Ruin Probabilities (Advanced Series on Statistical Science and Applied Probability Series)

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Advanced Series on Statistical Science & I Applied Probability

^^Â£J

Ruin Probabilities

Seren Asmussen

World Scientific

Ruin Probabilities

ADVANCED SERIES ON STATISTICAL SCIENCE & APPLIED PROBABILITY

Editor: Ole E. Barndorff-Nielsen

Published Vol. 1: Random Walks of Infinitely Many Particles by P. Revesz 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

Ruin P robabilities

Soren Asmussen Mathematical Statistics Centre for Mathematical Sciences Lund University

Sweden

World Scientific Singapore • NewJersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fatter Road , Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Asmussen, Soren

Ruin probabilities / Soren Asmussen. p. cm. -- (Advanced series on statistical science and applied probability ; vol. 2) Includes bibliographical references and index. ISBN 9810222939 (alk. paper) 1. Insurance--Mathematics. 2. Risk. I. Tide. II. Advanced series on statistical science & applied probability ; vol. 2. HG8781 .A83 2000 368'.01--dc2l 00-038176

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

First published 2000 Reprinted 2001

Copyright ® 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore

Contents Preface I

ix

Introduction 1 1 The risk process . . . . . . . . . . . . . .. . . . .. .. . . . . 1 2 Claim size distributions .. . . . . . . . .. . . . . . . . . . . . 5 3 The arrival process . . . . . . . . . . . . . . . . . . . . . . . . 11 4 A summary of main results and methods . . . . .. . . . . . . 13 5 Conventions . .. . .. .. . . . . . . . . . . . . . . . . . . . . 19

II Some general tools and results 23 1 Martingales . .. . .. .. . . . . . .. . . . . . . . . . . . . . 24 2 Likelihood ratios and change of measure . . .. . . . . . .. . 26 3 Duality with other applied probability models . . .. . . . . . 30 4 Random walks in discrete or continuous time . . . . . . . . . . 33 5 Markov additive processes . . . . . . . .. . . . . . . . . . . . 39 6 The ladder height distribution . . . .. . .. .. . . . . . . . . 47 III The compound Poisson model 57 1 Introduction . . . . . . . . .. .. .. . .. .. . . . . . . 58 2 The Pollaczeck-Khinchine formula

. . . . . . . . . . . . . . . 61 3 Special cases of the Pollaczeck-Khinchine formula . . . . . . . 62 4 Change of measure via exponential families . . . .... . .. . 67 5 Lundberg conjugation . .. . . . . . . . . . . . . . . . . . . . . 69 6 Further topics related to the adjustment coefficient .. . . . . 75 7 Various approximations for the ruin probability . . . . . . . . 79 8 Comparing the risks of different claim size distributions . . . . 83 9 Sensitivity estimates

. . . . . . . . . . . . . . . . . . . . . . . 10 Estimation of the adjustment coefficient . . . . . . . . . . . .

v

86 93

vi

CONTENTS

IV The probability of ruin within finite time 97 1 Exponential claims . . . . . . . . . . . . . . . . . . . . . . . . 98 2 The ruin probability with no initial reserve . . . . . . . . . . . 103 3 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . 108 4 When does ruin occur? . . . . . . . . . . . . . . . . . . . . . . 110 5 Diffusion approximations . . . . . . . . . . . . .. . . .. . . . 117 6 Corrected diffusion approximations . . . . . . . . . . .. . . . 121 7 How does ruin occur ? . . .. . . . . . . . . . . . . . . . . . . . 127 V Renewal arrivals 131 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2 Exponential claims. The compound Poisson model with negative claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3 Change of measure via exponential families . . . . . . . . . . . 137 4 The duality with queueing theory .. .. .. . . . .. . . . . . 141 VI Risk theory in a Markovian environment 145 1 Model and examples . . . . . . . . . . . .. . .. . . . . . . . 145 2 The ladder height distribution . . . . . . . . . .. . . . . . . . 152 3 Change of measure via exponential families ........... 160 4 Comparisons with the compound Poisson model ........ 168 5 The Markovian arrival process . . . . . . .. .. . . ... . . . 173 6 Risk theory in a periodic environment .. . . . .. . . . . . . . 176 7 Dual queueing models .... ... ................ 185 VII Premiums depending on the current reserve 189 1 Introduction . . . . . . . . . . . . . . . . . . . .. . . . . . . . 189 2 The model with interest . . . . . .. . . . . . . . . . .. . . . 196 3 The local adjustment coefficient. Logarithmic asymptotics . . 201 VIII Matrix-analytic methods 215 1 Definition and basic properties of phase-type distributions .. 215 2 Renewal theory . . . . . . . . . . . . . . . . . . . . . . . . . . 223 3 The compound Poisson model . . . . . . . . . .. . . . . . . . 227 4 The renewal model . . . . . . . . . . . . . . . .. . . . . . . . 229 5 Markov-modulated input . . .. . . . . . . . . . . . . . . . . . 234 6 Matrix-exponential distributions . . . . . . . . . . . .. . . . 240 7 Reserve-dependent premiums . . . . .. . . . .. . . . . . . . 244

vii

CONTENTS

IX Ruin probabilities in the presence of heavy tails 251 1 Subexponential distributions . . . . . . . . . . . . . . . . . . . 251 2 The compound Poisson model . .. . . . . . . . . . . . . . . . 259 3 The renewal model . . . . . . . . . . . . . . . . . . . . . . . . 261 4 Models with dependent input . . . . . . . . . . . . . . . . . . 264 5 Finite-horizon ruin probabilities . . . . .. . . . . . . . . . . . 271 6 Reserve-dependent premiums . . . . . . . . . . . . . . . . . . 279 X Simulation methodology 281 1 Generalities . .. . . . . . . . . . . .. . . . . . . . .. . .. . 281 2 Simulation via the Pollaczeck-Khinchine formula . . . . . . . 285 3 Importance sampling via Lundberg conjugation . . . . . . . . 287 4 Importance sampling for the finite horizon case . . . . . .. . 290 5 Regenerative simulation . .. . . . . . . . . . . . . . . . . . . 292 6 Sensitivity analysis . . . . .. . .. . . . . . . . . . . . . . . . 294 XI Miscellaneous topics 297 1 The ruin problem for Bernoulli random walk and Brownian motion. The two-barrier ruin problem . . . . . . . . . . . . . 297 2 Further applications of martingales . . . . . . . . . . . . . . . 304 3 Large deviations . . . . . ... . .. . . . . . . . . . . . . .. . 306 4 The distribution of the aggregate claims . . . . . . . . . .. . 316 5 Principles for premium calculation . . . .. . . . . . . . . . . . 323 6 Reinsurance . . . . . . . . . . . .. . . . . . . . . . . . . . . . 326 Appendix 331 Al Renewal theory . . . . .. .. . . . . . . . . . . . . .. . . . . 331 A2 Wiener-Hopf factorization .. . . . . . . . . . . . . . . . . . . 336 A3 Matrix-exponentials . . . . . . . . .. . . . . . . . .. . . . . 340 A4 Some linear algebra . . . . . . . . . . . . . . . . . . . . . . . . 344 AS Complements on phase-type distributions . . . . . . . . . .. . 350

Bibliography Index

363

383

This page is intentionally left blank

Preface The most important to say about the history of this book is: it took too long time to write it! In 1991, I was invited to give a course on ruin probabilities at the Laboratory of Insurance Mathematics, University of Copenhagen. Since I was to produce some hand-outs for the students anyway, the idea was close to expand these to a short book on the subject, and my belief was that this could be done rather quickly. The course was never realized, but the hand-outs were written and the book was started (even a contract was signed with a deadline I do not dare to write here!). But the pace was much slower than expected, and other projects absorbed my interest. As an excuse: many of these projects were related to the book, and the result is now that the book is much more related to my own research than the initial outline. Let me take this opportunity to thank above all my publisher World Scientific Publishing Co. and the series editor Ole Barndorff-Nielsen for their patience. A similar thank goes to all colleagues who encouraged me to finish the project and continued to refer to the book by Asmussen which was to appear in a year which continued to be postponed. Risk theory in general and ruin probablities in particular is 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. However, it would not be fair not to say that the practical relevance of the area has been questioned repeatedly. One reason for writing this book is a feeling that the area has in the recent years achieved a considerable mathematical maturity, which has in particular removed one of the standard criticisms of the area, that it can only say something about very simple models and questions. Apart from these remarks, I have deliberately stayed away from discussing the practical relevance of the theory; if the formulations occasionally give a different impression, it is not by intention. Thus, the book is basically mathematical in its flavour. It has obviously not been possible to cover all subareas. In particular, this applies to long-range dependence which is intensely studied in the neighboring ix

x

PREFACE

field of queueing theory. The main motivation comes from statistical data for network traffic (e.g. Willinger et al. [381]); for the effects on tail probabilities, see e.g. Resnick & Samorodnitsky [303] and references therein. Concerning ruin probabilities, see in particular Michna [259]. Another interesting area which is not covered is dynamic control. In the classical setting of Cramer-Lundberg models, some basic discussion can be found in the books by Biihlmann [82] and Gerber [157]; see also Schmidli [325] and the references in Asmussen & Taksar [52]. More recently, the standard stochastic control setting of diffusion models has been considered, e.g. Hojgaard & Taksar [206], Asmussen, Hojgaard & Taksar [35] and Paulsen & Gjessing [284]. The book does not go into the broader aspects of the interface between insurance mathematics and mathematical finance, an area which is becoming increasingly important. Finally, I regret that due to time constraints, it has not been possible to incorporate more numerical examples than the few there are. A book like this can be organized in many ways. One is by model, another by method. The present book is in between these two possibilities. Chapters III-VII introduce some of the main models and give a first derivation of some of their properties. Chapters IX-X then go in more depth with some of the special approaches for analyzing specific models and add a number of results on the models in Chapters III-VII (also Chapter II is essentially methodological in its flavor). Here is a suggestion on how to get started with the book. For a brief orientation, read Chapter I, the first part of 11.6 (to understand the PollaczeckKhinchine formula in 111.2 more properly), 111.1-5, IV.4a, VII.1, VIII.1-3 and IX.1-3. For a second reading, incorporate 11.1-4, 111.8-9, IV.2, IV.5, VI.1-3, VII.2, IX.4-5, X.1-3 and XI.3. The rest is up to your specific interests. Good luck! I have tried to be fairly exhaustive in citing references close to the text, In addition, some papers not cited in the text but judged to be of interest are included in the Bibliography. It is obvious that such a system involves a number of inconsistencies and omissions, for which I apologize to the reader and the authors of the many papers who ought to have been on the list. I intend to keep a list of misprints and remarks posted on my web page, http:// www.maths .lth.se/matstat / staff/asmus and I am therefore grateful to get relevant material sent by email to asmusfmaths .lth.se Lund February 2000 Soren Asmussen

PREFACE

xi

The second printing differs from the first only by minor corrections, many of which were pointed out by Hanspeter Schmidli . More substantial remarks, of which there are not many at this stage , as well as some additional references continue to be at the web page.

Lund September 2001 Soren Asmussen Acknowledgements Many of the figures , not least the more complicated ones, were produced by Lone Juul Hansen , Aarhus, supported by Center for Mathematical Physics and Stochastics (MaPhySto). A number of other figures were supplied by Christian Geisler Asmussen , Fig. 111 . 5.2 by Rafal Kulik , Fig. IV.6.1 by Bjarne Hojgaard and the table in Example 111.8 .6 by my 1999 simulation class in Lund. Section VII . 3 is reprinted from Asmussen & Nielsen [39] and parts of IX.4 from Asmussen, Schmidli & Schmidt [47] with the permission from Applied Probability Trust . Section VIII. 1 is almost identical to Section 2 of Asmussen [26] and reprinted with permission of Blackwell Publishers. Parts of II.6 is reprinted from Asmussen & Schmidt [49] and parts of IX. 5 from Asmussen & Kliippelberg [36] with the permission from Elsevier Science . Parts of X.1 and X.3 are reprinted from Asmussen & Rubinstein [46] and parts of VIII.5 from Asmussen [21] with permission from CRC Press.


Chapter I

Introduction 1 The risk process In this chapter , we introduce some general notation and terminology, and give a very brief summary of some of the models, results and topics to be studied in the rest of the book. A risk reserve process { Rt}t>o, as defined in broad terms , is a model for the time evolution of the reserves of an insurance company. We denote throughout the initial reserve by u = Ro. The probability O(u) of ultimate ruin is the probability that the reserve ever drops below zero, t/i(u) = P (infRt < 0) = P (infR t < 0 t>0 t>0

Ro=ul.

(1.1)

The probability of ruin before time T is t,i(u,T)

=

P inf Rt < 0 I . (1.2) (O 0 : Rt < 0} = inf It > 0 : St > u}, M =

(1.3)

sup St, MT = sup St, (1.4) O t} = max {n > 0: Un < t}• The size of the nth claim is denoted by Un. • Premiums flow in at rate p, say, per unit time. Putting things together, we see that Nt

Nt

Rt = u + pt - E Uk, St = E Uk - pt. (1.7) k=1 k=1

The sample paths of {Rt} and {St} and the connection between the two processes are illustrated in Fig. 1.1.

Figure 1.1

1. THE RISK PROCESS

3

Note that it is a matter of taste (or mathematical convenience) whether one allows {Rt} and/or {St} to continue its evolution after the time T(u) of ruin. Thus, for example, one could well replace Rt by Rtnr(u) or RtA,(,.) V 0. For the purpose of studying ruin probabilities this distinction is, of course, immaterial. Some main examples of models not incorporated in the above set-up are: • Models with a premium depending on the reserve (i.e., on Fig. 1.1 the slope of {Rt} should depend also on the level). We study this case in Ch. VII. • Brownian motion or more general diffusions. We shall discuss Brownian motion somewhat in Chapter IV, but as an approximation to the risk process rather than as a model of intrinsic merit. However, since any modeling involves some approximative assumptions, one may well argue that Brownian motion in itself could be a reasonable model, and the basic ruin probabilities are derived in XI.1. • General Levy processes (defined as continuous time processes with stationary independent increments) where the jump component has infinite Levy measure, allowing a countable infinity of jumps on Fig. 1.1. We shall not deal with this case either, though many results are straightforward to generalize from the compound Poisson model; a basic references is Gerber [127]. The models we consider will typically have the property that there exists a constant p such that Nt

a

E Uk

p,

t -* oo. (1.8)

k=1

The interpretation of p is as the average amount of claim per unit time. A further basic quantity is the safety loading (or the security loading) n defined as the relative amount by which the premium rate p exceeds p,

rl=

p-P P

It is sometimes stated in the theoretical literature that the typical values of the safety loading 77 are relatively small, say 10% - 20%; we shall, however, not discuss whether this actually corresponds to practice. It would appear obvious, however, that the insurance company should try to ensure 77 > 0, and in fact: Proposition 1.1 Assume that (1.8) holds. If 77 < 0, then M = oo a.s. and hence ,b(u) = 1 for all u. If 77 > 0, then M < oo a.s. and hence O(u) < 1 for all sufficiently large u.

4 CHAPTER I. INTRODUCTION Proof It follows from (1.8) that F _ N, St __ k =1 Uk

t

t

p

pt a4.

-

p'

t -^ oo.

If 77 < 0, then this limit is > 0 which implies St a$ oo and hence M = oo a.s. If ❑ -oo, M < oo a.s. rl > 0, then similarly limSt/t < 0, St In concrete models, we obtain typically a somewhat stronger conclusion, namely that M = oo a.s., tb(u) = 1 for all u holds also when rl = 0, and that ,b(u) < 1 for all u when rl > 0. However, this needs to be verified in each separate case. The simplest concrete example (to be studied in Chapter III) is the compound Poisson model, where {Nt} is a Poisson process with rate ,Q (say) and U1, U2, ... are i.i.d. and independent of {Nt}. Here it is easy to see that p = ,6EU (on the average, ,Q claims arrive per unit time and the mean of a single claim is EU) and that also Nt

lira EEUk = p.

t aoo t

(1.10)

k=1

Again, (1.10) is a property which we will typically encounter. However, not all models considered in the literature have this feature:

Example 1.2 (Cox PROCESSES) Here {Nt} is a Poisson process with random rate /3(t) (say) at time t. If U1, U2, ... are i.i.d. and independent of {(0(t), Nt)}, it is not too difficult to show that p as defined by (1.8) is given by

^t p = EU • lim it (3(s) ds

J

t-,oo t 0

(provided the limit exists). Thus p may well be random for such processes, namely, if {(3(t)} is non-ergodic. The simplest example is 3(t) = V where V is a r .v. This case is referred to as the mixed Poisson process, with the most notable special case being V having a Gamma distribution, corresponding to the Pdlya process. 0 We shall only encounter a few instances of a Cox process, in connection with risk processes in a Markovian or periodic environment (Chapter VI), and here (1.8), (1.10) hold with p constant. Proposition 1.3 Assume p 54 1 and define Rt = Rt1p. Then the connection between the ruin probabilities for the given risk process {Rt} and those ^(u), 0(u,T) for {Rt} is given by V)(u) = t/i (u), zP(u ,T) = i,i(u,Tp).

(1.11)

2. CLAIM SIZE DISTRIBUTIONS 5 The proof is trivial. Since { Rt } has premium rate 1, the role of the result is to justify to take p = 1, which is feasible since in most cases the process { Rt } has a similar structure as {Rt} (for example, the claim arrivals are Poisson or renewal at the same time). Note that when p = 1, the assumption > 0 is equivalent to p < 1; in a number of models, we shall be able to identify p with the traffic intensity of an associated queue, and in fact p < 1 is the fundamental assumption of queueing theory ensuring steady-state behaviour (existence of a limiting stationary distribution). Notes and references The study of ruin probabilities, often referred to as collective risk theory or just risk theory, was largely initiated in Sweden in the first half of the century. Some of the main general ideas were laid down by Lundberg [250], while the first mathematically substantial results were given in Lundberg [251] and Cramer [91]; another important early Swedish work is Tacklind [373]. The Swedish school was pioneering not only in risk theory, but in probability and applied probability as a whole; in particular, many results and methods in random walk theory originate from there and the area was ahead of related ones like queueing theory. Some early surveys are given in Cramer [91], Segerdahl [334] and Philipson [289]. Some main later textbooks are (in alphabetical order) Buhlmann [82], Daykin, Pentikainen & Pesonen [101], De Vylder [110], Gerber [157], Grandell [171], Rolski, Schmidli, Schmidt & Teugels [307] and Seal [326], [330]. Besides in standard journals in probability and applied probability, the research literature is often published in journals like Astin Bulletin , Insurance: Mathematics and Economics, Mitteilungen der Verein der Schweizerischen Versicherungsmathematiker and the Scandinavian Actuarial Journal. The term risk theory is often interpreted in a broader sense than as just to comprise the study of ruin probabilities. An idea of the additional topics and problems one may incorporate under risk theory can be obtained from the survey paper [273] by Norberg; see also Chapter XI. In the even more general area of non-life insurance mathematics, some main texts (typically incorporating some ruin theory but emphasizing the topic to a varying degree) are Bowers et al. [76], Buhlmann [82], Daykin et al. [101], Embrechts et al. [134], Heilmann [191], Hipp & Michel [198], Straub [353], Sundt [354], Taylor [364]. Note that life insurance (e.g. Gerber [159]) has a rather different flavour, and we do not get near to the topic anywhere in this book. Cox processes are treated extensively in Grandell [171]. For mixed Poisson processes and Polya processes, see e .g. the recent survey by Grandell [173] and references therein.

