1 The classical risk model.- 1.1 Ruin probabilities for the classical risk process.- 1.2 "Practical" evaluation of ruin probabilities.- 1.3 Inference for the risk process.- 2 Generalizations of the classical risk model.- 2.1 Models allowing for size fluctuation.- 2.2 Models allowing for risk fluctuation.- 3 Renewal models.- 3.1 Ordinary renewal models.- 3.2 Stationary renewal models.- 3.3 Numerical illustrations.- 4 Cox models.- 4.1 Markovian intensity: Preliminaries.- 4.2 The martingale approach.- 4.3 Independent jump intensity.- 4.3.1 An inbedded random walk.- 4.3.2 Ordinary independent jump intensity.- 4.3.3 Stationary independent jump intensity.- 4.4 Markov renewal intensity.- 4.5 Markovian intensity.- 4.5.1 Application of the basic approach.- 4.5.2 An alternative approach.- 4.6 Numerical illustrations.- 5 Stationary models.- Appendix. Finite time ruin probabilities.- A.1 The classical model.- A.2 Renewal models.- A.3 Cox models.- A.4 Diffusion approximations.- References and author index.- Inserted surveys.- Basic martingale theory.- Basic facts about weak convergence.- Point processes and martingales.- Point processes and random measures.- Basic definitions.- Superposition of point processes.- Thinning of point processes.- Basic Markov process theory.- Stationary point processes.
Risk theory, which deals with stochastic models of an insurance business, is a classical application of probability theory. The fundamental problem in risk theory is to investigate the ruin possibility of the risk business. Traditionally the occurrence of the claims is described by a Poisson process and the cost of the claims by a sequence of random variables. This book is a treatise of risk theory with emphasis on models where the occurrence of the claims is described by more general point processes than the Poisson process, such as renewal processes, Cox processes and general stationary point processes. In the Cox case the possibility of risk fluctuation is explicitly taken into account. The presentation is based on modern probabilistic methods rather than on analytic methods. The theory is accompanied with discussions on practical evaluation of ruin probabilities and statistical estimation. Many numerical illustrations of the results are given.
The book is intended for graduate or postgraduate students in actuarial science and probability as well as for working actuaries with background in probability. To make the book self-contained, surveys are given on Markov processes, martingales and point processes. It also contains an introduction to traditional risk theory.