Ruin Probabilities (2nd Edition) Book

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