2 Claim size distributions This section contains a brief survey of some of the most popular classes of distributions B which have been used to model the claims U1, U2,.... We roughly classify these into two groups , light-tailed distributions (sometimes the term

6 CHAPTER I. INTRODUCTION 'Cramer-type conditions' is used), and heavy-tailed distributions. Here lighttailed means that the tail B(x) = 1 - B(x) satisfies B(x) = O(e-8x) for some s > 0. Equivalently, the m.g.f. B[s] is finite for some s > 0. In contrast, B is heavy-tailed if b[s] = oo for all s > 0, but different more restrictive definitions are often used: subexponential, regularly varying (see below) or even regularly varying with infinite variance. On the more heuristical side, one could mention also the folklore in actuarial practice to consider B heavy-tailed if '20% of the claims account for more than 80% of the total claims', i.e. if 1 °O

x B(dx) > 0.8,

AB Jbos

where B(bo.2) = 0.2 and /LB is the mean of B.

2a Light-tailed distributions Example 2.1 (THE EXPONENTIAL DISTRIBUTION) Here the density is b(x) = be-ax

(2.1)

The parameter 6 is referred to as the rate or the intensity, and can also be interpreted as the (constant) failure rate b(x)/B(x). As in a number of other applied probability areas, the exponential distribution is by far the simplest to deal with in risk theory as well. In particular, for the compound Poisson model with exponential claim sizes the ruin probability ,O(u) can be found in closed form. The crucial feature is the lack of memory: if X is exponential with rate 6, then the conditional distribution of X - x given X > x is again exponential with rate b (this is essentially equivalent to the failure rate being constant). For example in the compound Poisson model, a simple stopping time argument shows that this implies that the conditional distribution of the overshoot ST(u) - u at the time of ruin given r(u) is again exponential ❑ with rate 8, a fact which turns out to contain considerable information. Example 2 .2 (THE GAMMA DISTRIBUTION) The gamma distribution with parameters p, 6 has density P

r(p)xP-le-ax b(x)

and m.g.f. P

B[s]= (8Is ) , s 1, > 1 for p < 1 and = 1 for p = 1 (the exponential case). The exact form of the tail B(x) is given by the incomplete Gamma function r(x; p),

r(bx; p) °°

B(x) = r(p)

where r (x; p) =

J

tP-le-tdt.

Asymptotically, one has

JP -1

) XP ie -ax

B(x) r(p In the sense of the theory of infinitely divisible distributions, the Gamma density (2.2) can be considered as the pth power of the exponential density (2.1) (or the 1/pth root if p < 1). In particular, if p is integer and X has the gamma distribution p, 0, then X v Xl + • • • + X, where X1, X2.... are i.i.d. and exponential with rate d. This special case is referred to as the Erlang distribution with p stages, or just the Erlang(p) distribution. An appealing feature is its simple connection to the Poisson process: B(x) = P(Xi + • • • + XP > x) is the probability of at most p - 1 Poisson events in [0, x] so that

B(x) = r` e- ate (b2 ): L i=o•

In the present text, we develop computationally tractable results mainly for the Erlang case (p = 1, 2, ...). Ruin probabilities for the general case has been studied, among others, by Grandell & Segerdahl [175] and Thorin [369]. ❑ Example 2 .3 (THE HYPEREXPONENTIAL DISTRIBUTION) This is defined as a finite mixture of exponential distributions, P

b(x) = r` aibie-a;y i=1 where >i ai = 1, 0 < ai < 1, i = 1, ... , p. An important property of the hyperexponential distribution is that its s.c.v. is > 1. ❑

8 CHAPTER I. INTRODUCTION Example 2 .4 (PHASE-TYPE DISTRIBUTIONS) A phase-type distribution is the distribution of the absorption time in a Markov process with finitely many states, of which one is absorbing and the rest transient. Important special cases are the exponential, the Erlang and the hyperexponential distributions. This class of distributions plays a major role in this book as the one within computationally tractable exact forms of the ruin probability z/)(u) can be obtained.

The parameters of a phase-type distribution is the set E of transient states, the restriction T of the intensity matrix of the Markov process to E and the row vector a = (ai)iEE of initial probabilities. The density and c.d.f. are b(x) = aeTxt,

resp. B(x) = aeTxe

where t = Te and e = (1 ... 1)' is the column vector with 1 at all entries. The couple (a, T) or sometimes the triple (E, a, T) is called the representation. We give a more comprehensive treatment in VIII.1 and defer further details to ❑ Chapter VIII. Example 2 .5 (DISTRIBUTIONS WITH RATIONAL TRANSFORMS) A distribution B has a rational m.g.f. (or, equivalently, a rational Laplace transform) if B[s] _ p(s)/q(s) with p(s) and q(s) being polynomials of finite degree. Equivalent characterizations are that the density b(x) has one of the forms q

b(x)

=

cjxienbx,

(2.7)

j=1 q1

q2

q3

b(x) = cjxieWWx + djxi cos(ajx)ea'x + > ejxi sin(bjx)eìx ,(2.8) j=1

j=1

j=1

where the parameters in (2.7) are possibly complex-valued but the parameters in (2.8) are real-valued. This class of distributions is popular in older literature on both risk theory and queues, but the current trend in applied probability is to restrict attention to the class of phase-type distributions, which is slightly smaller but more amenable to probabilistic reasoning. We give some theory for matrix❑ exponential distribution in VIII.6. Example 2 .6 (DISTRIBUTIONS WITH BOUNDED SUPPORT) This example (i.e. there exists a xo < oo such that B(x) = 0 for x > xo, B(x) > 0 for x < xo) is of course a trivial instance of a light-tailed distribution. However, it is notable from a practical point of view because of reinsurance: if excess-of-loss reinsurance has been arranged with retention level xo, then the claim size which is relevant from the point of view of the insurance company itself is U A xo rather than U ❑ (the excess (U - xo)+ is covered by the reinsurer). See XI.6.

2. CLAIM SIZE DISTRIBUTIONS 9

2b Heavy-tailed distributions Example 2.7 (THE WEIBULL DISTRIBUTION) This distribution originates from reliability theory. Here failure rates b(x) = b(x)/B(x) play an important role, the exponential distribution representing the simplest example since here b(x) is constant. However, in practice one may observe that b(x) is either decreasing or increasing and may try to model smooth (incerasing or decreasing) deviations from constancy by 6(x) = dx''-1 (0 < r < oo). Writing c = d/r, we obtain the Weibull distribution B(x) = e-Cx', b(x) = crx''-le-`xr, (2.9) which is heavy-tailed when 0 < r < I. All moments are finite.

❑

Example 2 .8 (THE LOGNORMAL DISTRIBUTION) The lognormal distribution with parameters a2, p is defined as the distribution of ev where V - N(p, a2), or equivalently as the distribution of a°U+µ where U - N(0,1). It follows that the density is

't (1ogX - pl = 1 W (logx -,u l b(x) = d dx or J ax lor 1 exp

f -1 (lox_P)2} (2.10)

Asymptotically, the tail is B (x )

2 x- p a 1 (2.11) ex log logx 2r p 1 1 2 ( a )

The loinormal distribution has moments of all orders. In particular, the mean ❑ is eµ+a /2 and the second moment is e2µ+2o2. Example 2 .9 (THE PARETO DISTRIBUTION) Here the essence is that the tail B(x) decreases like a power of x. There are various variants of the definition around, one being B(x) (1 + X)-b(x) (1 + x)a+1' x > 0. (2.12) Sometimes also a location parameter a > 0 and a scale parameter A > 0 is allowed, and then b(x) = 0, x < a, b(x) _ A(1 + (x a The pth moment is finite if and only if p < a - 1.

a)/A)-a+1'

x > a. (2.13) ❑

CHAPTER I. INTRODUCTION

10

Example 2.10 (THE LOGGAMMA DISTRIBUTION) The loggamma distribution with parameters p, 6 is defined as the distribution of et' where V has the gamma density (2.2). The density is

8p(log x)p-i b(x) - x6+lr(p) (2.14) The pth moment is finite if p < 5 and infinite if p > 5. For p = 1, the loggamma distribution is a Pareto distribution. ❑ Example 2 .11 (PARETO MIXTURES OF EXPONENTIALS) This class was introduced by Abate, Choudhury & Whitt [1] as the class of distributions of r.v.'s of the form YX, where Y is Pareto distributed with a = (p - 1)/p, A = 1 and X is standard exponential. The simplest examples correspond to p small and integer-valued; in particular, the density is

{

3 (1 - (1 + 2x + 2x2)e-2x) p = 2 (2.15) 3 (1 - (1 + Zx + $x2 + 16x3 ) a-3x/2)

p = 3.

In general, B(x) = O(x-P). The motivation for this class is the fact that the Laplace transform is explicit (which is not the case for the Pareto or other standard heavy-tailed distributions); in particular,

{

s ()

(2.16)

1-s+3s2-9s3log(1+2s I p=3.

11 Example 2.12 (DISTRIBUTIONS WITH REGULARLY VARYING TAILS) The tail B(x) of a distribution B is said to be regularly varying with exponent a if

B(x) - L(

x ), x

-+ 00,

(2.17)

where L (x) is slowly varying, i.e. satisfies L(xt)/L(x) -4 1, x -4 oo (any L having a limit in (0, oo) is slowly varying ; another standard example is (log x)'). Thus, examples of distributions with regularly varying tails are the Pareto distribution (2.12) (here L (x) -* 1) and ( 2.13), the loggamma distribution (with exponent 5) and a Pareto mixture of exponentials. ❑

11

3. THE ARRIVAL PROCESS

Example 2.13 (THE SUBEXPONENTIAL CLASS OF DISTRIBUTIONS) We say that a distribution B is subexponential if

lim

B

x-roo

`2^ = 2. (2.18) B(x)

It can be proved (see IX.1) that any distribution with a regularly varying tail is subexponential. Also, for example the lognormal distribution is subexponential (but not regularly varying), though the proof of this is non-trivial, and so is the Weibull distribution with 0 < r < 1. Thus, the subexponential class of distributions provide a convenient framework for studying large classes of heavy❑ tailed distributions. We return to a closer study in IX.1. When studying ruin probabilities, it will be seen that we obtain completely different results depending on whether the claim size distribution is exponentially bounded or heavy-tailed. From a practical point of view, this phenomenon represents one of the true controversies of the area. Namely, the knowledge of the claim size distribution will typically be based upon statistical data, and based upon such information it seems questionable to extrapolate to tail behaviour. However, one may argue that this difficulty is not resticted to ruin probability theory alone. Similar discussion applies to the distribution of the accumulated claims (XI.4) or even to completely different applied probability areas like extreme value theory: if we are using a Gaussian process to predict extreme value behaviour, we may know that such a process (with a covariance function estimated from data) is a reasonable description of the behaviour of the system under study in typical conditions, but can never be sure whether this is also so for atypical levels for which far less detailed statistical information is available. We give some discussion on standard methods to distinguish between light and heavy tails in Section 4f.

3 The arrival process For the purpose of modeling a risk process , the claim size distribution represents of course only one aspect (though a major one). At least as important is the specification of the structure of the point process {Nt } of claim arrivals and its possible dependence with the claims. By far the most prominent case is the compound Poisson (Cramer-Lundberg) model where {Nt} is Poisson and independent of the claim sizes U1, U2,.... The reason is in part mathematical since this model is the easiest to analyze, but the model also admits a natural interpretation : a large portfolio of insurance holders , which each have a ( time-homogeneous) small rate of experiencing a

12


claim , gives rise to an arrival process which is very close to a Poisson process, in just the same way as the Poisson process arises in telephone traffic (a large number of subscribers each calling with a small rate), radioactive decay (a huge number of atoms each splitting with a tiny rate ) and many other applications. The compound Poisson model is studied in detail in Chapters III, IV (and, with the extension to premiums depending on the reserve, in Chapter VII). To the author 's knowledge , not many detailed studies of the goodness-of-fit of the Poisson model in insurance are available . Some of them have concentrated on the marginal distribution of NT (say T = one year ), found the Poisson distribution to be inadequate and suggested various other univariate distributions as alternatives , e.g. the negative binomial distribution. The difficulty in such an approach lies in that it may be difficult or even impossible to imbed such a distribution into the continuous set-up of {Nt } evolving over time , and also that the ruin problem may be hard to analyze . Nevertheless , getting away from the simple Poisson process seems a crucial step in making the model more realistic, in particular to allow for certain inhomogeneities. Historically, the first extension to be studied in detail was {Nt } to be renewal (the interarrival times T1 , T2.... are i.i.d. but with a general not necessarily exponential distribution ). This model , to be studied in Chapter V, has some mathematically appealing random walk features , which facilitate the analysis. However , it is more questionable whether it provides a model with a similar intuitive content as the Poisson model. A more appealing way to allow for inhomogeneity is by means of an intensity ,3(t) fluctuating over time. An obvious example is 3(t) depending on the time of the year (the season), so that ,8 (t) is a periodic function of t; we study this case in VI .6. Another one is Cox processes, where {/3 (t)}too is an arbitrary stochastic process . In order to prove reasonably substantial and interesting results , Cox processes are, however, too general and one neeed to specialize to more concrete assumptions . The one we focus on (Chapter VI) is a Markovian environment : the environmental conditions are described by a finite Markov process {Jt }too, such that 8(t) = ,(3; when Jt = i. I.e., with a common term {Nt} is a Markov-modulated Poisson process ; its basic feature is to allow more variation (bursty arrivals ) than inherent in the simple Poisson process. This model can be intuitively understood in some simple cases like { Jt} describing weather conditions in car insurance , epidemics in life insurance etc. In others, it may be used in a purely descriptive way when it is empirically observed that the claim arrivals are more bursty than allowed for by the simple Poisson process. Mathematically, the periodic and the Markov -modulated models also have attractive features . The point of view we take here is Markov -dependent random walks in continuous time (Markov additive processes ), see 11. 5. This applies also to the case where the claim size distribution depends on the time of the year or

4. A SUMMARY OF MAIN RESULTS AND METHODS

13

the environment (VI.6) , and which seems well motivated from a practical point of view as well.

4 A summary of main results and methods 4a Duality with other applied probability models Risk theory may be viewed as one of many applied probability areas, others being branching processes, genetics models, queueing theory, dam/storage processes, reliability, interacting particle systems, stochastic differential equations, time series and Gaussian processes, extreme value theory, stochastic geometry, point processes and so on. Some of these have a certain resemblance in flavour and methodology, others are quite different. The ones which appear most related to risk theory are queueing theory and dam/storage processes. In fact, it is a recurrent theme of this book to stress this connection which is often neglected in the specialized literature on risk theory. Mathematically, the classical result is that the ruin probabilities for the compound Poisson model are related to the workload (virtual waiting time) process {Vt}too of an initially empty M/G/1 queue by means of ,0 (u,T) = P(VT > u), 0(u) = P(V > u),

(4.1)

where V is the limit in distribution of Vt as t -+ oo. The M/G/1 workload process { Vt } may also be seen as one of the simplest storage models, with Poisson arrivals and constant release rule p(x) = 1. A general release rule p(x) means that {Vt} decreases according to the differential equation V = -p(V) in between jumps, and here (4.1) holds as well provided the risk process has a premium rule depending on the reserve, R = p(R) in between jumps. Similarly, ruin probabilities for risk processes with an input process which is renewal, Markovmodulated or periodic can be related to queues with similar characteristics. Thus, it is desirable to have a set of formulas like (4.1) permitting to translate freely between risk theory and the queueing/storage setting. More generally, methods or modeling ideas developed in one area often has relevance for the other one as well. A stochastic process {Vt } is said to be in the steady state if it is strictly stationary (in the Markov case, this amounts to Vo having the stationary distribution of {Vt}), and the limit t -4 oo is the steady-state limit. The study of the steady state is by far the most dominant topic of queueing and storage theory, and a lot of information on steady-state r.v.'s like V is available. It should be noted, however, that quite often the emphasis is on computing expected values like EV. In the setting of (4.1), this gives only f0 O°i (u)du which is of limited

14 CHAPTER I. INTRODUCTION intrinsic interest . Similarly, much of the study of finite horizon problems (often referred to as transient behaviour) in queueing theory deals with busy period analysis which has no interpretation in risk theory at all. Thus , the two areas, though overlapping, have to some extent a different flavour. A prototype of the duality results in this book is Theorem 11.3.1 , which gives a sample path version of (4.1) in the setting of a general premium rule p(x): the events {VT > u} and {r (u) < T} coincide when the risk process and the storage process are coupled in a suitable way (via time-reversion ). The infinite horizon (steady state ) case is covered by letting T oo. The fact that Theorem H.3.1 is a sample path relation should be stressed : in this way the approach also applies to models having supplementary r.v.'s like the environmental process {Jt} in a Markov-modulated setting.

4b Exact solutions Of course , the ideal is to be able to come up with closed form solutions for the ruin probabilities 0(u), Vi(u,T). The cases where this is possible are basically the following for the infinite horizon ruin probability 0(u): • The compound Poisson model with constant premium rate p = 1 and exponential claim size distribution B, B(x) = e-bx. Here O(u) = pe-ryu where 3 is the arrival intensity, p = 0/8 and -y = 8 -,3. • The compound Poisson model with constant premium rate p = 1 and B being phase-type with a just few phases . Here Vi(u) is given in terms of a matrix-exponential function ( Corollary VIII. 3.1), which can be expanded into a sum of exponential terms by diagonalization (see, e .g., Example VIII. 3.2). The qualifier 'with just a few phases ' refers to the fact that the diagonalization has to be carried out numerically in higher dimensions. • The compound Poisson model with a claim size distribution degenerate at one point, see Corollary III.3.6. • The compound Poisson model with some rather special heavy-tailed claim size distributions, see Boxma & Cohen [74] and Abate & Whitt [3]. • The compound Poisson model with premium rate p(x) depending on the reserve and exponential claim size distribution B. Here ?P(u) is explicit provided that , as is typically the case, the functions w x f d 1 exdx () - p(y) y^ Jo p(x) can be written in closed form, see Corollary VII.1.8.

