Ruin Probabilities

Contents......Page 6
Preface......Page 10
Notation and conventions......Page 14
1 The risk process......Page 20
2a Light-tailed distributions......Page 25
2b Heavy-tailed distributions......Page 28
3 The arrival process......Page 30
4a Duality with other applied probability models......Page 32
4b Exact solutions......Page 33
4c Numerical methods......Page 35
4d Approximations......Page 36
4e Bounds and inequalities......Page 37
4g Simulation......Page 38
1 Wald martingales......Page 40
2 Gambler’s ruin. Two-sided ruin. Brownian motion......Page 42
3 Further simple martingale calculations......Page 48
4a Generators. The Dynkin martingale......Page 49
4b Diffusions and two-sided ruin......Page 52
4c The Kella-Whitt martingale......Page 53
1 Likelihood ratios and change of measure......Page 58
2 Duality with other applied probability models......Page 64
3 Random walks in discrete or continuous time......Page 67
4 Markov additive processes......Page 73
5 The ladder height distribution......Page 81
IV The compound Poisson model......Page 90
1 Introduction......Page 91
2 The Pollaczeck-Khinchine formula......Page 94
3a The ruin probability when the initial reserve is zero......Page 96
3b Exponential claims......Page 97
3c Some classical analytical results......Page 98
3d Deterministic claims......Page 100
4 Change of measure via exponential families......Page 101
5 Lundberg conjugation......Page 103
5a Alternative proofs......Page 108
6a On the existence of °......Page 110
6b Bounds and approximations for......Page 111
6c A refinement of Lundberg’s inequality......Page 112
7b De Vylder’s approximation......Page 114
7c The heavy tra±c approximation......Page 115
7d The light tra±c approximation......Page 117
7e Interpolating between light and heavy traffi......Page 118
8 Comparing the risks of different claim size distributions......Page 119
9 Sensitivity estimates......Page 122
10 Estimation of the adjustment coefficient......Page 129
V The probability of ruin within finite time......Page 134
1 Exponential claims......Page 135
2 The ruin probability with no initial reserve......Page 140
3 Laplace transforms......Page 145
4 When does ruin occur?......Page 147
4a Segerdahl’s normal approximation......Page 149
4b Gerber’s time-dependent version of Lundberg’s inequality......Page 151
4c Arfwedson’s saddlepoint approximation......Page 152
5 Diffusion approximations......Page 155
6 Corrected diffusion approximations......Page 158
7 How does ruin occur?......Page 165
1 Introduction......Page 170
2 Exponential claims. The compound Poisson model with negative claims......Page 173
3a The imbedded random walk......Page 176
3b Markov additive representations......Page 178
4 The duality with queueing theory......Page 180
1 Model and examples......Page 184
2 The ladder height distribution......Page 191
3 Change of measure via exponential families......Page 199
3a Lundberg conjugation......Page 200
3b Ramifications of Lundberg’s inequality......Page 203
4a Ordering of the ruin functions......Page 207
4b Ordering of adjustment coefficients......Page 210
4c Sensitivity estimates for the adjustment coefficient......Page 211
5 The Markovian arrival process......Page 213
6a The model......Page 215
6b Lundberg conjugation......Page 217
6c Markov-modulated approximations......Page 223
7 Dual queueing models......Page 224
1 Introduction......Page 228
1a Two-step premium functions......Page 234
1b Multi-step premium functions......Page 237
2 The model with constant interest......Page 241
3 The local adjustment coefficient. Logarithmic asymptotics......Page 246
3a Examples......Page 248
3b Proof of Theorem 3.2......Page 252
3c Proof of Theorem 3.3......Page 254
4 The model with tax......Page 258
5 Discrete-time ruin problems with stochastic investment......Page 261
6 Continuous-time ruin problems with stochastic investment......Page 267
1 Definition and basic properties of phase-type distributions......Page 272
1a Asymptotic exponentiality......Page 278
2 Renewal theory......Page 279
3a Phase-type claims......Page 283
4 The renewal model......Page 285
5 Markov-modulated input......Page 290
