## Modelling Extremal Events: For Insurance and FinanceBoth in insurance and in finance applications, questions involving extremal events (such as large insurance claims, large fluctuations, in financial data, stock-market shocks, risk management, ...) play an increasingly important role. This much awaited book presents a comprehensive development of extreme value methodology for random walk models, time series, certain types of continuous-time stochastic processes and compound Poisson processes, all models which standardly occur in applications in insurance mathematics and mathematical finance. Both probabilistic and statistical methods are discussed in detail, with such topics as ruin theory for large claim models, fluctuation theory of sums and extremes of iid sequences, extremes in time series models, point process methods, statistical estimation of tail probabilities. Besides summarising and bringing together known results, the book also features topics that appear for the first time in textbook form, including the theory of subexponential distributions and the spectral theory of heavy-tailed time series. A typical chapter will introduce the new methodology in a rather intuitive (tough always mathematically correct) way, stressing the understanding of new techniques rather than following the usual "theorem-proof" format. Many examples, mainly from applications in insurance and finance, help to convey the usefulness of the new material. A final chapter on more extensive applications and/or related fields broadens the scope further. The book can serve either as a text for a graduate course on stochastics, insurance or mathematical finance, or as a basic reference source. Its reference quality is enhanced by a very extensive bibliography, annotated by various comments sections making the book broadly and easily accessible. |

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### Contents

I | 1 |

II | 21 |

III | 22 |

IV | 28 |

V | 36 |

VI | 37 |

VII | 39 |

VIII | 44 |

LX | 323 |

LXI | 325 |

LXII | 327 |

LXIII | 345 |

LXIV | 348 |

LXV | 352 |

LXVI | 358 |

LXVII | 371 |

IX | 49 |

X | 53 |

XI | 59 |

XII | 60 |

XIII | 70 |

XIV | 82 |

XV | 88 |

XVI | 96 |

XVII | 103 |

XVIII | 106 |

XIX | 113 |

XX | 114 |

XXI | 120 |

XXII | 128 |

XXIII | 130 |

XXIV | 134 |

XXV | 138 |

XXVI | 152 |

XXVII | 168 |

XXVIII | 181 |

XXIX | 182 |

XXX | 196 |

XXXI | 204 |

XXXII | 209 |

XXXIII | 219 |

XXXIV | 220 |

XXXV | 225 |

XXXVI | 226 |

XXXVII | 232 |

XXXVIII | 237 |

XXXIX | 238 |

XL | 242 |

XLI | 247 |

XLII | 248 |

XLIII | 250 |

XLIV | 254 |

XLV | 260 |

XLVI | 263 |

XLVII | 264 |

XLVIII | 277 |

XLIX | 283 |

L | 290 |

LII | 294 |

LIII | 303 |

LIV | 305 |

LV | 307 |

LVI | 309 |

LVII | 316 |

LVIII | 317 |

LIX | 321 |

LXVIII | 372 |

LXIX | 378 |

LXX | 381 |

LXXI | 386 |

LXXII | 393 |

LXXIII | 403 |

LXXIV | 413 |

LXXV | 418 |

LXXVI | 424 |

LXXVII | 430 |

LXXVIII | 431 |

LXXIX | 433 |

LXXX | 436 |

LXXXI | 439 |

LXXXII | 444 |

LXXXIII | 449 |

LXXXIV | 454 |

LXXXV | 455 |

LXXXVI | 461 |

LXXXVII | 473 |

LXXXVIII | 481 |

LXXXIX | 483 |

XC | 486 |

XCI | 493 |

XCII | 498 |

XCIII | 503 |

XCIV | 507 |

XCV | 521 |

XCVI | 522 |

XCVII | 526 |

XCVIII | 527 |

XCIX | 532 |

C | 541 |

CI | 551 |

CIV | 552 |

CV | 553 |

CVII | 554 |

CIX | 555 |

CXI | 557 |

CXII | 559 |

CXIV | 561 |

CXV | 562 |

CXVI | 564 |

CXVII | 571 |

CXVIII | 583 |

CXIX | 587 |

CXX | 591 |

627 | |

CXXII | 643 |

### Common terms and phrases

a-stable applications ARCH(l assume asymptotic autocorrelations behaviour Borel sets Brownian motion Chapter condition constants cn corresponding Cramer-Lundberg defined Definition denotes density df F discussion domain of attraction Embrechts Example extremal events extremal index extreme value distribution extreme value theory Figure finite finite-dimensional distributions Gaussian generalised geometric Brownian motion Gumbel Gumbel distribution Haan heavy-tailed Hence Hill estimator homogeneous Poisson process iid rvs iid sequence instance integral Laplace-Stieltjes transform large deviations Lemma limit distribution limsup linear process maxima maximum domain MDA(yl mean excess function method normalised norming constants Notes and Comments P(Mn parameter Pareto Pareto distribution periodogram point process proof properties Proposition quantile random walk regular variation regularly varying reinsurance relation sample paths Section sequence Xn SLLN stationary process stationary sequence stochastic processes subexponential Suppose tail Theorem threshold upper order statistics vector weak convergence