4. A SUMMARY OF MAIN RESULTS AND METHODS

15

• The compound Poisson model with a two -step premium rule p(x) and B being phase-type with just a few phases, see VIII.7. • An a-stable Levy process with drift , where Furrer [150] recently computed ii(u) as an infinite series involving the Mittag- Lef$er function. Also Brownian models or certain skip -free random walks lead to explicit solutions (see XI . 1), but are somewhat out of the mainstream of the area . A notable fact ( see again XI.1) is the explicit form of the ruin probability when {Rt} is a diffusion with infinitesimal drift and variance µ(x), a2 (x): Ip (u) =

f °O exp {- ff 2µ(y)/a2(y) dy} dx - S(u) 1S(oo) f °D exp {- f f 2µ(y)/a2(y) dy} dx -

(4.2)

where S(u) =

f {eXp

LX 2,u(y)/a2(y) dy}

U

is the natural scale. For the finite horizon ruin probability 0(u, T), the only example of something like an explicit expression is the compound Poisson model with constant premium rate p = 1 and exponential claim size distribution . However, the formulas ( IV.1) are so complicated that they should rather be viewed as basis for numerical methods than as closed-form solutions.

4c Numerical methods Next to a closed-form solution, the second best alternative is a numerical procedure which allows to calculate the exact values of the ruin probabilities. Here are some of the main approaches: Laplace transform inversion Often, it is easier to find the Laplace transforms =

e8 f

,b(u)du ,

esu-Tb(

[-s,

u, T) du dT

0 TO 00

in closed form than the ruin probabilities z/'(u), (u, T) themselves. Given this can be done, Ab(u), O(u, T) can then be calculated numerically by some method for transform inversion, say the fast Fourier transform (FFT) as implemented in Grubel [179] for infinite horizon ruin probabilities for the renewal model. We don't discuss Laplace transform inversion much; relevant references are Grubel [179], Abate & Whitt [2], Embrechts, Grubel & Pitts [132] and Grubel & Hermesmeier [180] (see also the Bibliographical Notes in [307] p. 191).

CHAPTER L INTRODUCTION

16

Matrix-analytic methods This approach is relevant when the claim size distribution is of phase-type (or matrix-exponential), and in quite a few cases (Chapter VIII), 0(u) is then given in terms of a matrix-exponential function euu (here U is some suitable matrix) which can be computed by diagonalization, as the solution of linear differential equations or by some series expansion (not necessarily the straightforward Eo U'u/n! one!). In the compound Poisson model with p = 1, U is explicit in terms of the model parameters, whereas for the renewal arrival model and the Markovian environment model U has to be calculated numerically, either as the iterative solution of a fixpoint problem or by finding the diagonal form in terms of the complex roots to certain transcendental equations. Differential- and integral equations The idea is here to express 'O(u) or '(u, T) as the solution to a differential- or integral equation, and carry out the solution by some standard numerical method. One example where this is feasible is the renewal equation for tl'(u) (Corollary III.3.3) in the compound Poisson model which is an integral equation of Volterra type. However, most often it is more difficult to come up with reasonably simple equations than one may believe at a first sight, and in particular the naive idea of conditioning upon process behaviour in [0, dt] most often leads to equations involving both differential and integral terms. An example where this idea can be carried through by means of a suitable choice of supplementary variables is the case of state-dependent premium p(x) and phase-type claims, see VIII.7.

4d Approximations The Cramdr-Lundberg approximation This is one of the most celebrated result of risk theory (and probability theory as a whole). For the compound Poisson model with p = 1 and claim size distribution B with moment generating function (m.g.f.) B[s], it states that i/i(u) - Ce-"u,

u -* oo, (4.3)

where C = (1 - p)/(13B'[ry] - 1) and -y > 0 is the solution of the Lundberg equation (4.4) 00['Y]-1)-'Y = 0,

which can equivalently be written as

f3 [7] = 1 +13 .

4. A SUMMARY OF MAIN RESULTS AND METHODS 17 It is rather standard to call ry the adjustment coefficient but a variety of other terms are also frequently encountered. The Cramer-Lundberg approximation is renowned not only for its mathematical beauty but also for being very precise, often for all u > 0 and not just for large u. It has generalizations to the models with renewal arrivals, a Markovian environment or periodically varying parameters. However, in such cases the evaluation of C is more cumbersome. In fact, when the claim size distribution is of phase-type, the exact solution is as easy to compute as the Cramer-Lundberg approximation at least in the first two of these three models. Diffusion approximations Here the idea is simply to approximate the risk process by a Brownian motion (or a more general diffusion) by fitting the first and second moment, and use the fact that first passage probabilities are more readily calculated for diffusions than for the risk process itself. Diffusion approximations are easy to calculate, but typically not very precise in their first naive implementation. However, incorporating correction terms may change the picture dramatically. In particular, corrected diffusion approximations (see IV.6) are by far the best one can do in terms of finite horizon ruin probabilities '(u, T). Large claims approximations In order for the Cramer-Lundberg approximation to be valid, the claim size distribution should have an exponentially decreasing tail B(x). In the case of heavy-tailed distributions, other approaches are thus required. Approximations for O(u) as well as for 1(u, T) for large u are available in most of the models we discuss. For example, for the compound Poisson model ^(u) p

J

B dx, u -> oo. (4.6)

pu In fact , in some cases the results are even more complete than for light tails. See Chapter IX. This list of approximations does by no means exhaust the topic; some further possibilities are surveyed in 111 .7 and IV.2.

4e Bounds and inequalities The outstanding result in the area is Lundberg's inequality (u) < e-"lu.


18

Compared to the Cramer-Lundberg approximation (4.3), it has the advantage of not involving approximations and also, as a general rule, of being somewhat easier to generalize beyond the compound Poisson setting. We return to various extensions and sharpenings of Lundberg's inequality (finite horizon versions, lower bounds etc.) at various places and in various settings. When comparing different risk models, it is a general principle that adding random variation to a model increases the risk. For example, one expects a model with a deterministic claim size distribution B, say degenerate at m, to have smaller ruin probabilities than when B is non-degenerate with the same mean m. This is proved for the compound Poisson model in 111.8. However, empirical evidence shows that the general principle holds in a broad variety of settings, though not too many precise mathematical results have been obtained.

4f Statistical methods Any of the approaches and results above assume that the parameters of the model are completely known. In practice, they have however to be estimated from data, obtained say by observing the risk process in [0, T]. This procedure in itself is fairly straightforward; e.g., in the compound Poisson model, it splits up into the estimation of the Poisson intensity (the estimator is /l3 = NT/T) and of the parameter(s) of the claim size distribution, which is a standard statistical problem since the claim sizes Ui, ... , UNT are i.i.d. given NT. However, the difficulty comes in when drawing inference about the ruin probabilities. How do we produce a confidence interval? And, more importantly, can we trust the confidence intervals for the large values of u which are of interest? In the present author's opinion, this is extrapolation from data due to the extreme sensitivity of the ruin probabilities to the tail of the claim size distribution in particular (in contrast, fitting a parametric model to U1, . . . , UNT may be viewed as an interpolation in or smoothing of the histogram). For example, one may question whether it is possible to distinguish between claim size distributions which are heavy-tailed or have an exponentially decaying tail. The standard suggestion is to observe that the mean residual life

E[U - x U > x] = B(x)

f '(y-x)B(dx)

typically has a finite limit (possibly 0) in the light-tailed case and goes to oo in the heavy-tailed case, and to plot the empirical mean residual life 1

N - k (U(`) - U(k)) i =k+ i

5. CONVENTIONS

19

as function of U(k), where U(1) < ... < U(N) are the order statistics based upon N i.i.d. claims U1, ... , UN, to observe whether one or the other limiting behaviour is apparent in the tail. See further Embrechts, Klnppelberg & Mikosch [134].

4g Simulation The development of modern computers have made simulation a popular experimental tool in all branches of applied probability and statistics, and of course the method is relevant in risk theory as well. Simulation may be used just to get some vague insight in the process under study: simulate one or several sample paths, and look at them to see whether they exhibit the expected behaviour or some surprises come up. However, the more typical situation is to perform a Monte Carlo experiment to estimate probabilities (or expectations or distributions) which are not analytically available. For example, this is a straightforward way to estimate finite horizon ruin probabilities. The infinite horizon case presents a difficulty, because it appears to require an infinitely long simulation. Truncation to a finite horizon has been used, but is not very satisfying. Still, good methods exist in a number of models and are based upon representing the ruin probability zb(u) as expected value of a r.v. (or a functional of the expectation of a set of r.v's) which can be generated by simulation. The problem is entirely analogous to estimating steady-state characteristics by simulation in queueing/storage theory, and in fact methods from that area can often be used in risk theory as well . We look at a variety of such methods in Chapter X, and also discuss how to develop methods which are efficient in terms of producing a small variance for a fixed simulation budget. A main problem is that ruin is typically a rare event (i.e., having small probability) and that therefore naive simulation is expensive or even infeasible in terms of computer time.

5 Conventions Numbering and reference system The basic principles are just as in the author's earlier book Applied Probability and Queues (Wiley 1987; reference [14], in this book referred to as [APQ]). The chapter number is specified only when it is not the current one. Thus Proposition 4.2, formula (5.3) or Section 3 of Chapter VI are referred to as Proposition VI.4.2, formula VI.(5.3) and Section VI.3 (or just VI.3), respectively, in all other chapters than VI where we just write

20

CHAPTER L INTRODUCTION Proposition 4.2, formula (5.3) or Section 3. References like Proposition A.4, (A.29) refer to the Appendix.

Abbreviations

c.d.f. cumulative distribution function P(X < x) c.g.f. cumulant generating function, i.e. log E[s] where b[s] is the m.g.f. i.i.d. independent identically distributed

i.o. infinitely often l.h.s. left hand side (of equation) m.g.f. moment generating function, see under b[s] below. r.h.s. right hand side (of equation) r.v. random variable s.c.v. squared coefficient of variation, EX2/(EX)2. w.r.t. with respect to w.p. with probability Mathematical notation P probability. E expectation.

- Used in asymptotic relations to indicate that the ratio between two expressions is 1 in the limit. E.g. n!

27r nn+1/2e-n, n -i oo.

A different type of asymptotics: less precise, say a heuristic approxi1 + h + h2/2, h -+ 0. mation, or a more precise one like eh - The same symbol B is used for a probability measure B(dx) = P(X E dx) and its c.d.f. B(x) = P(X < x) = fx. B(dy). B[s] the m.g.f. (moment generating function) fm e82B(dx) of the distribution B. If, as for typical claim size distributions, B is concentrated on [0, oo), b[s] is defined always if Rs < 0 and sometimes in a larger strip (for example, if B(x) - ce-ax, then for 1s < 5). The Laplace transform is b[-s].

B(x) the tail 1 - B(x) = P(X > x) of B. IIGII the total mass (variation ) of a (signed ) measure G . In particular, for a probability distribution IIGII = 1, and for a defective probability distribution IIGII < 1.

5. CONVENTIONS

21

{6B the mean EX = f xB(dx) of B ABA' the nth moment EXn = f x"B(dx) of B. I(A) the indicator function of the event A, E[X;A] means E[XI(A)]. R(s) the real part of a complex number s. 0 marks the end of a proof, an example or a remark. Xt_ the left limit limstt X8f i.e. the value just before t. D [0, oo) the space of R-valued functions which are right-contionuous and have left limits. Unless otherwise stated, all stochastic processes considered in this book are assumed to have sample paths in this space. Usually, the processes we consider are piecewise continuous, i.e. only have finitely many jumps in each finite interval. Then the assumption of D-paths just means that we use the convention that the value at each jump epoch is the right limit rather than the left limit. In the French-inspired literature, often the term 'cadlag' (continues a droite avec limites a gauche) is used for the D-property. N(it, a2) the normal distribution with mean p and variance oa2. Matrices and vectors are denoted by bold letters. Usually, matrices have uppercase Roman or Greek letters like T, A, row vectors have lowercase Greek letters like a, 7r, and column vectors have lowercase Roman letters like t, a. In particular: I is the identity matrix e is the column vector with all entries equal to 1 ei is the ith unit column vector, i.e. the ith entry is 1 and all other 0. Thus, the ith unit row vector is e'i. (the dimension is usually clear from the context and left unspecified in the notation). F o r a given set x1, ... , xa, of numbers, (xi)diag denotes the diagonal matrix with the xi on the diagonal (xi)row denotes the row vector with the xi as components (xi),oi denotes the column vector with the xi as components Special notation for risk processes /3 the arrival intensity (when the arrival process is Poisson). Notation like f3i and 3(t) in Chapter VI has a similar , though slightly more complicated, intensity interpretation.

22

CHAPTER L INTRODUCTION B the claim size distribution. Notation like BE and B(t) in Chapter VI has a similar, though slightly more complicated, interpretation.

J the rate parameter of B for the exponential case B(x) = e-by. p the net amount /3pB of claims per unit time, or quantities with a similar time average interpretation, cf. I.1. 'q the safety loading , cf. I.1. ry The adjustment coefficient. FL, EL the probability measure and its corresponding expectation corresponding to the exponential change of measure given by Lundberg conjugation, cf. e.g. 111.5, VI.5.

Chapter II

Some general tools and results The present chapter collects and surveys some topics which repeatedly show up in the study of ruin probabilities. Due to the generality of the theory, the level of the exposition is, however, somewhat more advanced than in the rest of the book. The reader should therefore observe that it is possible to skip most of the chapter, in particular at a first reading of the book. More precisely, the relevance for the mainstream of exposition is the following: The martingale approach in Section 1 is essentially only used here. All results are proved elsewhere , in most cases via likelihood ratio arguments. The likelihood ratio approach in Section 2 is basic for most of the models under study. When encountered for the first time in connection with the compound Poisson model in Chapter III, a parallel self-contained treatment is given of the facts needed there. The general theory is, however, used in Chapter VI on risk processes in a Markovian (or periodic) environment. The duality results in Section 3 (and, in part, Sections 4, 5) are, strictly speaking, not crucial for the rest of the book. The topic is, however, fundamental ( at least in the author' s opinion) and the probability involved is rather simple and intuitive. Sections 4, 5 on random walks and Markov additive processes can be skipped until reading Chapter VI on the Markovian environment model.

23

24 CHAPTER II. SOME GENERAL TOOLS AND RESULTS The ladder height formula in Theorem 6.1 is basic for the study of the compound Poisson model in Chapter III. The more general Theorem 6.5 can be skipped.

1 Martingales We consider the claim surplus process {St} of a general risk process. As usual, the time to ruin r(u) is inf It > 0 : St > u}, and the ruin probabilities are

ip(u) = P (T(u) < oo), V) (u, T) = P(T(u) < T). Our first result is a representation formula for O(u) obtained by using the martingale optional stopping theorem . Let e(u) = ST(u) - u denote the overshoot. Proposition 1.1 Assume that (a) for some ry > 0, {e'YS° }t>0 is a martingale, (b) St a$ -oo on {T(u) = oo}. Then e-7u (u) = E[e74(u)j7-(u) < oo]

Proof We shall use optional stopping at time r(u)AT (we cannot use the stopping time T(u) directly because P(T(u) = oo) > 0 and also because the conditions of the optional stopping time theorem present a problem; however, using r(u) A T invokes no problems because r(u) A T is bounded by T). We get

1 = Ee7So = E e'Y S-(,.)AT = E [e7ST(°); T(u) < T] + E [eryST ; T(u) > T] .

(1.2)

As T -> oo, the second term converges to 0 by (b) and dominated convergence (e7ST < eryu on {r(u) > T}), and in the limit (1.2) takes the form 1 = E [e'ys-(-); T(u) < oo] + 0 = eryuE [e7Vu);T(u) < cc] = e7uE {e7f(u) I T(u) < cc] z/,(u).

Example 1 .2 Consider the compound Poisson model with Poisson arrival rate ,0, claim size distribution B and p = ,QµB < 1. Thus N,

StUi-t, f-1

1. MARTINGALES 25 where {Nt} is a Poisson process with rate ,Q and the U; are i.i.d. with common distribution B (and independent of {Nt}). A simple calculation (see Proposition III.1.1) shows that Eels- = e"(') where K(a) = ,Q(B[a] - 1) - a. From this it is readily seen (see III.6a for details) that typically a solution to the Lundberg equation K(y) = 0 exists, and thus Ee'rs° = 1. Since {St} has stationary independent increments, it follows that E [e7st+v I J] = e"rstE [e7(st+v-St) I Ft] = e7StEe"rs° = elst where .Ft = a(S" : v < t). Thus, condition (a) of Proposition 1.1 is satisfied, and (b) follows from p < 1 and the law of large numbers (see Proposition III.1.2(c)).

Example 1 .3 Assume that {Rt} is Brownian motion with variance constant o.2 and drift p > 0. Then {St } is Brownian motion with variance constant o2 and drift -p < 0. By standard formulas for the m.g.f. of the normal distribution, Eeas° = e"(°) where n(a) = a2a2/2 - ap. From this it is immediately seen that the solution to the Lundberg equation ic(y) = 0 is -y = 2p/a2, and thus Ee7s° = 1. Since {St} has stationary independent increments, the martingale property now follows just as in Example 1.2. Thus, the conditions of Proposition 1.1 are satisfied. ❑ Corollary 1.4 (LUNDBERG ' S INEQUALITY ) tion 1 . 1, O(u ) < e-7".

Under the conditions of Proposi-

❑

Proof Just note that C(u) > 0.

Corollary 1.5 For the compound Poisson model with B exponential, B(x) _ e-dx, and p =,3/6 < 1, the ruin probability is O(u) = pe- r" where -y = S - /3.

Proof Since c(a) = /3 (B[a] - 1) - a = -a - a it is immediately seen that y = S - ,Q. Now at the time r(u) of ruin {St} upcrosses level u by making a jump . The available information on this jump is that the distribution given r(u) = t and S,-(„)_ = x is that of a claim size U given U > u - x, and thus by the memoryless property of the exponential distribution , the conditional distribution of the overshoot e(u) = U - u + x is again just exponential with rate S. Thus 00

e5e -

E [e'rt (") I T(u) < oo] = I

dx = f

5edx

26 CHAPTER IL SOME GENERAL TOOLS AND RESULTS Corollary 1.6 If {Rt} is Brownian motion with variance constant a2 and drift p > 0, then z/'(u) = e-7" where 'y = 21A/a2. Proof Just note that ^(u) = 0 by continuity of Brownian motion.

❑

Notes and references The first use of martingales in risk theory is due to Gerber [156], and is further exploited in his book [157]. More recent references are Dassios & Embrechts [98], Grandell [171], [172], Embrechts, Grandell & Schmidli [131], Delbaen & Haezendonck [103] and Schmidli [320].

