Generalized Poisson Models and their Applications in Insurance and Finance

Abstract

Preface xiii 1 Basic notions of probability theory 1 1.1 Random variables, their distributions and moments 1 1.2 Generating and characteristic functions 11 1.3 Random vectors. Stochastic independence 21 1.4 Weak convergence of random variables and distribution functions ... 24 1.5 Poisson theorem 30 1.6 Law of large numbers. Central limit theorem. Stable laws 35 1.7 The Berry-Esseen inequality 45 1.8 Asymptotic expansions in the central limit theorem 47 1.9 Elementary properties of random sums 56 1.10 Stochastic processes 62 2 Poisson process 69 2.1 The definition and elementary properties of a Poisson process 69 2.2 Poisson process as a model of chaotic displacement of points in time. . 72 2.3 The asymptotic normality of a Poisson process 74 2.4 Elementary rarefaction of renewal processes 76 3 Convergence of superpositions of independent stochastic processes 83 3.1 Characteristic features of the problem 83 3.2 Approximation of distributions of randomly indexed random sequences by special mixtures 85 3.3 The transfer theorem. Relations between the limit laws for random sequences with random and non-random indices 89 3.4 Necessary and sufficient conditions for the convergence of distributions of random sequences with independent random indices 91 3.5 Convergence of distributions of randomly indexed sequences to identifiable location or scale mixtures. The asymptotic behavior of extremal random sums 98 3.6 Convergence of distributions of random sums. The central limit theorem and the law of large numbers for random sums 105 3.7 A general theorem on the asymptotic behavior of superpositions of independent stochastic processes 115

Precise large deviations for sums of random variables with consistently varying tails

“…For an overview on Cox processes and their applications to actuarial and financial mathematics, we refer the reader to Grandell (1976) and Bening and Korolev (2002); see also Korolev (1999Korolev ( , 2001. In this subsection we prove that the precise large-deviation result obtained in Theorem 4.1 can be applied to the compound process…”

Extremal limit theorems for observations separated by random power law waiting times

“…Many authors have contributed to the asymptotic theory of sums, maxima, and general random sequences with a random number of terms. See, for example, Korolev (1994Korolev ( , 1995, the books of Kruglov and Korolev (1990), Bening and Korolev (2002), and the references therein. Here, however, our focus is on convergence of processes and their connections to fractional calculus.…”

“…Fractional differential equations are useful to model various problems in physics (see a comprehensive review by Metzler and Klafter, 2004), finance (see a recent review by Scalas, 2006), and hydrology Benson et al, 2001Benson et al, , 2000Schumer et al, 2001). Time-fractional derivatives are connected with physical models of particle sticking and trapping .…”

Stochastic resonance and the trade arrival rate of stocks

“…where the event arrival rate r is itself a time dependent random process independent of N (t) [3]. Bezrukov and Vodyanoy have shown that the following arrival rate r(t) of a doubly stochastic Poisson process exhibits SR:…”