 Exponential inequalities for the negative values of the surplus in collective risk models with stochastic intensity of insurance payments
• 9.7 Limit theorems for extrema of compound Cox processes with nonzero mean
• 9.8 Convergence rate estimates for extrema of compound Cox processes with nonzero mean
• 9.9 Minimum admissible reserve of an insurance company with stochastic intensity of insurance payments
• 9.10 Optimization of the initial capital of an insurance company in a static insurance model with random portfolio size
• 10 Functional limit theorems for compound Cox processes
• 10.1 Functional limit theorems for non-centered compound Cox processes
• 10.2 Functional limit theorems for nonrandomly centered compound Cox processes
• 11 Generalized risk processes
• 11.1 The definition of generalized risk processes
• 11.2 Conditions of convergence of the distributions of generalized risk processes
• 11.3 Convergence rate estimates for generalized risk processes
• 11.4 Asymptotic expansions for the distributions of generalized risk processes
• 11.5 Asymptotic expansions for the quantiles of generalized risk processes
• 11.6 Exponential inequalities for the probabilities of negative values of generalized risk processes
• 12 Statistical inference concerning the parameters of risk processes
• 12.1 Statistical estimation of the ruin probability in classical risk processes
• 12.2 Specific features of statistical estimation of ruin probability for generalized risk processes
• 12.3 A nonparametric estimator of the ruin probability for a generalized risk process
• 12.4 Interval estimator of the ruin probability for a generalized risk process
• 12.5 Computational aspects of the construction of confidence intervals for the ruin probability in generalized risk processes
• Bibliography
• Index