2 Likelihood ratios and change of measure We consider stochastic processes {Xt} with a Polish state space E and paths in the Skorohod space DE = DE[0, oo), which we equip with the natural filtration {.Ft}too and the Borel a-field F. Two such processes may be represented by probability measures F, P on (DE, F), and in analogy with the theory of measures on finite dimensional spaces one could study conditions for the RadonNikodym derivative dP/dP to exist. However, as shown by the following example this set-up is too restrictive: typically', the parameters of the two processes can be reconstructed from a single infinite path, and F, P are then singular (concentrated on two disjoint measurable sets). Example 2 .1 Let F, P correspond to the claim surplus process of two compound Poisson risk processes with Poisson rates /3, 0 and claim size distributions B, B. The number Nt F) of jumps > e before time t is a (measurable) r.v. on (DE,F), hence so is Nt = limfyo N2ì. Thus the sets S = I tlim

N

-+oot t =,6

S = { lim Nt I t +00 t

gJ

are both in F. But if a $ ^ , then S and S are disjoint , and by the law of large numbers for the Poisson process , F(S) = P(S) = 1. A somewhat similar ❑ argument gives singularity when B $ B. The interesting concept is therefore to look for absolute continuity only on finite time intervals (possibly random, cf. Theorem 2.3 below). I.e., we look for a process {Lt} (the likelihood ratio process) such that

P(A) = E[Lt; A], A E Ft,

(2.1)

'though not always: it is not difficult to construct a counterexample say in terms of transient Markov processes.

2. LIKELIHOOD RATIOS AND CHANGE OF MEASURE 27 (i.e, that the restriction of P to (DE,.Tt) is absolutely continuous w.r.t. the restriction of P to (DE, .Pt)) The following result gives the connection to martingales. Proposition 2.2 Let {Ft}t>o be the natural filtration on DE, F the Borel o•field and P a given probability measure on (DE,F). (i) If {Lt}t> o is a non-negative martingale w.r.t. ({Ft} , F) such that ELt = 1, then there exists a unique probability measure Pon .F such that (2.1) holds. (ii) Conversely, if for some probability measure P and some {.Pt}-adapted process {Lt}t>o (2.1) holds, then {Lt} is a non-negative martingale w.r.t. ({.Ft}, P) such that LLt = 1. Proof Under the assumptions of (i), define P by Pt (A) = E[Lt; A], A E F. Then Lt > 0 and ELt = 1 ensure that Pt is a probability measure on (DE, Ft). Lets < t, A E F8. Then

Ft (A) = E[Lt;A] = EE[LtI(A)IF8] = EI(A)E[LtIFB] = EI(A)L8 = PS(A), using the martingale property in the fourth step. Hence the family {Pt} is t>o consistent and hence extendable to a probability measure F on (DE,Y) such that P(A) = Pt(A), A E Ft . This proves (i). Conversely, under the assumptions of (ii) we have for A E rg and s < t that A E Ft as well and hence E[L8; A] = E[Lt; A]. The truth of this for all A E Y. implies that E[LtI.F8] = L8 and the martingale property. Finally, ELt = 1 follows by taking A = DE in (2.1) and non-negativity by letting A = {Lt < 0}. Then P(A) = E[Lt; Lt < 0] can only be non-negative if P(A) = 0. ❑ The following likelihood ratio identity (typically with r being the time r(u) to ruin) is a fundamental tool throughout the book: Theorem 2 .3 Let {Lt}, P be as in Proposition 2.2(i). If r is a stopping time and G E PT, G C {T < oo}, then { 1

P(G) = EG _; 1 .

J

(2.2)

Proof Assume first G C {T < T} for some fixed deterministic T < oo. By the martingale property, we have E [ LTIFT]1 = LT on {T < T}. Hence

E

[_ ; G

]

= E [LT ; G

l

]

= E [_I(G)E[LTIFT] ]

= E { _I(G)Lr

]

= P(G).

28

CHAPTER II. SOME GENERAL TOOLS AND RESULTS

In the general case , applying (2.3) to G of{r < T} we get 1111

F(Gn {r rt°) for all t when u > v.

32

CHAPTER IL SOME GENERAL TOOLS AND RESULTS

Suppose first VT > u (this situation corresponds to the solid path of {Rt} in Fig. 3.1 with Ro = u = ul). Then Vo, = r(VT) - U1 > roil - U1 = Rol. If VaN > 0, we can repeat the argument and get VoN_1 > Ra2 and so on. Hence if n satisfies VVN_n+1 = 0 (such a n exists, if nothing else n = N), we have RQ„ < 0 so that indeed r(u) < T.

Suppose next VT < u (this situation corresponds to the broken path of {Rt} in Fig. 3.1 with Ro = u = u2). Then similarly VVN = r0,T l - Ul

< roil -

Ul

=

RQ„

Va1V_1

0 for all n < N, and since ruin can only occur at the times of claims, we have r(u) > T. ❑ A basic example is when {Rt} is the risk reserve process corresponding to claims arriving at Poisson rate ,3 and being i.i.d. with distribution B, and a general premium rule p(r) when the reserve is r. Then the time reversibility of the Poisson process ensures that {At } and {At } have the same distribution (for finite-dimensional distributions, the distinction between right- and left continuity is immaterial because the probability of a Poisson arrival at any fixed time t is zero). Thus we may think of {Vt} as having compound Poisson input and being defined for all t < oo. Historically, this represents a model for storage, say of water in a dam though other interpretations like the amount of goods stored are also possible. The arrival epochs correspond to rainfalls, and in between rainfalls water is released at rate p(r) when Vt (the content) is r. We get: Corollary 3.2 Consider the compound Poisson risk model with a general premium rule p(r). Then the storage process {Vt} has a proper limit in distribution, say V, if and only if O(u) < 1 for all u, and then '0 (u) = P(V > u). Proof Let T -^ oo in (3.3).

❑

Notes and references Some main reference on storage processes are Harrison & Resnick [187] and Brockwell, Resnick & Tweedie [79]. Theorem 3.1 and its proof is from Asmussen & Schock Petersen [50], Corollary 3.2 from Harrison & Resnick [188]. The results can be viewed as special cases of Siegmund duality, see Siegmund [344]. Some further relevant more general references are Asmussen [21] and Asmussen & Sigman [51]. Historically, the connection between risk theory and other applied probability areas appears first to have been noted by Prabhu [293] in a queueing context. Nevertheless, one may feel that the interaction between the different areas has been surprisingly limited even up to today.

4. RANDOM WALKS IN DISCRETE OR CONTINUOUS TIME

33

4 Random walks in discrete or continuous time A random walk in discrete time is defined as X, = Xo + Y1 + • • • + Y, where the Yi are i . i.d., with common distribution F (say). Here F is a general probability distribution on R (the special case of F being concentrated on {-1, 1} is often referred to as simple random walk or Bernoulli random walk). Most often, Xo =

0. For discrete time random walks , there is an analogue of Theorem 3.1 in terms of Lindley processes . For a given i.i.d. R -valued sequence Z1, Z2 ,..., the Lindley process Wo, W1, W2,... generated by Z1, Z2, ... is defined by assigning Wo some arbitrary value > 0 and letting Wn+1 = (Wn + Zn+1)+•

(4.1)

Thus {Wn}n=o,1,... evolves as a random walk with increments Z1i Z2, ... as long as the random walk only takes non-negative values, and is reset to 0 once the r.w. hits (-oo, 0). I.e., {Wn}n=0,1,.., can be viewed as the reflected version of the random walk with increments Z1, Z2,.... In particular, if Wo = 0 then (Z1+•••+Zn) WN = Zl+•••+ZN- min n=0,1,...,N

(4.2)

(for a rigorous proof, just verify that the r .h.s. of (4.2) satisfies the same recursion as in (4.1)). Theorem 4.1 Let r(u) = inf In: u + Y1 + • • • + Yn < 0}. Let further N be fixed and let Wo, W1, . . . , WN be the Lindley process generated by Z1 = -YN, Z2 = -YN_1 i ..., ZN = - Y1 according to Wo = 0. Then the events {r(u) < N} and {WN > u} coincide.

Proof By (4.2), WN = -YN - ... - Yl -

min (-YN - ... - YN-n+1) n=0,1,..., N

min (Y1 + • • • + YN-n) n=0,1,...,N

min (Y1 + + Yn). n=0,1,..., N

From this the result immediately follows. 0 Corollary 4.2 The following assertions are equivalent: (a) 0(u) = P(r(u) < oo) < 1 for all u > 0; (b) 1/i(u) = P(•r(u) < oo) -> 0 as u -* oo; (c) The Lindley process {WN} generated by Zl = -Y1, Z2 = -Y2, ... has a proper limit W in distribution as n -+ oo;

34 CHAPTER II. SOME GENERAL TOOLS AND RESULTS (d) m = inf.=o,l.... (Yi + • • • + Yn) > -oo a.s.; (e)Yi+•••+Yn -74 - ooa.s. In that case , W v -m and P(W > u) = P (-m > u) = 0(u). Proof Since (YN,... , Y1) has the same distribution as (Y1, . .. , YN), the Lindley processes in Corollary 4.2 and Theorem 4.1 have the same distribution for n = 0,1, ... , N. Thus the assertion of Theorem 4.1 is equivalent to WN D MN =

(Z1 + ... + Z.) sup n=0,1,...,N

so that WN _P4 M = supra=0,1,... (Z1 + • • • + Zn) = -m and P(W > u) = P(M > u) = i (u ). By Kolmogorov's 0-1 law, either M = oo a.s . or M < oo a.s. Combining these facts gives easily the equivalence of (a)-(d). Clearly, (d) #. (e). The converse follows from general random walk theory since it is standard that lim sup (Y1 + • • + Yn) = oo when Y1 + • • • + Yn 74 -oo. 0

By the law of large numbers, a sufficient condition for (e) is that EY is welldefined and > 0. In general, the condition 00 F(YI+•••+ Yn 0). ❑

5 Markov additive processes A Markov additive processes, abbreviated as MAP in this section2, is defined as a bivariate Markov process {Xt} = {(Jt, St)} where {Jt} is a Markov process with state space E (say) and the increments of {St} are governed by {Jt} in the sense that E [f (St+8 - St)g(Jt+s)I.Ft] = Ejt,o[f (S8)g(J8)].

(5.1)

For shorthand , we write Pi, Ei instead of P2,0, Ei,0 in the following. As for processes with stationary independent increments , the structure of MAP's is completely understood when E is finite: 2and only there ; one reason is that in parts of the applied probability literature, MAP stands for the Markovian arrival process discussed below.

40 CHAPTER H. SOME GENERAL TOOLS AND RESULTS In discrete time, a MAP is specified by the measure-valued matrix (kernel) F(dx) whose ijth element is the defective probability distribution Fij(dx) = Pi,o(Ji = j, Y1 E dx) where Y„ = S„ - Sr_1. An alternative description is in terms of the transition matrix P = (piA,jEE (here pij = Pi(J1 = j)) and the probability measures Hij(dx)=P(Y1 EdxlJo=i,J1=j)=

Fij (dx) Pij

In simulation language, this means that the MAP can be simulated by first simulating the Markov chain {J„} and next the Y1, Y2, ... by generating Yn according to Hij when J„_1 = i, Jn = j. If all Fij are concentrated on (0, oo), a MAP is the same as a semi-Markov or Markov renewal process, with the Y„ being interpreted as interarrival times. In continuous time (assuming D-paths), {Jt} is specified by its intensity matrix A = (Aij)i,jEE• On an interval [t, t+s) where Jt - i, {St} evolves like a process with stationary independent increments and the parameters pi, v;, vi(dx) in (4.6) depending on i. In addition, a jump of {Jt} from i to j # i has probability qij of giving rise to a jump of {St} at the same time, the distribution of which has some distribution Bij. (That a process with this description is a MAP is obvious; the converse requires a proof, which we omit and refer to Neveu [272] or cinlar [87].) If E is infinite a MAP may be much more complicated. As an example, let {Jt} be standard Brownian motion on the line. Then a Markov additive process can be defined by letting t St = lim 1 I(IJB1 < e)ds E1o 2d o

be the local time at 0 up to time t. As a generalization of the m.g.f., consider the matrix Ft [a] with ijth element least Ei ;it = A.

Proposition 5.1 For a MAP in discrete time and with E finite, Fn[a] = F[a]n where

P[a]

=

P ,[a)

= (Ei[easl; 1

J1 ='^])iJEE = (Fij[a])i

,.9 EE =

(iii&ij[a])i

j EE

5. MARKOV ADDITIVE PROCESSES

41

Proof Conditioning upon (Jn, Sn ) yields Ei[easn+

';

Ei[ e 5„;

Jn+1 = A] =

Jn =

k]Ek[e"Y"

; J1 = A

kEE

which in matrix formulation is the same as Fn+1 [a] = Fn[a]F[a].

❑

Proposition 5.2 Let E be finite and consider a continuous time Markov additive process with parameters A, pi, 013 , vi(dx) (i E E), qij, Bij (i, j E E) and So = 0. Then the matrix Pt[a] with ijth element Ei [east; Jt = j] is given by etK[a], where K[a] = A+ (r.(')(a)) diag

+ (),ijgij(Bij[a] - 1)) ,

00 r(i) (a) = api + a2ot /2 + f

Proof Let

{Stt)

(e°

- 1 )v(dx).

} be a process with stationary independent increments and pa-

rameters pi , a= , vi(dx). Then, up to o( h) terms, aSt h

= (1 + Ajjh) Ei [east ; Jt = j] Ejesh'^

+ E Ak j hEi [ease ; Jt = k] { 1 - qkj +

qkj Bkj [a] }

k?^j

= Ei [east;

Jt = j] (1 + htc (j) (a))

+h E Ei [east ; Jt = k] { xk kEE

j

+ Ak j qk j (Bk

j la] - 1) }

(recall that qjj = 0). In matrix formulation , this means that F't+h [a] =

Ft[a] II+h(rc(i)(a)) +hA+h(Aijgij(Bij[a]-1)) I, \ diag

Ft[a] = Ft[a]K,

which in conjunction with Fo[a] = I implies Ft[a] = etK[a) according to the standard solution formula for systems of linear differential equations. ❑ In the following, assume that the Markov chain/process {Jt} is ergodic. By Perron-Frobenius theory (see A.4c), we infer that in the discrete time case the

42 CHAPTER II. SOME GENERAL TOOLS AND RESULTS matrix F[a] has a real eigenvalue ic(a) with maximal absolute value and that in the continuous time case K[a] has a real eigenvalue K(a) with maximal real part. The corresponding left and right eigenvectors v("), h(") may be chosen with strictly positive components. Since v("), h(") are only given up to a constants, we are free to impose two normalizations, and we shall take V(a)h(a) = 1,

Yrh(a ) = 1, (5.2)

where 7r = v(°) is the stationary distribution. Then h(°) = e.

The function ic(a) plays in many respects the same role as the cumulant g.f. of a random walk, as will be seen from the following results. In particular, its derivatives are 'asymptotic cumulants', cf. Corollary 5.7, and appropriate generalizations of the Wald martingale (and the associated change of measure) can be defined in terms of ,c(a) (and h(")), cf. Proposition 5.4. Corollary 5.3

Ei [east, Jt = j] - h(a)vva)etw(a).

Proof By Perron-Frobenius theory (see A.4c). We also get an analogue of the Wald martingale for random walks: Proposition 5.4 Eie"sth(a) = h=a)et?("). Furthermore, Jeast- tK(a)h(a) J jj it L o is a martingale. Proof For the first assertion, just note that [a]h(a) = eietK (a)h(a) = etK(a)h(a). Eie"sthâ) = e'Pt[a]h( a) = e,etx It then follows that

E feast+^-(t+v)K(a)h(a) I ^tl l .Jt+v = east-tK( a)E [ee (st+v-st)-vK(a)h(a) jt+v I ^tJ = east-tt(a)EJt (eases-vK(a )hâ)1 = east-tK(a)hâ). ❑ Let k(a) denote the derivative of h() w.r.t. a, and write k = k(°).

Corollary 5.5 EiSt = tK'(0) + ki - Eikjt = ttc'(0) + ki - e=e°tk.


43

Proof By differentiation in Proposition 5.4, Ei [Steast h(a) + east kâ)1 = et"(a) (kia) + tic (a)hia)) . (5.3) ❑

Let a = 0 and recall that h(°) = e so that 0=°) = h(o) = 1.

The argument is slightly heuristic (e.g., the existence of exponential moments is assumed ) but can be made rigorous by passing to characteristic functions. In the same way, one obtains a generalization of Wald's identity EST = E-r • ES, for a random walk: Corollary 5.6 For any stopping time T with finite mean, E=ST = tc'(0)E7- + k; - Eikjr . Corollary 5.7 No matter the initial distribution v of Jo,

tam E tSt

a (0),

t im v^"St = '(0)

Proof The first assertion is immediate by dividing by tin Corollary 5.5. For the second , we differentiate (5.3) to get

Ej [St a " st h i(a ) + 2Ste"st k(a) + e"st kâ) J etI(a) (kia )' + ttc (a)ki") + t {ic"(a)h;a) + ttc (a)2hia )

+ W (a)k;") }) .

Multiplying by v=, summing and letting a = 0 yields E„ [St + 2Stkj, ] = t2tc (0)2 + 2tK'(0)vk + ttc"(0) + O(1) . Squaring in Corollary 5.5 yields

[E,St]2 =

t2/c'(0 ) 2

+ 2ttc (0)vk - 2ttc (0)Evkjt + 0(i).

Since it is easily seen by an asymptotic independence argument that E„ [Stkjt] ❑ = trc'(0) E„kjt + O(1), subtraction yields Vary St = tic"(0) + O(1). Remark 5 . 8 Also for E being infinite (possibly uncountable ), Ee"st typically grows asymptotically exponential with a rate ic(a) independent of the initial condition (i.e., the distribution of Jo). More precisely, there is typically a function h = h(") on E and a ^c(a) such that Ey a"st -t" (") -* h(x), t --a oo, (5.4)

44 CHAPTER IL SOME GENERAL TOOLS AND RESULTS for all x E E. From (5.1) one then ( at least heuristically) obtains lim Ex eaSv -v a) K(

v-+oo

nEx east-tK(a)EJt eas-t-(v-t)K(a) u[J = Ex east-tk(a)h(Jt)

It then follows as in the proof of Proposition 5.4 that

{ h(Jt) east-tK(a) L

(5.5) o

is a martingale . In view of this discussion , we take the martingale property as our basic condition below (though this is automatic in the finite case). An example beyond the finite case occurs for periodic risk processes in VI.6, where {Jt} is deterministic period motion on E = [0, 1) (i.e., Jt = (s+t) mod 1 P8-a.s. ❑ forsEE). Remark 5.9 The condition that (5.5) is a martingale can be expressed via the infinitesimal generator G of {Xt } = { (Jt, St) } as follows. First, G is defined as Gf (x) = lim

Exf (Xt) - f (x)

tyo t

provided the limit exists. Usually, some extra conditions are imposed, in particular that f is bounded;,for the present purposes, this is, however, inconvenient due to the unboundedness of ea8 so we shall not aim for complete rigour but interpret C in a broader sense. Given a function h on E, let ha(i, s) = ea8h(i). We then want to determine h and x(a) such that Ejeasth (Jt) = etK(a)h(i). For t small , this leads to h(i) + tcha( i, 0) = h(i )( 1 + ttc(a)), gha(i, 0) = n(a) h(i).

