Lectures on Risk Theory

Twenty-five years ago, Hans Blihlmann published his famous monograph Mathe­ matical Methods in Risk Theory in the series Grundlehren der Mathematischen Wis8enschaften and thus established nonlife actuarial mathematics as a recognized subject of probability theory and statistics with a glance towards economics. This book was my guide to the subject when I gave my first course on nonlife actuarial mathematics in Summer 1988, but at the same time I tried to incorporate into my lectures parts of the rapidly growing literature in this area which to a large extent was inspired by Blihlmann's book. The present book is entirely devoted to a single topic of risk theory: Its subject is the development in time of a fixed portfolio of risks. The book thus concentrates on the claim number process and its relatives, the claim arrival process, the aggregate claims process, the risk process, and the reserve process. Particular emphasis is laid on characterizations of various classes of claim number processes, which provide alternative criteria for model selection, and on their relation to the trinity of the binomial, Poisson, and negativebinomial distributions. Special attention is also paid to the mixed Poisson process, which is a useful model in many applications, to the problems of thinning, decomposition, and superposition of risk processe8, which are important with regard to reinsurance, and to the role of martingales, which occur in a natural way in canonical situations.

Table of Contents:
1 The Claim Arrival Process.- 1.1 The Model.- 1.2 The Erlang Case.- 1.3 A Characterization of the Exponential Distribution.- 1.4 Remarks.- 2 The Claim Number Process.- 2.1 The Model.- 2.2 The Erlang Case.- 2.3 A Characterization of the Poisson Process.- 2.4 Remarks.- 3 The Claim Number Process as a Markov Process.- 3.1 The Model.- 3.2 A Characterization of Regularity.- 3.3 A Characterization of the Inhomogeneous Poisson Process.- 3.4 A Characterization of Homogeneity.- 3.5 A Characterization of the Poisson Process.- 3.6 A Claim Number Process with Contagion.- 3.7 Remarks.- 4 The Mixed Claim Number Process.- 4.1 The Model.- 4.2 The Mixed Poisson Process.- 4.3 The Pólya-Lundberg Process.- 4.4 Remarks.- 5 The Aggregate Claims Process.- 5.1 The Model.- 5.2 Compound Distributions.- 5.3 A Characterization of the Binomial, Poisson, and Negativebinomial Distributions.- 5.4 The Recursions of Panjer and DePril.- 5.5 Remarks.- 6 The Risk Process in Reinsurance.- 6.1 The Model.- 6.2 Thinning a Risk Process.- 6.3 Decomposition of a Poisson Risk Process.- 6.4 Superposition of Poisson Risk Processes.- 6.5 Remarks.- 7 The Reserve Process and the Ruin Problem.- 7.1 The Model.- 7.2 Kolmogorov’s Inequality for Positive Supermartingales.- 7.3 Lundberg’s Inequality.- 7.4 On the Existence of a Superadjustment Coefficient.- 7.5 Remarks.- Appendix: Special Distributions.- Auxiliary Notions.- Measures.- Generalities on Distributions.- Discrete Distributions.- Continuous Distributions.- List of Symbols.- Author Index.