Generalized Poisson Models and their Applications in Insurance and Finance

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; necessary and sufficient conditions for the convergence of distributions of random sequences with independent random indices; convergence of distributions of randomly indexed sequences to identifiable location or scale mixtures; the asymptotic behaviour of extremal random sums; 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 behaviour 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 distribution; 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 derivations 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 distributions 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 behaviour of random sums