(5.6)

We shall not exploit this approach systematically; see, however, V.3b and Remark VI.6.5. 0 Proposition 5.10 Let {(Jt, St)} be a MAP and let 0 be such that h(Jt) OSt-t,(9) {Lt}t>o = . h(Jo) Lo

is a Px -martingale for each x E E. Then {Lt } is a multiplicative functional, and the family {f LEE given by Theorem 2.5 defines a new MAP. xEE


45

Proof That { Lt} is a multiplicative functional follows from L8 ogt = h(Jt+s) es(St+ .- St)-sl(e) h(Jt) The proof that we have a MAP is contained in the proof of Theorem 5.11 below in the finite case. In the infinite case , one can directly verify that (5.1) holds for the P. We omit the details. ❑ Theorem 5.11 Consider the irreducible case with E finite. Then the MAP in Proposition 5.10 is given by P = e-K(e) Oh e) F[e]Oh('),

ea' f ij (dx) = Hij (dx) Hij [0]

in the discrete time case, and by A = Oh(°) K [0]Oh(e ) vi(dx) = e"xvi

(dx), qij =

r.(0)j, Ai = µi + 0Q, , î = of eft

qij Bij [0] 1 + qij ( Bij [0] - 1)

, Bi.7(dx) Bij [0] Bij(dx)

in the continuous time case . Here Oh(e) is the diagonal matrix with the h=e) on the diagonal. In particular, if vi(dx) is compound Poisson, vi (dx) = f3 Bi(dx) with ,Qi < oo and Bi a probability measure, then also vi (dx) is compound Poisson with e Ox

î = /3iBi[0], Bi(dx) = Bi(dx). Bi [0] Remark 5.12 The expression for A means h(e)

Aij = hie)

Aij [1 + gij(Bij[0] i 0 j. (5.7)

In particular, this gives a direct verification that A is an intensity matrix: the off-diagonal elements are non-negative because Aij > 0, 0 < qij < 1 and Bij [0] > 0. That the rows sum to 1 follows from Ae = Oh(e) K[O]h(B) - ic(0)e = ic(0)Oh e) h(e) - ,c(0)e = tc(0)e - tc(0)e = 0 . That 0 < qij < 1 follows from the inequality qb 0_1 > a_2 > ... in (-oo, 0); the mark at time Qk is denoted by Uk. Let p(t) be the conditional probability that ST+ E A, T+ = t given the event At that an arrival at t occurs . Then clearly

* (A) = P(ST+ E A) = G+

f

p(t)f3dt.

, which makes an upwards jump at time - o,-A; (k = St}t>o 1, 2, ...), moves down linearly at a unit rate in between jumps and starts from

Consider a process {

S0 = U. Now conditionally upon At , { Su}0 0 is (1 - p) E pn S n x n- 1 e -ax = 00

n-1 (n - 1)1

( 1 - p)pSe-

a ( l -v)x = p( S - O)e-(b-0)x.

Integrating from u to oo, the result follows . Alternatively, use Laplace transforms. The result can, however , also be seen probabilistically without summing infinite series . Let r ( x) be the failure rate of M at x > 0. For a failure at x, the current ladder step must terminate which occurs at rate S and there must be no further ones which occurs w.p. 1 - p. Thus r(x) = S(1 - p) = S -,3 so that the conditional distribution of M given M > 0 is exponential with rate S -'3 and 0(u) = P(M > u) = P(M > 0)P(M > uIM

> 0) = pe-(6-Mu. 0

64

CHAPTER III. THE COMPOUND POISSON MODEL

In VIII.3, we show that expression for /'(u) which are explicit (up to matrix exponentials) come out in a similar way also when B is phase-type. E.g. (Example VIII.3.2), if 3 = 3 and B is a mixture of two exponential distributions with rates 3 and 7, and weights 1/2 for each, then 24 1 V, (u) 35e-u + 35e-6u. (3.1) For a heavy-tailed B, we use the Pollaczeck-Khinchine formula in Chapter IX to show that b(u) -- 1 p pBo(u), u -+ oo. (3.2) Notes and references Corollary 3. 2 is one of the main classical early results in the area. A variety of proofs are available . We mention in particular the following: (a) check that ip (u) = pe -(6-0)u is solution of the renewal equation (3.3) below; (b) use stopped martingales , cf. II.1.

3c Some classical analytical results Recall the notation G+(u) = f^°° G+(dx). Corollary 3.3 The ruin probability Vi(u) satisfies the defective renewal equation ik (u) = 6+ (u) + G+ * 0(u) = Q f B(y) dy + u

f

u

0(u - y)f3 (y) dy. (3.3)

0

Equivalently, the survival probability Z(u) = 1 - i(u) satisfies the defective renewal equation Z(u) = 1 - p + G+ * Z(u) = 1 - p + f u Z(u - y)/3B (y) dy. 0

(3.4)

Proof Write o (u) as P(M>u) = P(S,+ >u,T+ u,S,+ u] = /3iE(U - u)+. (7.5)

u

Proof According to the Pollaczeck-Khinchine formula, ao

00

(u) P) anllBBon(U) onPaBon(u) • n=1

n=1

Asymptotically, En'=2 • • • = O(/32) so that only the first terms matters, and hence 00 /3pBBo (u) = 0 / B(x)dx. 10 (U) ( u

The alternative expressions in (7.5) follow by integration by parts.

❑

Note that heuristically the light traffic approximation in Proposition 7.3 is the same which comes out by saying that basically ruin can only occur at the F(U - T > u). Indeed, by monotone time T of the first claim , i.e. z/' (u) convergence

P(U - T > u) =

J o" B(x + u)/3e-ax dx , ( 3 J O B dx. 0

u

Notes and references Light traffic limit theory for queues was initiated by Bloomfield & Cox [69]. For a more comprehensive treatment, see Daley & Rolski [96], [97], Asmussen [19] and references there. Again, the Poisson case is much easier than the renewal case. Another way to understand that the present analysis is much simpler than in these references is the fact that in the queueing setting light traffic theory is much easier for virtual waiting times (the probability of the conditioning event {M > 0} is explicit) than for actual waiting times , cf. Sigman [347]. Light traffic does not appear to have been studied in risk theory.

7e Interpolating between light and heavy traffic We shall now outline an idea of how the heavy and light traffic approximations can be combined. The crude idea of interpolating between light and heavy traffic leads to 0 (u) C1 - Q limIP ( u) + Q lim z/'(u) Amax &0 amax ATAm.=

1-

aJ 1 a 0+

1 = = p,

Omax max m.

83

8. COMPARISONS OF CLAIM SIZE DISTRIBUTIONS

which is clearly useless . Instead, to get non-degenerate limits , we combine with our explicit knowledge of ip(u) for the exponential claim size distribution E whith the same mean PB as the given one B, that is, with rate 1/µB = /3max. Let OLT) (u) denote the light traffic approximation given by Proposition 7.3 and use similar notation for -%(B) (u) = (u), z/i(E) (u) = pe-(Qmax-Q)u, _(E) (u), (U). Substituting v = u(,3n, -,3), we see that the following limits HT) (u'), ÎE)

exist:

hm

1 (B) HT

Qmsx-Q

e-6"

J

Q1Qm.x ,VHT) ( ax Qm-Q )

e

h (B) ( . ^ LT Q max-Q m"^ Qlo V LT) (

= e(1 -6)" =

CHT(v)

(say),

2µE/µE2)'"

00

-

f / Qmax B(x)dx 00 e-Qmaxxdx 4/ Qmax

Qmax-Q

amaze"

J

B(x) dx = cLT(v)

(say),

"/Qmex

and the approximation we suggest is

Cu)

CLT(u ( /3max -0) + O16 CHT( U(Qmaz - /3)) ,O(E)(u) 1 (1 - O0 M. ) M.

Al - Wmax f(x ) dx + pee6mQ. (7.6) (1-p)

The particular features of this approximation is that it is exact for the exponential distribution and asymptotically correct both in light and heavy traffic. Thus , even if the safety loading is not very small, one may hope that some correction of the heavy traffic approximation has been obtained. Notes and references In the queueing setting , the idea of interpolating between light and heavy traffic is due to Burman & Smith [83 ], [84]. Another main queueing paper is Whitt [380], where further references can be found . The adaptation to risk theory is new; no empirical study of the fit of (7.6) is , however, available.

8 Comparing the risks of different claim size distributions Given two claim size distributions B(1), B(2), we may ask which one carries the larger risk in the sense of larger values of the ruin probability V(') (u) for a fixed value of 0.

84 CHAPTER III. THE COMPOUND POISSON MODEL To this end, we shall need various ordering properties of distributions, for more detail and background on which we refer to Stoyan [352] or Shaked & Shantikumar [337]. Recall that B(') is said to be stochastically smaller than B(2) (in symbols, B(') 0 (9.2) (see Remark 9.6 below for some discussion of this assumption). Consider first the adjustment coefficient y as function of 3, 9, (, and write -yp = 8-y/8/3 and so on . Similar notation for partial derivatived are used below, e.g. for the ruin probabilities ,0 = t/'(u) and the Cramer-Lundberg constant C. Proposition 9.3

7

70 = 'Ye

= =

/3(1-

we(e +'y,()(0 +'0) '

(Q+'Y)[we(0+7,O-we (9,^)] 1-(/3+y)we(9+'y,

(3+'y)PC (0+7,()-wC(e,()^ 1 - (/3 + y)we(9 + 7, ()

(9 . 3) ( 9 . 4) (9 . 5)

Proof According to (9.10) below, we can rewrite the Lundberg equation as w(9+ -y, ^) - w(6, () = log(1 + -y//3). Differentiating w.r.t. /3 yields w e(e + Y,()YC = 1

+y/ /3

\ Q2

From this (9.3) follows by straightforward algebra, and the proofs of (9.4), (9.5) are similar. ❑ Now consider the ruin probability 0 = 0 (u) itself. Of course, we cannot expect in general to find explicit expressions like in Example 9.1 or Proposition 9.3, but must look for approximations for the sensitivities 0,3, Viei '0(. The most intuitive approach is to rely on the accuracy of the Cramer-Lundberg approximation , so that heuristically we obtain '00 50-ryu = Coe-"u - u-ypCe-7u -urypO.

(9.6)

As will be seen below, this intuition is indeed correct. However , mathematically a proof is needed basically to show that two limits (u -* oo and the differentiation as limit of finite differences) are interchangeable. Consider first the case of 8/8/3:

9. SENSITIVITY ESTIMATES Proposition 9.4 As u

89

oo, it holds that

a

7C2

ue -ryu a/3 Q(1 P)

Proof We shall use the renewal equation (3.3) for z/'(u),

0(u) = /3

0(u - x),3(x) dx.

Ju"O B(x) dx + f

(9.8)

Letting cp = e0/e/3 and differentiating (9.8), we get p(u) =

J "O B(x) dx + J U O(u - x)B(x) dx + J U W(u - x),QB(x) dx. u

0

0

Proceeding in a similar way as in the proof of the Cramer-Lundberg approximation based upon (9.8) (Section 5), we multiply by e7" and let Z(u) = elt" cp(u), F(dx) = e'yy/3B(x)dx, Z= zl + z2

where zl (u) = e7u

J m B(x)dx, z2(U) = e7" J u b(u - x)B(x)dx. u

0

Then Z = z + F * Z and F is a proper probability distribution . By dominated convergence, z2(u) _ 1 ûe'riì7i( u - x) F(dx ) --f

J

C F(dx) = C

as u -4 oo, and alsoo zl(u) -+ 0 because of B['y ] < oo. Hence by a variant of the key renewal theorem (Proposition A1.2 of the Appendix ), Z(u)/u -a C//3PF where PF is the mean of F. But from the proof of Theorem 5.3 (see in particular (5.12)), PF = (1 - p)/C'y. Combining these estimates , the proof is complete.

11 For the following, we note the formulas

(9.9)

Ee,St (U)

=

wS(O,C),

Ee,(e"U

=

Be,([a] = exp {w(9 + a, () - w(O, ()} ,

(9.10)

w((9 + a, () exp {w(9 + a, () - w(9, O}

(9.11)

Ee,4t (U)e°Ù =

which are well-known and easy to show (see e.g. Barndorff-Nielsen [58]). Further write

de = [we (9 +'y, () - we(9 , ()] exp {w (O + -y, () - w(9, ()}


90

[we(e+7, 01 (i+) do = +'Y, ^) - w(0, ^)} [wc (0 + 7, C) - we (0, C)] (1 + 7 ) Proposition 9.5 Assume that (9.2) holds. Then as u -> oo, 2 z 07P N ue-7u (3C de , 8^ ue-7u,6C do 89 1-p 8( 1-p Proof By straightforward differentiation, 8 8() 8(

f

exp

[t(y)

= f

{O y + (t(y) - w (9, ()} 1z(dy)

- wc (O, )}B(dy)•

Letting cp it thus follows from (9.8) that cp(u)

- x),lB(x) dx = e-7uzl(u) + e-7°zz(u) + V(u T

where zl (u) = ,6e7u f "o f[t(y) - w(e, ()]B(dy) dx, u Z2(U) = e7° f u ^/i(u - x)f3 f ^[t(y) - wc(O, ())B(dy) dx. 0

x

Multiplying by e7" and letting z = zl + z2,

Z(u) = e"uV(u),

F(dx) = e7x,QB(x)dx,

this implies Z = z + F * Z. By dominated convergence and (9.9)-(9.11), oo z2 (u) f

C - e7x/3 f 00 [t(y) - w( (0, ()]B(dy) dx

0

x

0C T ON O - wc(9, ()]--(e7v - 1) B(dy)

'f '[t(y) - w( (0, ()]e7vB(dy) 'fCd 7 c

(9.12)

9. SENSITIVITY ESTIMATES

91

as u -3 oo, and also zj (u) -4 0 because of

o c'o e11(t (y) - w((9, ())B(dy)

< oo.

f Hence,

Z(u) /3C U 7µF from which the second assertion of (9.12) follows, and the proof of the first one ❑ is similar. Example 9.6 Consider the gamma density b (x )

= Sa xa- 1e-dz = 1 exp {-Sx + a log x - (log r(a) -a log S)} • r(a) 1.

Here (9.2) holds with

p(dx) = x-ldx, 9 = -S, < = a, t(x) = logx, w(e, () = log r(a) - a log S = log r(c) - C log(-9). We get w( (0, () ='I'(t;) -log(-9) = %F(a) -logs where %1 = F'/]F is the Digamma function, we (9, () = -C/9 = a/S. It follows after some elementary calculus that p = a)3/5 and, by inserting in the above formulas, that

C

(9.13)

, = a,QS a 1 /(S -- pa+1 ry) - 1 5a-1

de = d( 7!3 76 = -7e =

(9.14)

cry (5 - ,Y)a+1 '

log ( \ ( \5a_ / \SSry ) 72 - Sry

'

a/32 + a/37 + /37 - /35' a/i'y + aryl 62-5ry- a/35-a&y'

(05 + 57 _'3_y - rye) S 5-ry-a,3-ary tog(F 'y)- inally, ( 9.12) takes the form

alp a

,yu/3C2do u86 89 1-p'

az/) = 8z/, ,,, ue-_Yu 'C2d( 8a 8( 1 -p

(9.15) (9.16)

(9.17) (9.18)


92

Example 9.7 Consider the inverse Gaussian density ( b(x) Zx37 exp

This has the form (9.2) with

µ(dx) = 2x3zrdx, 9 = - 22, C = - 2 , t(x) _ -, w(e, () = -Cc - log c = -2

(- 9) (-() - 2 -log (-0- 21og 2.

In particular, for a < a* = z

Be,S[a] = exp {w (9 + a, C) - w(9, ()} = exp {c (C - "62 - 2a) } Thus the condition B[a*] > 1 + a* /,l3 of Section 6a needed for the existence of ry becomes e^Q > 1+62 / 2,3. Straightforward but tedious calculations , which we omit in part , further yield ,3Ee,([Y] - 1 = eXP {c(C - CZ -try)} 1 C C2 -try - 1

16 +ry c C2-2ry

2( = + 70

-Yc =

de =

do =

We (e, C)

= B

=

93

10. ESTIMATION OF THE ADJUSTMENT COEFFICIENT Finally, (9.12) takes the form a = a

ar; ae

_

t 1lEY u -S

,3C2de 1-p'

z a = -c - -cue_7u)3C P

Remark 9.8 The specific form of (9.2) is motivated as follows. In general, the exponent of the density in an exponential family has the form 01 tl (x) + • • • + 9ktk (x). Thus, we have assumed k = 2 and ti (x) = x. That it is no restriction to assume one of the ti(x) to be linear follows since the whole set-up requires exponential moments to be finite (thus we can always extend the family if necessary by adding a term Ox). That it is no restriction to assume k < 2 follows since if k > 2, we can just fix k - 2 of the parameters. Finally if k = 1, the exponent is either Ox, in which case we can just let t(x) = 0, or Ct(x), in ❑ which case the extension just described applies. Notes and references The general area of sensitivity analysis (gradient estimation) is currently receiving considerable interest in queueing theory. However, the models there (e.g. queueing networks) are typically much more complicated than the one considered here, and hence explicit or asymptotic estimates are in general not possible. Thus, the main tool is simulation, for which we refer to X.7 and references there. Comparatively less work seems to have been done in risk theory; thus, to our knowledge, the results presented here are new. Van Wouve et al. [379] consider a special problem related to reinsurance.

10 Estimation of the adjustment coefficient We consider a non-parametric set-up where /3, B are assumed to be completely unknown, and we estimate -y by means of the empirical solution ryT to the Lundberg equation. To this end, let NT 16T = ^T , BT [a]= NT ^` e"U;, kT (a) = /T (BT [a] - 1) - a, sj=1

and let -YT be defined by IKT('ryT) = 0.

Note that if NT = 0, then BT and hence ryT is undefined. Also, if 1 PT = /3TNT(U1+...+UNT) > 1, then ryT < 0. However , by the LLN both F (NT = 0) and F (PT > 1) converge to 0 as T -- oo.

94 CHAPTER III. THE COMPOUND POISSON MODEL Theorem 10 .1 As T -4 oo, 7T

a4' 7.

