Ruin Probabilities

621 Pages·2010·2.885 MB·English

by  Søren Asmussen

RUIN PROBABILITIES
Second Edition
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ADVANCED SERIES ON STATISTICAL SCIENCE &
APPLIED PROBABILITY
Editor: Ole E. Barndorff-Nielsen
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Vol. 1 Random Walks of Infinitely Many Particles
by P. Révész
Vol. 2 Ruin Probabilities
by S. Asmussen
Vol. 3 Essentials of Stochastic Finance: Facts, Models, Theory
by Albert N. Shiryaev
Vol. 4 Principles of Statistical Inference from a Neo-Fisherian Perspective
by L. Pace and A. Salvan
Vol. 5 Local Stereology
by Eva B. Vedel Jensen
Vol. 6 Elementary Stochastic Calculus — With Finance in View
by T. Mikosch
Vol. 7 Stochastic Methods in Hydrology: Rain, Landforms and Floods
eds. O. E. Barndorff-Nielsen et al.
Vol. 8 Statistical Experiments and Decisions: Asymptotic Theory
by A. N. Shiryaev and V. G. Spokoiny
Vol. 9 Non-Gaussian Merton–Black–Scholes Theory
by S. I. Boyarchenko and S. Z. Levendorskiĭ
Vol. 10 Limit Theorems for Associated Random Fields and Related Systems
by A. Bulinski and A. Shashkin
Vol. 11 Stochastic Modeling of Electricity and Related Markets
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Vol. 12 An Elementary Introduction to Stochastic Interest Rate Modeling
by N. Privault
Vol. 13 Change of Time and Change of Measure
by O. E. Barndorff-Nielsen and A. Shiryaev
Vol. 14 Ruin Probabilities (2nd Edition)
by S. Asmussen and H. Albrecher
EH - Ruin Probabilities.pmd 2 7/26/2010, 4:56 PM
Advanced Series on
Statistical Science &
Vol. 14
Applied Probability
RUIN PROBABILITIES
Second Edition
Søren Asmussen
Aarhus University, Denmark
Hansjörg Albrecher
University of Lausanne, Switzerland
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Library of Congress Cataloging-in-Publication Data
Asmussen, Søren.
Ruin probabilities / by Søren Asmussen & Hansjörg Albrecher
p. cm. -- (Advanced series on statistical science and applied probability ; v. 14)
Includes bibliographical references and index.
ISBN-13: 978-981-4282-52-9 (hardcover : alk. paper)
ISBN-10: 981-4282-52-9 (hardcover : alk. paper)
1. Insurance--Mathematics. 2. Risk. I. Albrecher, Hansjörg. II. Title.
HG8781.A83 2010
368'.01--dc22
2010023280
British Library Cataloguing-in-Publication Data
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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
v
vi CONTENTS
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 neg-
ative 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
CONTENTS vii
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
viii CONTENTS
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
Preface
This book is a second edition of the book of the same title by the first author
which was published in 2000. The subject of ruin probabilities and related top-
icshassincethenundergoneaconsiderabledevelopment,nottosayboom. This
much expanded and revised second edition aims at covering a substantial part
of these developments as well as the classical topics.
Risk theory in general and ruin probabilities in particular are traditionally
considered as part of insurance mathematics, and has been an active area of
research from the days of Lundberg all the way up to today. One reason for
writing this book is a feeling that the area has in recent years achieved a con-
siderable mathematical maturity, which has in particular removed one of the
standard criticisms of the area, namely that it can only say something about
very simple models and questions. Although in insurance practice, usually sim-
pler(andcoarser)riskmeasureslikeValue-at-Riskareused,itiswidelybelieved
that the thinking advocated by ruin theory is still important for modern risk
management. For instance, in times of market-consistent valuation principles,
the role of the time diversification effect of insurance portfolios, which is one of
the core elements of ruin theory, should not be forgotten. In addition, ruin the-
ory has fruitful methodological links and applications to other fields of applied
probability, like queueing theory and mathematical finance (pricing of barrier
options, credit products etc.). Apart from these remarks, we have deliberately
stayed away from discussing the practical relevance of the theory; if the formu-
lations occasionally give a different impression, it is not by intention. Thus, the
book is basically mathematical in its flavor.
The present second edition is more than 50% longer than the first and has
more than double the number of references. The longer parts of the new mate-
rial,reflectingsubareasthathavebeenparticularlyactiveinthelastdecade,are
collected in Chapters XI–XIV, which treat L´evy processes, Gerber-Shiu func-
tions,dependenceandstochasticcontrol,respectively. Shorteradditionsinclude
ix