5a Calculations via fluid models. Diagonalization......Page 291
5b Computations via K......Page 295
6 Matrix-exponential distributions......Page 296
7 Reserve-dependent premiums......Page 300
7a Computing  (u) via differential equations......Page 301
7b Two-step premium rules......Page 304
8 Erlangization for the finite horizon case......Page 306
1 Subexponential distributions......Page 312
2 The compound Poisson model......Page 321
3 The renewal model......Page 324
4 Finite-horizon ruin probabilities......Page 328
4a Excursion theory for Markov processes......Page 329
4b The time to ruin......Page 332
5 Reserve-dependent premiums......Page 337
6a The mean excess plot......Page 339
6b Extreme values and POT......Page 341
6c The Hill estimator......Page 343
1 Preliminaries......Page 348
1a Special Levy processes......Page 351
1b Exponential change of measure......Page 354
2 One-sided ruin theory......Page 355
3 The scale function and two-sided ruin problems......Page 359
4 Further topics......Page 364
4a Local time at the maximum......Page 365
4b The ladder height process......Page 366
4c Excursions......Page 367
4d The Wiener-Hopf factorization......Page 368
4e A quintuple identity......Page 369
5 The scale function for two-sided phase-type jumps......Page 372
1 Introduction......Page 376
2 The compound Poisson model......Page 379
2a A Laplace transform approach......Page 380
2b Change of measure......Page 386
2c Martingales......Page 388
2d Further ruin-related quantities......Page 391
3 The renewal model......Page 393
3a Change of measure......Page 394
3b A modified random walk......Page 395
3c Integro-differential equations......Page 396
4a Spectrally negative L´evy processes......Page 403
4b The compound Poisson model with two-sided jumps......Page 407
XIII Further models with dependence......Page 416
1 Large deviations......Page 417
2 Heavy-tailed risk models with dependent input......Page 429
3 Linear models......Page 436
4 Risk processes with shot-noise Cox intensities......Page 438
5 Causal dependency models......Page 443
6 Dependent Sparre Andersen models......Page 446
7 Gaussian models. Fractional Brownian motion......Page 447
8 Ordering of ruin probabilities......Page 452
9 Multi-dimensional risk processes......Page 454
1 Introduction......Page 464
2 Stochastic dynamic programming......Page 466
3 The Hamilton-Jacobi-Bellman equation......Page 467
1a The crude Monte Carlo method......Page 480
1b Variance reduction techniques......Page 481
1c Rare events simulation......Page 482
2 Simulation via the Pollaczeck-Khinchine formula......Page 484
2a Light tails: importance sampling......Page 485
2b Heavy tails: conditional Monte Carlo......Page 486
2c Heavy tails: importance sampling......Page 488
3 Static importance sampling via Lundberg conjugation......Page 489
4 Static importance sampling for the finite horizon case......Page 493
5 Dynamic importance sampling......Page 494
5a An algorithm by Dupuis, Leder and Wang......Page 495
6 Regenerative simulation......Page 501
7 Sensitivity analysis......Page 503
1 More on discrete-time risk models......Page 506
2 The distribution of the aggregate claims......Page 512
2a The saddlepoint approximation......Page 513
2b The NP approximation......Page 515
2c Panjer’s recursion......Page 517
2d The distribution of dependent sums......Page 520
3 Principles for premium calculation......Page 529
4 Reinsurance......Page 532
1a Renewal processes and the renewal theorem......Page 536
1b Renewal equations and the key renewal theorem......Page 537
1c Regenerative processes......Page 538
1e Residual and past lifetime......Page 539
1f Markov renewal theory......Page 540
A2 Wiener-Hopf factorization......Page 541
A3 Matrix-exponentials......Page 545
4a Generalized inverses......Page 549
4b The Kronecker product   and the Kronecker sum......Page 551
4c The Perron-Frobenius theorem......Page 554
5a Asymptotic exponentiality......Page 555
5b Discrete phase-type distributions......Page 557
5c Closure properties......Page 558
5d Phase-type approximation......Page 561
5e Phase-type fitting......Page 564
A6 Tauberian theorems......Page 567
Bibliography......Page 568
Index......Page 616