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