Book Details
Book：Ruin Probabilities: 2nd Edition
Authors：Soren Asmussen,Hansjorg Albrecher
Publisher：World Scientific Publishing Company
Total Pages：500
Language：English

The book gives a comprehensive treatment of the classical and modern ruin probability theory. Some of the topics are Lundberg's inequality, the CramrLundberg approximation, exact solutions, other approximations (e.g., for heavy-tailed claim size distributions), finite horizon ruin probabilities, extensions of the classical compound Poisson model to allow for reserve-dependent premiums, Markov-modulation, periodicity, change of measure techniques, phase-type distributions as a computational vehicle and the connection to other applied probability areas, like queueing theory. In this substantially updated and extended second version, new topics include stochastic control, fluctuation theory for Levy processes, GerberShiu functions and dependence.



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 . . . . . . . . . . . . 110
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 L´evy 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 L´evy 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