If furthermore B[27] < oo, then

'YT - 'Y ,: N 0,T a2y ,

(10.1)

where a2 = /3r.(27)/K'(7)2. For the proof, we need a lemma. Lemma 10 .2 As T -* oo, N

r-T(7) N

(

n[7], B[2'Y] -

B [7]2

(10.2)

/3T ) ,

(10.3)

N (0,

Proof Since Var(eryU) = we have B[7], B[27] - B[7]2 n Hence ( 10.2) follows from NT/T a4' ,Q and Anscombe 's theorem. More generally, since NT /T ,: N ()3,,3/ T), it is easy to see that we can write \ V1 1 l _ ,a BT[7] I B[7] I + , ,/^ B[27] - B[7]2 V2 , 16T

where V1, V2 are independent N (0, 1) r.v.'s. Hence KT(7)

= (F' +

0))((BT [7] - B[7]) + B [7] - 1) - If - 1) - 7 + (,3T - i3)(B[7] -1) + (3(BT[7]

(OT

a(B[7l

-

0+ Iv/o-(b[-y]-,)vl+ N CO,

(

vfo-VFB[2-y]- b[-Yp'V21

T { (E[7] - 1)2 + E[27] - B[7]2

T

0

- B[7])

})

95

10. ESTIMATION OF THE ADJUSTMENT COEFFICIENT

❑

which is the same as (10.3). Proof of Theorem 10.1 By the law of large numbers,

4 /3,

BT[a] -3 B[a],

OT a

Let 0
1/2 or 11 < 100%. For this reason , various alternatives have been developed. One (see Schmidli [321]) is to let {Vt} be the workload process of an M /G/1 queue with the same arrival epochs as the risk process and service times U1, U2,..., i .e. Vt = St - info< „< t S,,. Letting Wo = 0, wn = inf{t > W.-1 : Vt = 0, V. > 0 for some t E [Wn_ 1, t]}, the nth busy cycle is then [Wn-1, Wn), and the known fact that the Y„ = max Vt tE[W„-1,Wn)

are i .i.d. with a tail of the form P(Y > y) - C1e-"a ( see e.g. Asmussen [23]) can then be used to produce an estimate of ry. This approach in fact applies also for many models more general than the compound Poisson one.

Further work on estimation of -y with different methods can be found in Csorgo & Steinebach [94], Csorgo & Teugels [95], Deheuvels & Steinebach [102], Embrechts & Mikosch [133], Herkenrath [192], Hipp [196], [197], Frees [146], Mammitzsch [253] and Pitts, Griibel & Embrechts [292].

Chapter IV

The probability of ruin within finite time This chapter is concerned with the finite time ruin probabilities

0(u, T) = P( /r(u) 0. Using 5 = 6 - 0, the result follows. Note that it follows from Proposition 1.3 that we can write Ee-aT( u) = e-euEe -017(o). (1.4) The interpretation of this that T(u) can be written as the independent sum of T(0) plus a r.v. Y(u) belonging to a convolution semigroup . More precisely, M(u) T(u) = T + E Tk

k=1

where T = T(0) is the length of the first ladder segment , Ti, T2 , ... are the lengths of of the ladder segments 2,3,..., and M(u)+1 is the index of the ladder segment corresponding to T(u). Cf. Fig. 1.1 where Y1, Y2, ... are the ladder heights which form a terminating sequence of exponential r.v.'s with rate 5.

St

Ti

a

F- T+ Ti

t

T I U

1 a

i

F- Y1

-Y2

Figure 1.1

101

1. EXPONENTIAL CLAIMS

For numerical purposes , the following formula is convenient by allowing t,1(u, T) to be evaluated by numerical integration: Proposition 1.3 Assume that claims are exponential with rate b = 1. Then V(u,T) 1 I fl(O)h(0) fdO

(1.6)

where

fl(9)

=

fexp {2iTcos9-(1+/3)T+u(/cos9-1)

f2(0)

=

cos (uisin9) - cos (u/,3 sin0 + 29)

f3(0)

=

1+/3-2/cos9.

Note that the case 6 # 1 is easily reduced to the case S = 1 via the formula V,0,6(u) = Vfl/j l(Su,ST). Proof We use the formula ,i (u,T) = P(VT > u) where {Vt } is the workload process in an initially empty M/M/1 queue with arrival rate 0 and service rate S = 1, cf. Corollary 11.4.6. Let {Qt} be the queue length process of the queue (number in system, including the customer being currently served). If QT = N > 0, then VT = U1,T + • • • + UN,T, where U1,T is the residual service time of the customer being currently served and U2 , T, ... , UN,T the service times of the customers awaiting service . Since U1 ,T, U2,T, ... , UN,T are conditionally i.i.d. and exponential with rate S = 1, the conditional distribution of VT given QT = N is that of EN where the r.v. EN has an Erlang distribution with parameters (N, 1), i.e. density xN -le-x/(N - 1 )!. Hence 00

F(VT > u ) P(QT = N)P(EN > u) N=1 N-1 k

00

F(QT = N) e-u

k!

N=1 k=1 °O -u

Ee

k

1t P(QT - k + 1).

k=0

For j = 0,1, 2, ..., let (cf. [4]) 00 (x/2)2n+3 Ij (x)

- I ex

OnI(n+j)!

fo "

cos B cos j O dB

102

CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME

denote the modified Bessel function of order j, let I _ j (x) = Ij (x), and define tj = e-(1+R)Taj/2Ij(2vT T). Then (see Prabhu [294] pp. 9-12, in particular equations (1.38), (1.44); similar formulas are in [APQ] pp. 87-89) 00 E aj j= -00

= 1, k

g'(QT >- k + 1) = 1 -

bj

-k-2 + $k+1 E bj

j=-00 j=-00

00 k +1 +

00 t j - )3k +1 tj

j=-k-1

j=kk+1

By Euler 's formulas, 00 E '3j/2 cos(je) j=k+1 00

_

^j/ zeij

,13(k +l)/2ei(k +1)9

=

R E ,(31/2eie - 1 j=k+1

R

[,3(k +1)/ 2ei(k + l)6 (,31 /2e-ie I/31/2eie -l 112

L

1( k +1)/2

1)] 1

[ 31/ 2 cos(kO) - cos((k + 1)0)] f3(0)

00 flk +1 >

3j/2 COS(jB)

j=-k-1

00 ok+lR 00 j=-k-1

l)/2e-i(k+1)e )3j/2eije = R)3(k+ (31 /2eie - 1

k +1)/2e - i(k +1)e (01 /2 e - ie - 1)] R [/3( L lal/2e:0 _112

/(k+1)/2 [,(31/2 cos (( k + 2)9) - cos (( k + 1)0)]

f3(9) Hence the integral expression in (1.8 ) yields F(QT > k + 1) - )3k+1

= e-(1+0)T

e201/2Tcos 7r ✓ 0

e )3(k

+l)/2 [31/ 2

cos ( kO) - cos((k + 2)9)]

f3(0)

d9.

2. THE RUIN PROBABILITY WITH NO INITIAL RESERVE

103

Since P(QOO > k + 1) = flk+1, it follows as in (1.7) that

Cu)

_

[^ a-u ak+l (30 k L. k=0

A further application of Euler's formulas yields cc k =0

OI

k 'ese)k __ (u^1 U #kJ2 cos((k + 2)9) = R eNO ^` L k=

L

= eup i/z

COS a cos(u(31/2 sin 9 + 20),

Co Uk

oo (u)31/2eê)k = )3k z cos(k9) = R k. E Fk! k. k=O

=ateU161/2 e '0+2iO

k-0 i/z

ate" o'/z e

,e

=

= e' COS a cos(uf31/2 sin 0). ❑

The rest of the proof is easy algebra.

Notes and references Proposition 1.3 was given in Asmussen [12] (as pointed out by Barndorff-Nielsen & Schmidli [59], there are several misprints in the formula there; however, the numerical examples in [12] are correct). Related formulas are in Takacs [359]. Seal [327] gives a different numerical integration fomula for 1 - 0(u,T) which, however, is numerically unstable for large T.

2 The ruin probability with no initial reserve In this section , we are concerned with describing the distribution of the ruin time T(0) in the case where the initial reserve is u = 0. We allow a general claim size distribution B and recall that we have the explicit formula z/i(0) _ P(7(0) Goo) = p. We first prove two classical formulas which are remarkable by showing that the ruin probabilities can be reconstructed from the distributions of the St, or, equivalently, from the accumulated claim distribution N, F(x, t) = P

,

Ui < x I /

(note that P(St < x ) = F(x + t, t )). The first formula, going back to Cramer, expresses V)(0, T) in terms of F(., T), and the next one (often called Seal's formula but originating from Prabhu [293]) shows how to reduce the case u 54 0 to this.


104

f

Theorem 2 . 1 1 - i,b (0,T)

T

T F(x,T)dx.

Proof For any v E [0,T], we define a new claim surplus process

lStv)} 0 u and otherwise that of T(v) with v = u - ST( u') = u1/4 - ^(T(u')). Using ( 4.3), we get

E[ T (u)

- T ( u')]

=

E[ T ( ul /4

- VU T; S( u

)

< ul/4]

< ET(ul / 4) = O(ul/4),

and thus in (4.6), we can replace T(u) by r(u'). Let h(u) = E f (^(u)). Then h(u) -4 h(oo) = E f (6(oo)), and similarly as above we get

E[f(^(u)) I -Fr(u,))I h(ul /4

-

C < ul /4 ) f + f(e(u') - u1/4)I(S(u') >

^(u)) I(6 (u')

u1 /4)

h(oo) + 0, using that ul/4 - e(u') oo w .r.t. P because of ^(u') - l:(oo) (recall that rt < 0). Hence

Ef (Vu ))

9 (T(u,)-mu \

h(oo)Eg (r(ul) - mul h(oo)Eg(Z). O

4. WHEN DOES RUIN OCCUR?

113

Proof of Corollary 4.3 ery"z/i(u , umL + ywL f) = e"P(T (u) < umL + ywL) = EL [e-7V "); T(u) < umL + ywL f, ELe-7E (") . PL(T(u ) < umL + ywL)

-4 C4(y), where we used Stain's lemma in the third step and (4.4) in the last.

❑

For practical purposes , Segerdahl 's result suggests the approximation b(u,T) Ce-7"4 (T - umL wI V"U

(4.7)

To arrive at this , just substitute T = umL + ywL in (4.5) and solve for y = y(T). The precise condition for (4.7) to be valid is that T varies with u in such a way that y(T) has a limit in (- oo, oo ) as u -* oo. Thus , in practice one would trust (4.7) whenever u is large and ly(T)l moderate or small (numerical evidence presented in [12 ] indicates , however , that for the fit of (4.7) to be good, u needs to be very large). Notes and references Corollary 4 . 3 is due to Segerdahl [333]. The present proof is basically that of Siegmund [342]; see also von Bahr [55 ] and Gut [182]. For refinements of Corollary 4.3 in terms of Edgeworth expansions , see Asmussen [12] and Malinovskii [254]. Cf. also Hoglund [204].

4b Gerber's time- dependent version of Lundberg's inequality For y > 0, define ay, yy by 1 7y = ay - yK(ay)•

K,(ay) =

(4.8)

17

Note that ay > 7o and that 7y > •y (unless for the critical value y = 1/ML), CL Fig. 0.1. Theorem 4.5 '(u , y u)
k'(7) .

(4 .9) ( 4 . 10 )


114

Proof Consider first the case y < 1/K'(y). Then ic(ay) > 0 (see Fig . 0.1), and hence t,h(u,yu

)

=

e-ayuEav

[e - ay4(u)+ T(u)K(ay ); T(u) < yu]

Y < e-ayuEav [ eT(u)K(av )L T(u) < yu}

< e-ayu + yUr-(ay)

Similarly, if y > 1/ic'(y), we have rc(ay) < 0 and get (u) - Y' (u, yu ) =
yo, Proposition 4.2 yields EaT(u) u u r,. ' (0) r1 (a)

I.e., if we want EaT(u) ,: T, then the relevant choice is precisely a = ay where y = T/u. We thereby obtain that T is 'in the center' of the Pa-distribution of T(u). This idea is precisely what characterizes the saddlepoint method. The traditional application of the saddlepoint method is to derive approximations, not inequalities, and in case of ruin probabilities the approach leads to the following result: Theorem 4 .8 If y < 1/ic'(ry), then the solution &y < ay of ,c(&) = ic(ay) is < 0, and ay - ay a-.yyu

b(u,yu) c

y l ay I

21ry/3B" [ay]

fU_

V

u -+ 00. (4.11) '

If y > 1/ r .'(-y ), then ay > 0, and ay-ay e -ryyu

ii(u) - z,i(u, yu )

, u -4 oo. (4.12)

ayay 27ry/3B"[ay] u Proof In view of Stam 's lemma, the formula 0(u, yu) = e- ayuEay f e-ay^ ( u)+T(u)K(ay); T(u)

< yu]

suggests heuristically that l e-aauEaye - ayC(-) . Ea ,, [eT(u )K( ay); T(u) < yu] .

t/,(u, yu )

(4.13)

Here the first expectation can be estimated similarly as in the proof of the Cramer-Lundberg ' s approximation in Chapter III. Using Lemma 111.5.6 with P replaced by Pay and FL by Pay, we have ryas = ay - ay and get

Ea

e -ayf (00)

_

'Ya(

y ayKal lay

C

1 - ^3

]-1/ Bay [lay - ay y 'Yay -

ay

116

CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME , a nr=. i B[7ay - ay + ayl /BLay] - I

ry I i

(1

ay -&y

+,1-B[ay]1

a ^c'(ay) a

)

y(ay - ay) ay +,(j (1 - B[ay] /ay

&y

-y(ay - ay ) r.(ay) _ y(ay - ay)K(ay) ay

ayI&YI

For the second term in (4.13), it seems tempting to apply the normal approxiyu + ul/2wV, where V is normal(0,1) under Pay mation (4.4). Writing r(u) and W2 = I3ay{.l'B)y /(Pay - 1)3 =

(jB"[ay]l (Pay - 1)3 = y3/3B"[ay],

we get heuristically that Eay Ler

(u)r-(ay);

T(U) < yu]

= eyuk (ay)E''ay (ek(ay )"1/2WV; V < 01 Ir

eyur. (ay)

00 e-r(ay)"1'2"'x

J0

c2(x)

dx

00

1

/2 w

K(ay )u 1

1 e-zcp(z /( k(ay)u1 /2w)) dz

/O° _ 1 1 Je Z

dz

,c(ay)ul/2W p 2ir = eyu-(ay)

1 rc(ay ) 2,7ruw2

Inserting these estimates in (4.13), (4.11) follows. The proof of (4.12) is 0 entirely similar. The difficulties in making the proof precise is in part to show (4.13) rigorously, and in part that for the final calculation one needs a sharpened version of the CLT for t(u) (basically a local CLT with remainder term). Example 4.9 Assume that B(x) = e-ay. Then ic(a) = ,3(5/(S - a) - 1) - a, ,c'(a) _ /3a/(8 - a)2 - 1, and the equation ic'(a) = 1/y is easily seen to have

5. DIFFUSION APPROXIMATIONS

117

solution ay=5-

V

1

(the sign of the square root is - because the c.g.f. is undefined for a > 5). It follows that 5^y =5-ay =

/«y =f3+ay=l3+d-

1+1/y'

ay -ay =Qay -say =,3+5-2

V 1+^1/y /35 1+1/y -/3'

1+/351/y' sy

25 _ 251/2(1 + y)3/2

7 B ii[ay]

(5

0 3/2

- ay)3

and (4.11) gives the expression '31/4

('3

+ s _2 /

( - i )( v s

vc

L

'3 _

fl

, y)

)

a-''y"

51 /4(1 +1IY)3/4 \,/4

^y

for 1/i (u, yu) when y < 1/ic'('y) = p/1 - p. 0 Notes and references Theorem 4.8 is from Arfwedson [9]. A related result appears in Barndorff-Nielsen & Schmidli [59].

5 Diffusion approximations The idea behind the diffusion approximation is to first approximate the claim surplus process by a Brownian motion with drift by matching the two first moments, and next to note that such an approximation in particular implies that the first passage probabilities are close. The mathematical result behind is Donsker's theorem for a simple random walk {Sn}n=o,1,... in discrete time: if p = ES, is the drift and o, 2 = Var(Si ) the variance, then {

__ ,= (s, - tcp) Lo

{Wo ( t)}t>0 ,

c -a

00,

(5.1)


118

where {W( (t)} is Brownian motion with drift S and variance (diffusion constant) 1 (here 2 refers to weak convergence in D = D[0, oo)). It is fairly straightforward to translate Donsker's theorem into a parallel statement for continuous time random walks (Levy processes), of which a particular case is the claim surplus process (see the proof of Theorem 5.1 below). However, for the purpose of approximating ruin probabilities the centering around the mean (the tcp term in (5.1)) is inconvenient. We want an approximation of the claim surplus process itself, and this can be obtained under the assumption that the safety loading rt is small and positive. This is the regime of the diffusion approximation (note that this is just the same as for the heavy traffic approximation for infinite horizon ruin probabilities studied in III.7c). Mathematically, we shall represent this assumption on 77 by a family {StP) L

o

of claim surplus processes indexed by the premium rate p, such that the claim size distribution B and the Poisson rate a are the same for all p (i.e., St = EN` U= - tp), and consider the limit p j p, where p is the critical premium rate APBTheorem 5 .1 As p J, p, we have

{i!t s: ,,z } {W_1(t )}t>o (5.2) t>o where p = pp = p - p, a2 =/3µB2) Proof The first step is to note that { WC (St P) - tcpp) y = { WC (Sct) -pct) }

{Wo( t)}t>o

(5.3)

whenever c = cp f oo as p 1 p. Indeed , this is an easy consequence of (5.1) with S;a = Snp) and the inequalities Sn )C - p/c < St(p) < S((n+l)/ c + Pp/c, n/c < t < (n + 1)/c, cf. Lemma 111.1.3. Letting c = a2/pp, (5.3) takes the form LI S(P) { a2 to2/µ2 + t LI S (P) { a2 ta2/µ2

{W0(t)}, + {Wo(t ) - t} _ {W_1(t)} .

0

5. DIFFUSION APPROXIMATIONS

119

Now let Tp(u) = inf{t>0: S?)>u},

TS(u)=inf{t>0: WW(t)>u}.

