Preface Notation and conventions I Introduction 1 The risk process 2 Claim size distributions 3 The arrival process 4 A summary of main results and methods II Martingales and simple ruin calculations 1 Wald martingales 2 Gambler's ruin. Two-sided ruin. Brownian motion 3 Further simple martingale calculations 4 More advanced martingales III Further general tools and results 1 Likelihood ratios and change of measure 2 Duality with other applied probability models 3 Random walks in discrete or continuous time 4 Markov additive processes 5 The ladder height distribution IV The compound Poisson model 1 Introduction 2 The Pollaczeck-Khinchine formula 3 Special cases of the Pollaczeck-Khinchine formula 4 Change of measure via exponential families 5 Lundberg conjugation 6 Further topics related to the adjustment coefficient 7 Various approximations for the ruin probability 8 Comparing the risks of different claim size distributions 9 Sensitivity estimates 10 Estimation of the adjustment coefficient V The probability of ruin within finite time 1 Exponential claims 2 The ruin probability with no initial reserve 3 Laplace transforms 4 When does ruin occur? 5 Diffusion approximations 6 Corrected diffusion approximations 7 How does ruin occur? VI Renewal arrivals 1 Introduction 2 Exponential claims. The compound Poisson model with negative claims 3 Change of measure via exponential families 4 The duality with queueing theory VII Risk theory in a Markovian environment 1 Model and examples 2 The ladder height distribution 3 Change of measure via exponential families 4 Comparisons with the compound Poisson model 5 The Markovian arrival process 6 Risk theory in a periodic environment 7 Dual queueing models VIII Level-dependent risk processes 1 Introduction 2 The model with constant interest 3 The local adjustment coefficient. Logarithmic asymptotics 4 The model with tax 5 Discrete-time ruin problems with stochastic investment 6 Continuous-time ruin problems with stochastic investment IX Matrix-analytic methods 1 Definition and basic properties of phase-type distributions 2 Renewal theory 3 The compound Poisson model 4 The renewal model 5 Markov-modulated input 6 Matrix-exponential distributions 7 Reserve-dependent premiums 8 Erlangization for the finite horizon case X Ruin probabilities in the presence of heavy tails 1 Subexponential distributions 2 The compound Poisson model 3 The renewal model 4 Finite-horizon ruin probabilities 5 Reserve-dependent premiums 6 Tail estimation XI Ruin probabilities for Levy processes 1 Preliminaries 2 One-sided ruin theory 3 The scale function and two-sided ruin problems 4 Further topics 5 The scale function for two-sided phase-type jumps XII Gerber-Shiu functions 1 Introduction 2 The compound Poisson model 3 The renewal model 4 Levy risk models XIII Further models with dependence 1 Large deviations 2 Heavy-tailed risk models with dependent input 3 Linear models 4 Risk processes with shot-noise Cox intensities 5 Causal dependency models 6 Dependent Sparre Andersen models 7 Gaussian models. Fractional Brownian motion 8 Ordering of ruin probabilities 9 Multi-dimensional risk processes XIV Stochastic control 1 Introduction 2 Stochastic dynamic programming 3 The Hamilton-Jacobi-Bellman equation XV Simulation methodology 1 Generalities 2 Simulation via the Pollaczeck-Khinchine formula... 3 Static importance sampling via Lundberg conjugation 4 Static importance sampling for the finite horizon case 5 Dynamic importance sampling 6 Regenerative simulation 7 Sensitivity analysis XVI Miscellaneous topics 1 More on discrete-time risk models 2 The distribution of the aggregate claims 3 Principles for premium calculation 4 Reinsurance Appendix A1 Renewal theory A2 Wiener-Hopf factorization A3 Matrix-exponentials A4 Some linear algebra A5 Complements on phase-type distributions A6 Tauberian theorems Bibliography Index
