Table of Contents
Preface ix
Notation and conventions xiii
I Introduction 1
1 The risk process 1
2 Claim size distributions 6
3 The arrival process 11
4 A summary of main results and methods 13
II Martingales and simple ruin calculations 21
1 Wald martingales 21
2 Gambler's ruin. Two-sided ruin. Brownian motion 23
3 Further simple martingale calculations 29
4 More advanced martingales 30
III Further general tools and results 39
1 Likelihood ratios and change of measure 39
2 Duality with other applied probability models 45
3 Random walks in discrete or continuous time 48
4 Markov additive processes 54
5 The ladder height distribution 62
IV The compound Poisson model 71
1 Introduction 72
2 The Pollaczeck-Khinchine formula 75
3 Special cases of the Pollaczeck-Khinchine formula 77
4 Change of measure via exponential families 82
5 Lundberg conjugation 84
6 Further topics related to the adjustment coefficient 91
7 Various approximations for the ruin probability 95
8 Comparing the risks of different claim size distributions 100
9 Sensitivity estimates 103
10 Estimation of the adjustment coefficient 100
V The probability of ruin within finite time 115
1 Exponential claims 116
2 The ruin probability with no initial reserve 121
3 Laplace transforms 126
4 When does ruin occur? 128
5 Diffusion approximations 136
6 Corrected diffusion approximations 139
7 How does ruin occur? 146
VI Renewal arrivals 151
1 Introduction 151
2 Exponential claims. The compound Poisson model with negative claims 154
3 Change of measure via exponential families 157
4 The duality with queueing theory 161
VII Risk theory in a Markovian Environment 165
1 Model and examples 165
2 The ladder height distribution 172
3 Change of measure via exponential families 180
4 Comparisons with the compound Poisson model 188
5 The Markovian arrival process 194
6 Risk theory in a periodic environment 196
7 Dual queueing models 205
VIII Level-dependent risk processes 209
1 Introduction 209
2 The model with constant interest 222
3 The local adjustment coefficient. Logarithmic asymptotics 227
4 The model with tax 239
5 Discrete-time ruin problems with stochastic investment 242
6 Continuous-time ruin problems with stochastic investment 248
IX Matrix-analytic methods 253
1 Definition and basic properties of phase-type distributions 253
2 Renewal theory 260
3 The compound Poisson model 264
4 The renewal model 266
5 Markov-modulated input 271
6 Matrix-exponential distributions 277
7 Reserve-dependent premiums 281
8 Erlangization for the finite horizon case 287
X Ruin probabilities in the presence of heavy tails 293
1 Subexponential distributions 293
2 The compound Poisson model 302
3 The renewal model 305
4 Finite-horizon ruin probabilities 309
5 Reserve-dependent premiums 318
6 Tail estimation 320
XI Ruin probabilities for Lévy processes 329
1 Preliminaries 329
2 One-sided ruin theory 336
3 The scale function and two-sided ruin problems 340
4 Further topics 345
5 The scale function for two-sided phase-type jumps 353
XII Gerber-Shiu functions 357
1 Introduction 357
2 The compound Poisson model 360
3 The renewal model 374
4 Lévy risk models 384
XIII Further models with dependence 397
1 Large deviations 398
2 Heavy-tailed risk models with dependent input 410
3 Linear models 417
4 Risk processes with shot-noise Cox intensities 419
5 Causal dependency models 424
6 Dependent Sparre Andersen models 427
7 Gaussian models. Fractional Brownian motion 428
8 Ordering of ruin probabilities 433
9 Multi-dimensional risk processes 435
XIV Stochastic control 445
1 Introduction 445
2 Stochastic dynamic programming 447
3 The Hamilton-Jacobi-Bellman equation 448
XV Simulation methodology 461
1 Generalities 461
2 Simulation via the Pollaczeck-Khinchine formula 465
3 Static importance sampling via Lundberg conjugation 470
4 Static importance sampling for the finite horizon case 474
5 Dynamic importance sampling 475
6 Regenerative simulation 482
7 Sensitivity analysis 484
XVI Miscellaneous topics 487
1 More on discrete-time risk models 487
2 The distribution of the aggregate claims 493
3 Principles for premium calculation 510
4 Reinsurance 513
Appendix 517
A1 Renewal theory 517
A2 Wiener-Hopf factorization 522
A3 Matrix-exponentials 526
A4 Some linear algebra 530
A5 Complements on phase-type distributions 536
A6 Tauberian theorems 548
Bibliography 549
Index 597