It is well-known (Corollary XI.1.8 or [APQ] p. 263) that the distribution IG(•; ('; u) of r( (u) (often referred to as the inverse Gaussian distribution) is given by IG(x; C; u) =PIT( (u) < x) = 1 - 1 I 7= - ( ^ I + e2(

\\\

J

\ I - - f I

\

(5.4)

Note that IG(.; (; u) is defective when < 0. Corollary 5.2 As p j p, (ua2 To-2 op

\

IPI

^

-> IG ( T ;- 1 ;u).

p2

Proof Since f -4 SUP0 0. One has (6.1) IG(x; C; u) = IG(x/u2; (U;1) . The corrected diffusion approximation to be derived is 1+u2 (6.2) ,;; IGu+u2,-2'

(u,T)

where as ususal ry > 0 is the adjustment coefficient for the given risk process, i.e. the solution of r.(-y) = 0, and

= Q B"'['Yo

Si = QoEoU2

Eo U3 ],

S2 =

Bier [Yo]

3E0U2

3B"[Yo]

Write the initial reserve u for the given risk process as u = C/Oo ( note that C < 0) and, for brevity, write r = T(u), _ ^(u) = ST - u. The first step in the derivation is to note that

µ = k (0) = r-0, (00) ,., 9otc0" (0) = 0061 = ul, Vargo S, - Varo S1 = f30Eo U2 = S1, 0o to. Theorem 5.3 applies and yields 1061 U61 Stdlu2/CZdi

{W_1(t)}t>0 t>0

which easily leads to 1 StU2

{W( J(t)1t>0

{ u S1 t>o Y'(u, tu2 )

-i IG (t; (01; bl IG(t81; (; 1) •

Since

L

e-atIG (dt; (, u) =

e-uh(a , () where h (A, C) =

2A + (2 - C, (6.3)

this implies (take u = 1) Ego exp { -.S,7-(u)/u2}

e-h(A,()

The idea of the proof is to improve upon this by an O (u-1) term (in the following, means up to o(u-1) terms):

123

6. CORRECTED DIFFUSION APPROXIMATIONS

Proposition 6.1 As u -+ oo, 9o T 0 in such as way that C = Sou is fixed, it holds for any fixed A > 0 that

Ego exp { -Ab1rr(u)/u2} -- exp { -h(A, - 'yu /2)(1 + b2/u)}

+ Aug 1I

J . (6.5)

Once this is established , we get by formal Laplace transform inversion that

C

2 u,

bl

I

IG I t +2 ; - ry2 ;1 + u2 I

Indeed, the r.h.s. is the c.d.f. of a (defective) r.v. distributed as Z - 52/u where Z has distribution IG (•; - z ;1 + -629. u But the Laplace transform of such a r.v. is

Ee-azead2/++ Ee-az[1 + ab2/u]

where the last expression coincides with the r.h.s. of (6.5) according to (6.3). To arrive at (6.2 ), just replace t by Tb1/u2. Note, however , that whereas the proof of Proposition 6.1 below is exact, the formal Laplace transform inversion is heuristic: an additional argument would be required to infer that the remainder term in (6.2) is indeed o(u-1), The justification for the procedure is the wonderful numerical fit which has been found in numerical examples and which for a small or moderate safety loading 77 is by far the best amoung the various available approximations [note, however, that the saddlepoint approximation of Barndorff-Nielsen & Schmidli [59] is a serious competitor and is in fact preferable if 77 is large] . A numerical illustration is given in Fig. 6 . 1, which is based upon exponential claims with mean µB = 1. The solid line represents the exact value , calculated using numerical integration and Proposition 1.3, and the dotted line the corrected diffusion approximation (6.2). In ( 1) and (2), we have p =,3 = 0.7, in (3) and (4), p = 0.4. The initial reserve u has been selected such that the infinite horizon ruin probability b(u) is 10% in (1) and (3), 1% in (2) and (4).


124 0.1

0.111

W IU.TI

W(U.T)

a.aa1

0.08

0s-

. (Inc

0.OOIi

0.114

O.W21 0.02

I

90

120

160

2W A0

Z WT

40

80

120

160

100

240

280 T

111

111 00..T)

WI..T1

0.01 0.

OM 0.199 0.08

0.(061

0.07

0.0

0.00 0.05{

0.011

11.19)2 11

20

L1

60

T

20 i0

IM

T 1n0

Figure 6.1 It is seen that the numerical fit is extraordinary for p = 0.7. Note that the ordinary diffusion approximation requires p to be close to 1 and '0 (u) to be not too small, and all of the numerical studies the author knows of indicate that its fit at p = 0.7 or at values of Vi(u) like 1% is unsatisfying. Similarly, the fit at p = 0.4 may not be outstanding but nevertheless, it gives the right order of magnitude and the ordinary diffusion approximation hopelessly fails for this value of p. For further numerical illustrations, see Asmussen [12], BarndorffNielsen & Schmidli [59] and Asmussen & Hojgaard [34]. The proof of Proposition 6.1 proceeds in several steps.

h(X,() 0 p Lemma 6.2 e- EB ex p

(h7^)u S

A51 7-

(SAT 3 3

- u2 2u3 (e - ^) .

125

6. CORRECTED DIFFUSION APPROXIMATIONS Proof For a>0, 1 = PB(T < oo) = Eo0 exp

{(B

- 00)(u +C) -

'r

(,co (e) - r-0 (00)) }

Replacing B by 8/u and Bo by C/u yields e-(B-()

= E eo exp { (e - C)C/u - T (co (8/u) - ,co ((/u)) }

Let 8 = (2a + (2)1/2 = h(), () + C and note that 2

KO (0) = 102,3 EoU2 + 103OoEoU3 + " 2 6

+ a1b2

+ .... (6.6) ❑

Using d2 - C2 = 2), the result follows. U3 Ep Lemma 6 . 3 lim Eof (u) = EoC(oo) = a2 = u-roo 3EoU2 Proof By partial integration , the formulas

Po(C(0) > x) Po(C(co) > x)

1 °° Po(ST(o) > x) = EIU fIP0 (U>y)dy , 1 / Po(C(0) > y) dy EoC(0) x

imply k EDUk + 1 k Eo[(0)k+1

EoC(0) _ (k + 1)EoU' EoC(^) _ (k + 1) Eo£(0)

Lemma 6 .4 Ea, exp ue al }

1J

3

exP I-

CZ Z h (A, C) 1 1 + u2/ 111 + 2u[2),+

(2A + ()1/2 J 1

Proof It follows by a suitable variant of Stam's lemma (Proposition 4.4) that the r.h.s. in Lemma 6.2 behaves like C l Eeo eXp r _ ^81T 1 [1+h(AC) S - 61a2T (B3 - (3)

Sl u2 1 u 2u3

J

t _ aa1T l + e-h(A,1) h(A, () 62 Eeo exp u u2 J - 2u (B3

exp --i 3J . (6.7) - (3)Eea LauT 2 2


126

The last term is approximately (e 3

27.6

(3)

-

d h(A,S) d e-

()

62 - 2,\+ (2 (3 e

2u [ (2,\ +

- 2u [2A+

(2

/

1 2

3

- (2A + ()1/21

exp S -h(A, C)

( 1+ u2

The result follows by combining Lemma 6 . 2 and (6.7) and using e-h(a.() - e -h(aS)h (^^ 262

exp {_h(.x,() I 1 + u2 ) y . 0

The last step is to replace h(A, () by h(\, -yu/2). There are two reasons for

this : in this way, we get the correct asymptotic exponential decay parameter ^/ in the approximation ( 6.2) for O(u) (indeed , letting formally T -* oo yields 7/)(u) C'e-7u where C' = e-7a2); and the correction terms which need to be added cancels conveniently with some of the more complicated expressions in Lemma 6.4. l

Lemma 6 . 5 exp { _h(A) (1 + / y u J)) exp 1- h (A, --yu/2) 11+ 62 I} S 1 \\\ u/11 l

62 (3 2u 2A

Proof Use first (6.6) and 7co (Oo) = ico('y + Bo) to get 0 = 21 (^/2 + 2y90) + 1112 (_Y3 + 3_Y200 + 3y9o) + O(u-4),

a2

2 + 00 = - 2 (^/2 + 3y9o + 390) + O(u-3). Thus

-y = -290 + O (u-2), and inserting this and 9o 2

+90 62 0

+ O(u -3)

= S/u on the r.h.s. yields

2u2

+O(u -3).

Thus by Taylor expansion around ( = 90u, we get h(A, -yu/2) h(A, () - - 1 (y/2 + Oo)u , [2+ (2

127

7. HOW DOES RUIN OCCUR? exp { -h (x, -'yu/2)

pz^

(

i+

M

exP { -h (A, () I 1 + u 2 ) } S 1 - (i+ 62 exP{ -h(A, () (i+a

(i

Pt^

exP

{

+

U

)

)

2A + (2 - 1 (-y/2 + Oo)u

)}

S

-1

J

62(2

[2+ C2 2u 62

-h (A, () I 1 + u2 )I 2u

exp { -h(A, ()} 3

L

1

2A+C2_(2 exp { _h, ()} . 0

Proof of Proposition 6.1: Just insert Lemma 6.5 in Lemma 6.4.

❑

Notes and references Corrected diffusion approximations were introduced by Siegmund [345] in a discrete random walk setting, with the translation to risk processes being carried out by the author [12]; this case is in part simpler than the general random walk case because the ladder height distribution G+ can be found explicitly (as pBo) which avoids the numerical integration involving characteristic functions which was used in [345] to determine the constants. In Siegmund's book [346], the approach to the finite horizon case is in part different and uses local central limit theorems. The adaptation to risk theory has not been carried out. The corrected diffusion approximation was extended to the renewal model in Asmussen & Hojgaard [34], and to the Markov-modulated model of Chapter VI in Asmussen [16]; Fuh [148] considers the closely related case of discrete time Markov additive processes. Hogan [200] considered a variant of the corrected diffusion approximation which does not require exponential moments. His ideas were adapted by Asmussen & Binswanger [27] to derive approximations for the infinite horizon ruin probability 'i(u) when claims are heavy-tailed; the analogous analysis of finite horizon ruin probabilities O(u,T) has not been carried out and seems non-trivial.

7 How does ruin occur? We saw in Section 4 that given that ruin occurs, the 'typical' value (say in sense of the conditional mean) was umL, that is, the same as for the unconditional Lundberg process. We shall now generalize this question by asking what a sample path of the risk process looks like given it leads to ruin. The answer is similar: the process behaved as if it changed its whole distribution to FL, i.e.

128


changed its arrival rate from 0 to /3L and its claim size distribution from B to BL. Recall that .FT(u) is the stopping time o-algebra carrying all relevant information about r(u) and {St}o2 I(Tk < x)

1 - e-aLx,

M(tu) p(u) M(u) >2 I(Uk < x) BL(x).

Proof For the first assertion, take

I(Tk < x) - (1 - e-ALx) M(u)

k=1

The proof of the second is similar.

❑

Notes and references The results of the present section are part of a more general study carried out by the author [11]. A somewhat similar study was carried out in the queueing setting by Anantharam [6], who also treated the heavy-tailed case; however, the queueing results are of a somewhat different type because of the presence of reflection at 0. From a mathematical point of view, the subject treated in this section leads into the area of large deviations theory. This is currently a very active area of research, see further XI.3.


Chapter V

Renewal arrivals 1 Introduction The basic assumption of this chapter states that the arrival epochs O'1, D'2, .. . of the risk process form a renewal process: letting Tn = Qn - Q,,-1 (T1 = a1), the Tn are independent, with the same distribution A (say) for T2, T3,. . .. In the so-called zero-delayed case, the distribution Al of T1 is A as well. A different important possibility is Al to be the stationary delay distribution A° with density A(x)/µA. Then the arrival process is stationary which could be a reasonable assumption in many cases (for these and further basic facts from renewal theory, see A.1). We use much of the same notation as in Chapter I. Thus the premium rate is 1, the claim sizes U1, U2,... are i.i.d. with common distribution B, {St} is the claim surplus process given by I.(1.7), with

Nt = # {n:

Un 0. If the environment is Markovian with transition rate A from on to off and u from OFF to ON, the

133

1. INTRODUCTION

interarrival times become i.i.d. (an arrival occurs necessarily in the ON state, and then the whole process repeats itself). More precisely, A is phase-type (Example 1.2.4) with phase space {oN,oFF}, initial vector (1 0) and phase generator

11 However, in general the mechanism generating a renewal arrival process appears much harder to understand. Therefore, the relevance of the model has been questioned repeatedly, and the present author agrees to a large extent to this criticism. However, we feel it reasonable to present at least some basic features of the model, if for nothing else then for the mathematical elegance of the subject, the fundamental connections to the theory of queues and random walks, and for historical reasons. The following representation of the ruin probability will be a basic vehicle for studying the ruin probabilities: Proposition 1.4 The ruin probabilities for the zero-delayed case can be represented as 0(u) = P(M(d) > u) where M(d) = Max {Snd) : n = 0,1,...} with {S(d)} a discrete time random walk with increments distributed as the independent difference U - T between a claim U and an interarrival time T.

Proof The essence of the argument is that ruin can only occur at claim times.

{Snd^

The values of the claim surplus process just after claims has the same distri-

bution as

}• Since the claim surplus process {St} decreases in between

arrival times, we have max St = max ^d^. S o 0 is the unique solution of 1 = Ee'Y(u-T ) = S 8 A[- Y] (2.2) 7 and7r+=1Proof We can couple the renewal model { St} and the compound Poisson model {St*} with negative claims in such a way the interarrival times of { St*} are To ,,Ti = U1, T2 = U2..... Then B* = A, 3* = 6, and (2 . 1) means that 8(A[-ry] - 1) + ry = 0 which is easily seen to be the same as (2.2). Now the value of {St*} just before the nth claim is

To +T1* +...+Tn -U1 Un, and from Fig . 2.1 it is seen that ruin is equivalent to one of these values being > u. Hence M*

max {To + Ti + • • • + Tn - Ui - • • • - Un } = max St = t>0 n=0,1,... To + max {Ul+•••+Un-TI-.•.-Tn} n=0,1,...

To + M(d) in the notation of Proposition 1.4. Taking m.g.f.'s and noting that V)*(u) = P(M* > u) so that Theorem 2.1 means that M* is exponentially distributed with rate ry, we get Ee'M(d) = Ee°M*

Ee-To

_ -Y/(-- a) = 1 - 7r+ 7r

b/(S-a) + +,Y -a

I.e., the distribution of M(d) is a mixture of an atom at zero and an exponential distribution with rate parameter ry with weights 1 - u+ and lr+, respectively. ❑ Hence P(M(d) > u) _ 1r+e-'r". A variant of the last part of the proof, which has the advantage of avoiding transforms and leading up to the basic ideas of the study of the phase-type case in VIII.4 goes as follows: define 7r+ = P(M(d) > 0) and consider {St*} only when the process is at a maximum value. According to Theorem 2.1, the failure rate of this process is y. However, alternatively termination occurs at a jump time (having rate 8), with the probability that a particular jump time is not followed by any later maximum values being 1 - Tr+, and hence the failure rate

3. CHANGE OF MEASURE VIA EXPONENTIAL FAMILIES

137

is b(1- 7r+). Putting this equal to -y, we see that ry = 6(1- 7r+) and hence r+ = 1- -y/b. However, consider instead the failure rate of M(d) and decompose M(d) into ladder steps as in II.6, 111.2. The probability that the first ladder step is finite is 7r+. Furthermore, a ladder step is the overshoot of a claim size, hence exponential with rate b. Thus a ladder step terminates at rate b and is followed by one more with probability 7r+. Hence the failure rate of M(') is 6(1 - 7r+) = ry and hence P(M(d) > u) = P(M(d) > 0)e-7u = 7r+e-'r". 0

3 Change of measure via exponential families We shall discuss two points of view, the imbedded discrete time random walk and Markov additive processes.

3a The imbedded random walk The key steps have already been carried out in Corollary 11.4.5, which states that for a given a, the relevant exponential change of measure corresponds to changing the distribution F(d) of Y = U - T to

F(d)(x) = e-K^d^(«) ^x e"vFidi(dy) 00 K(d) (a) = log F(d) [a] = log B[a] + log A[-a] . It only remains to note that this change of measure can be achieved by changing the interarrival distribution A and the service time distribution B to Aad^, resp. B^d) where

Aad> (dt) = ^[ a] A(dt), Bads (dx) = - - B(dx). This follow since, letting P(d) refer to the renewal risk model with these changed parameters , we have

+,3 ] A[-a -)3] E«d'efl' = Bad> [a] A ad> [-Q] = B[a B[a] A[-a] F( d) [a +)3] F(d) [a]

= Fad) [^]

Letting M(u) = inf in = 1,2.... : S(d) > u}

CHAPTER V. RENEWAL ARRIVALS

138

be the number of claims leading to ruin and u

^(u)

= SM(u) - u

the overshoot , we get: Proposition 3.1 For any a such that k(d)' (a) > 0, O(u) = e-auE

(d)e-a{ (u)+M(u)K (d)(a) .

Consider now the Lundberg case, i .e. let 7 > 0 be the solution of r. (d) (7) _ 0. We have the following versions of Lundberg' s inequality and the CramerLundberg approximation: Theorem 3 .2 In the zero-delayed case, (a) '(u) < e-ryu, (b) V)(u) - Ce-"u where C = limu.,,. E(d)e -1' (u), provided the distribution F of U - T is non-lattice. Proof Proposition 3.1 implies Cu) = e-«uE ( 7d)e-«^(u) , and claim (a) follows immediately from this and e (u) > 0. For claim (b), just note that F7(d) is non-lattice when F is so . This is known to be sufficient for ^(O) ]p (d) ([APQ] Proposition 3.2 p. 187) and thereby for ^(u) to be non-lattice w.r.t. to converge in distribution since p(yd) (r(0) < oo) = 1 because of r (d)' (-y) > 0.

It should be noted that the computation of the Cramer-Lundberg constant C is much more complicated for the renewal case than for the compound Poisson case where C = (1 - p)/($B'[7] - 1) is explicit given 7. In fact, in the easiest non-exponential case where B is phase-type, the evaluation of C is at the same level of difficulty as the evaluation of i/i(u) in matrix-exponential form, cf. VIII.4. Corollary 3.3 For the delayed case Tl = s, ik.(u) - C8e-7u where Cs = Ce-78B[7]. For the stationary case, 00)(u) - C(°)e-ryu where C(O) = C0[7] - 1). 7µA


139

Proof Using (1.4), B(x) = o(e-7x) and dominated convergence, we get r u +8 e7uB(u + s) +

e"8(u)

--4 0 +

L

00

J

e7(v-8)e7(u+8-v), (u + s - y) B(dy)

0

For the stationary case, another use of dominated convergence combined with Ao[s] = (A[s] -1)/SPA yields

e7u,(°) (

00 u) e7u iP8(u) Ao(ds) -+ f 0

Ce-8B[7] Ao(ds)

= CB['Y](A[-y] - 1) = C(O). IPA 0 Of course, delayed version of Lundberg's inequality can be obtained in a

similar manner. The expressions are slightly more complicated and we omit the details.

3b Markov additive processes We take the Markov additive point of view of II.5. The underlying Markov process {Jt} for the Markov additive process {Xt} = {(Jt,St)} can be defined by taking Jt as the residual time until the next arrival. According to Remark 11.5.9, we look for a function h(s) and a k (both depending on a) such that Gh,,(s, 0) = tc(a)h(s), where G is the infinitesimal generator of {Xt} = {(Jt, St)} and h,, (s, y) = e°yh(s). Let P8f E8 refer to the case Jo = s. For s > 0, E8h0 (Jdt, Sdt) = h(s - dt ) e-adt = h ( s) - dt(ah ( s) + h'(s)) so that Gha ( s, 0) = -ah (s) - h'(s). Equating this to tch(s) and dividing by h(s) yields h'(s)/h(s) _ - a - /c,

h(s) = e-(a

+x( a))8

(normalizing by h(0) = 1). To determine boundary 0. Here

K,

(3.1)

we invoke the behavior at the

1 = h«(0,0 ) = Eo[ha ( Jdt,Sdt] = Ee'uh(T) means 1 = f ' e°^B(dy) f ' h( s)A(ds), 0 0


140

B[a]A[-a - rc(a)] =

1.

(3.2)

As in 11.5, we can now for each a define a new probability measure Pa;s

governing {(Jt, St)}too by letting the likelihood ratio Lt restricted to Yt = a((J,,,S„):0 0 or, equivalently, p < 1. Then: (a) as n -+ oo, Wn converges i n distribution to a random variable W, and we have P(W > u) = V, (u); (4.2) (b) as t -* oo, Vt converges in distribution to a random variable V, and we have P(V > u) = ?/iiol(u).

(4.3)

Proof Part (a) is contained in Theorem 11.4.1 and Corollary 11.4.2, but we shall present a slightly different proof via the duality result given in Theorem II.3.1. Let the T there be the random time UN. Then P(r(u) < T) is the probability z/iiNi (u) of ruin after at most N claims, and obviously z/'(u) = limN-,"^ Vi(N) (u). Also {Zt}o 0. (4.5)

Proof Letting n - oo in Proposition 4.1, we get W = (W + U* - T*)+, where U*,T* are independent and distributed as U1, resp . T1. Hence for x > 0, conditioning upon U* - T* = y yields K(x) = P ((W + U* - T*)+ < x) = P(W + U* - T* < x) fK(x_y)F(dy) (x > 0 is crucial for the second equality!).

❑

Now return to the Poisson case . Then the corresponding queue is M/G/1, and we get:

144


Corollary 4.6 For the M/G/1 queue with p < 1, the actual and the virtual waiting time have the same distribution in the steady state. That is, W v V. Proof For the Poisson case, the zero-delayed and the stationary renewal processes are identical. Hence '(u) = Ali(°)(u), implying P(W > u) = P(V > u) for all u. 0 Notes and references The GI/G/1 queue is a favourite of almost any queueing book (see e .g. Cohen [88] or [APQ] Ch. VIII), despite the fact that the extension from M/G/1 is of equally doubtful relevance as we argued in Section 1 to be the case in risk theory. Some early classical papers are Smith [350] and Lindley [246]. Note that (4.5) looks like the convolution equation K = F * K but is not the same (one would need (4.5) to hold for all x E R and not just x > 0). The equation (4.5) is in fact a homogeneous Wiener-Hopf equation, see e.g. Asmussen [24] and references there.

Chapter VI

Risk theory in a Markovian environment 1 Model and examples We assume that arrivals are not homogeneous in time but determined by a Markov process {Jt}0 u}, M = supt>o St. The ruin probabilities with initial environment i are '+ki(u) = pi(T(u ) < oo) = Pi (M > u), 145

Oj( u,T) = Pi (T(u) < T),

146

CHAPTER VI. MARKOVIAN ENVIRONMENT

where as usual Pi refers to the case Jo = i. Unless otherwise stated, we shall assume that pi = 1; this is no restriction when studying infinite horizon ruin probabilities, cf. the operational time argument given in Example 1.5 below. We let p Pi = /ji/AB;, P = E 7riPi, r^ = P (1.1) iEE

Then pi is the average amount of claims received per unit time when the environment is in state i, and p is the overall average amount of claims per unit time, cf. Proposition 1.11 below. An example of how such a mechanism could be relevant in risk theory follows. Example 1 .1 Consider car insurance, and assume that weather conditions play a major role for the occurence of accidents. For example, we could distinguish between normal and icy road conditions, leading to E having two states n, i and corresponding arrival intensities Qn, f3i and claim size distributions Bn, Bi; one expects that 3i > on and presumably also that Bn # Bi, meaning that accidents occuring during icy road conditions lead to claim amounts which are different from the normal ones. Cl The versatility of the model in terms of incorporating (or at least approximating) many phenomena which look very different or more complicated at a first sight goes in fact much further: Example 1.2 (ALTERNATING RENEWAL ENVIRONMENT) The model of Example 1.1 implicitly assumes that the sojourn times of the environment in the normal and the icy states are exponential, with rates Ani and Ain, respectively, which is clearly unrealistic. Thus, assume that the sojourn time in the icy state has a more general distribution A(i). According to Theorem A5.14, we can approximate A(i) with a phase-type distribution (cf. Example 1.2.4) with representation (E(i), a(i), T(=)), say. Assume similarly that the sojourn time in the normal state has distribution A(n) which we approximate with a phase-type distribution with representation (E('),a('),T(n)), say. Then the state space for the environment is the disjoint union of E(n) and E(i), and we have f3, = iii when j E E(i), /3 = Nn when j E E(n); in block-partitioned form, the intensity matrix is A OW-) T(i) T(n) t(n)a(i) where t(n) = -T(n)e, t(i) = -T(')e are the exit rates.

❑

1. MODEL AND EXAMPLES 147 Example 1 .3 Consider again the alternating renewal model for car insurance in Example 1.2, but assume now that the arrival intensity changes during the icy period, say it is larger initially. One way to model this would be to take A(') to be Coxian (cf. Example VIII.1.4) with states i1, ... , iq (visited in that order) and letfOil >...>,3i,. ❑ Example 1 . 4 (SEMI-MARKOVIAN ENVIRONMENT) Dependence between the length of an icy period and the following normal one (and vice versa) can be modelled by semi-Markov structure. This amounts to a family (A(")) ?CH Of sojourn time distributions, such that a sojourn time of type rt is followed by one of type c w.p. w,,, where W = (w,J017,tEH is a transition matrix. Approximating each A('?) by a phase-type distribution with representation (E('l),a(n),T(n)), say, the state space E for the environment is { ('q, i ) : n E H, i E E(n) }, and . T(1) +w11t(1)a(1) w12t (1)a(2) w21t(2)a(1)

T(2) +w22t( 2)a(2)

wg1t(9)a(1)

wg2t(9)a(2)

w1gt(1)a(9) w2gt ( 2)a(q)

A = ... T(9) +wggt(9)0, (9)

where q = CHI, t(n) = -T("i)e. The simplest model for the arrival intensity amounts to ,3,,,j = ,Q,, depending only on 77. In the car insurance example, one could for example have H = {i1, i8f n1, n8}, such that the icy period is of two types (long and short) each with their sojourn time distribution A('L), resp. A('^), and similarly for the normal period. Then for example wi,n, is the probability that a long icy period is followed by a short normal one. ❑ Example 1.5 (MARKOV-MODULATED PREMIUMS) Returning for a short while to the case of general premium rates pi depending on the environment i, let T

9(T) = f pi, dt, it = Je-l(t), St = SB-=(t). 0 Then (by standard operational time arguments) {St } is a risk process in a Markovian environment with unit premium rate, and 1/ii(u) = t/ii(u). Indeed, the parameters are îj = aid/pi, Qi = ,3i/pi. ❑ From now on, we assume again pi = 1 so that the claim surplus is Nt

St = ?Ui_t. 1


148

We now turn to some more mathematically oriented basic discussion. The key property for much of the analysis presented below is the following immediate observation: Proposition 1.6 The claim surplus process {St} of a risk process in a Markovian environment is a Markov additive process corresponding to the parameters µi = -pi, o = 0, vi(dx) = ,(3iBi(dx), qij = 0 in the notation of Chapter 11.5. In particular, the Markov additive structure will be used for exponential change of measure and thereby versions of Lundberg's inequality and the CramerLundberg approximation. Next we note a semi-Markov structure of the arrival process: Proposition 1.7 The Pi-distribution of T1 is phase-type with representation (ei,A - (Qi)diag)• More precisely, Pi (Ti E dx, JT1 = j) = Qj

• e;e(A-(Oi)d'sg)xe, . dx.

Proof The result immediately follows by noting that T1 is obtained as the lifelength of {Jt} killed at the time of the first arrival and that the exit rate obvi❑ ously is f3j in state j. A remark which is fundamental for much of the intuition on the model consists in noting that to each risk process in a Markovian environment, one can associate in a natural way a standard Poisson one by averaging over the environment. More precisely, we put )3* = E 7fi/3i, B* = 1 /^* Bi.

iEE iEE )3

These parameters are the ones which the statistician would estimate if he ignored the presence of Markov-modulation: Proposition 1.8 As t

oo,

Nt Nt a . )3*, N > 1(Ul < x) a4 B*(x). t l=1

Note that the last statement of the proposition just means that in the limit, the empirical distribution of the claims is B*. Note also that (as the proof shows) 7ri/3i//3* gives the proportion of the claims which are of type i (arrive in state i).

149

1. MODEL AND EXAMPLES

Proof Let ti = f1 I(JJ = i) ds be the time spent in state i up to time t and Nti) the number of claim arrivals in state i . Then it is standard that ti lt '4' iri as t -> oo. However , given {Jt}0 0 is arbitrary. That is, we may imagine that we have two types of claims such that the claim size distributions are E3 and E7. Claims of type E3 arrive with intensity 2 - s = o in state 1 and with intensity 1 - s = 1o in state 2, those of type E7 with intensity z s = 5 in state 1 and with intensity z . s 5 in state 2. Thus, since E3 is a more dangerous claim size distribution than E7 (the mean is larger and the tail is heavier), state 1 appears as more dangerous than state 2, and in fact P1 =

31AB1 =

P2 =

,31µB 2 =

9 3 2 (5

1 3

2 5

1 7

81 70 '

3

1

1

4

1

_ 19

2

5

3

5

7

70

Thus in state 1 where p, > 1, the company even suffers an average loss, and (at least when a is small such that state changes of the environment are infrequent), the paths of the surplus process will exhibit the type of behaviour in Fig. 1.1 with periods with positive drift alternating with periods with negative drift; the overall drift is negative since it = (2 2) so that p = 71P1 + 112P2 = 7. On Fig. 1.1, there are p = 2 background states of {Jt}, marked by thin, resp. thick, lines in the path of {St}. Computing the parameters of the averaged compound Poisson model, we first get that 3 (3* = 2.2 9 +2 2 = 3.

1. MODEL AND EXAMPLES

151

Figure 1.1 Thus, a fraction r, 01 /,3* = 3/4 of the claims occur in state 1 and the remaining fraction 1/4 in state 2. Hence

B* = 415E3+5E7/ +4

(

3

51E3 +5 E7) = 1E

+2E7.

That is, the averaged compound Poisson model is the same as in III.(3.1). 0 The definition (1.1) of the safety loading is (as for the renewal model in Chapter V) based upon an asymptotic consideration given by the following result: Proposition 1.11 (a ) ESt/t -* p - 1, t -* oo; (b) St/t -* p - 1 a.s., t -+ oo. Proof In the notation of Proposition 1.8, we have

E[St + t I

(t(i))iE EI

= E

t(i)OW =

iEE

t(i)Pi•

iEE

Taking expectations and using the well -known fact Et(i)/t -* 7ri yields (a). For (b), note first that EN Uk')/N a4' µgi. Hence (i)

Nt

St + t =

t iEE

Nti) 1 U(i) k' N(i) E t k=1

-4

1: 7ri Qi µs; = P. iEE


152

Corollary 1.12 If 77 < 0, then M = 00 a.s., and hence 1/ii(u ) = 1 for all i and u. If 77 > 0, then M < oo a.s., and ,0i(u) < 1 for all i and u. Proof The case 77 < 0 is trivial since then the a. s. limit p - 1 of St / t is > 0, and hence M = 00. The case 77 > 0 is similarly easy. Now let r) = 0, let some state i be fixed and define w=wl=inf{t >0:Jt_#i,Jt=i},

w2=inf {t>w1:Jt_#i,Jt=i},

X 1 =Sty,, X2 =SW2 -So,,, and so on. Then by standard Markov process formulas (e.g. [APQ], Theorem II.4.2(a) p. 38) Eiw1 = -1/ir,\ i and EiX1

Ei f 13 J, PB,, dt - Eiw o'o Eiw • E îfjµs; - 1 jEE

= (p - 1)Eiw = 0.

Now obviously the w,a form a renewal process , and hence wn /n a4. Eiw. Since the X„ are independent , with X2, X3, ... having the Pi-distribution of X, also + ... + Xn SWn ](1 a . EiX = 0. n n Thus {SWn l is a discrete time random walk with mean zero, and hence oscillates between -0o and oo so that also here M = oo. 0 Notes and references The Markov-modulated Poisson process has become very popular in queueing theory during the last decade, see the Notes to Section 7. In risk theory, some early studies are in Janssen & Reinhard [211], [302], [212], and a more comprehensive treatment in Asmussen [16]. The mainstream of the present chapter follows [16], with some important improvements being obtained in Asmussen [17] in the queueing setting and being implemented numerically in Asmussen & Rolski [43]. Statistical aspects are not treated here. See Meier [258] and Ryden [314], [315]. There seems still to be more to be done in this area. Proposition 1.1 and the Corollary are standard. The proof of Proposition 1.1(b) is essentially the same as the proof of the strong law of large numbers for cumulative processes, see [APQ] p. 136 or A.ld.

2 The ladder height distribution Our mathematical treatment of the ruin problem follows the model of Chapter III for the simple compound Poisson model, and involves a version of the

2. THE LADDER HEIGHT DISTRIBUTION

153

Pollaczeck-Khinchine formula (see Proposition 2.2(a) below ) where the ladder height distribution is evaluated by a time reversion argument. Define the ladder epoch T+ by T+ = inf It : St > 0} = r(0), let G+(i,j;A) = Pt(ST+ E A,Jr+ =j, T+ < oo) and let G+ be the measure-valued matrix with ijth element G+(i, j; •). The form of G+ turns out to be explicit (or at least computable), but is substantially more involved than for the compound Poisson case . However , by specializing results for general stationary risk processes (Theorem II . 6.5; see also Example II.6.4) we obtain the following result , which represents a nice simplified form of the ladder height distribution G+ when taking certain averages : starting {Jt} stationary, we get the same ladder height distribution as for the averaged compound Poisson model, cf. the definition of ,6*, B* in Section 1. Proposition 2.1 irG+(dy)e =,3*B *(y)dy. For measure-valued matrices, we define the convolution operation by the same rule as for multiplication of real-valued matrices, only with the product of real numbers replaced by convolution of measures. Thus, e.g., G+ is the matrix whose ijth element is

E G +(i, k; •) * G +(k,j; •)• kEE

Also, IIG+ II denotes the matrix with ijth element

IIG+(i, j; •) II = JG+(i,i;dx). Let further R denote the pre-T+ occupation kernel,

T R(i, j;A) =ZI(St E;EA,Jt=j)dt, and S (dx) the measure -valued diagonal matrix with /3 Bj(dx) as ith diagonal element. Proposition 2.2 (a) The distribution of M is given by 00

1 - a/i;(u) = Pi(M < u) = e' E G+ (u)(I - IIG +II)e.

(2.1)

n=0

0 (b) G+ (y, oo) = J

R(dx)S((y - x, oo)). That is, for i,j E E, ao 0

G+(i, j; (y, oo)) = f R(i, j; dx)/jBj(y - x). 00

(2.2)


154

Proof The probability that there are n proper ladder steps not exceeding x and (x)ej, and that the environment is j at the nth when we start from i is e ; G+ the probability that there are no further ladder steps starting from environment j is e^ ( I - IIG+II)e. From this (2.1) follows by summing over n and j. The ❑ proof of (2.2) is just the same as the proof of Lemma 11.6.3. To make Proposition 2.2 useful , we need as in Chapters II, III to bring R and G+ on a more explicit form . To this end , we need to invoke the time-reversed version {Jt } of {Jt} ; the intensity matrix A* has ijth element * 7r î3

7ri

and we have

Pi(JT = j) = 7rj P2(JT = i)-

(2.3)

7ri

We let {St*} be defined as {St}, only with {Jt} replaced by {Jt } (the /3i and Bi are the same ), and let further {my} be the E-valued process obtained by observing {Jt } only when {St*} is at a minimum value. That is, mx = j when for some (necessarily unique) t we have St = -x, JJ = j, St < S* for u < t; see Figure 2.1 for an illustration in the case of p = 2 environmental states of {Jt}, marked by thin, resp. thick, lines in the path of {St}.

0

----------------------------

x

Figure 2.1 The following observation is immediate: Proposition 2.3 When q > 0, {mx} is a non -terminating Markov process on E, hence uniquely specified by its intensity matrix Q (say).


155

Proposition 2.4 Q satisfies the non-linear matrix equation Q = W(Q) where 0 S(dx) eQx,

co(Q) = n* - (/3i)diag + T

and S(dx) is the diagonal matrix with the f3iBi(dx) on the diagonal. Furthermore, the sequence

{Q(n)}

Q(O) =

A*

defined by

- (/3i) diag,

Q( n+l) _ ^,

(

Q(

n))

converges monotonically to Q. Note that the integral in the definition of W(Q) is the matrix whose ith row is the ith row of e2Bi(dx).

_ 3 f

Proof The argument relies on an interpretation in terms of excursions. An excursion of {St*} above level -x starts at time t if St = -x, {S,*, } is a minimum value at v = t- and a jump (claim arrival) occurs at time t, and the excursion ends at time s = inf {v > t : S;, = -x}. If there are no jumps in (t, s], we say that the excursion has depth 0. Otherwise each jump at a minimum level during the excursion starts a subexcursion, and the excursion is said to have depth 1 if each of these subexcursions have depth 0. In general, we recursively define the depth of an excursion as 1 plus the maximal depth of a subexcursion. The definitions are illustrated on Fig. 2.2 where there are three excursions of depth 1,0,2. For example the excursion of depth 2 has one subexcursion which is of depth 1, corresponding to two subexcursions of depth 0.

0

mms1

-

Ô \ -T.

----------------------------

Figure 2.2


156

Let p=7) be the probability that an excursion starting from Jt = i has depth at most n and terminates at J8 = j and pij the probability that an excursion starting from Jt = i terminates at J8 = j. By considering minimum values within the excursion, it becomes clear that pij = r

[eQh]

0

ij Bi (dy) •

(2.4)

To show Q = cp(Q), we first compute qij for i $ j. Suppose mx = i. Then a jump to j (i. e., mx+dx = j) occurs in two ways , either due to a jump of {Jt } which occurs with intensity A= j, or through an arrival starting an excursion terminating with J, = j. It follows that qij = A;j +/3ipij. Similarly,

Fi(mh =i ) = 1 + =h-flh+Qihpii+o(h) implies qii = 'iii -/i +)3ipii. Writing out in matrix notation , Q = W(Q) follows. Now let {m ( n) } be {mx } killed at the first time i7n (say) a subexcursion of depth at least n occurs . It is clear that { mini } is a terminating Markov process and that { mio) } has subintensity matrix A* - (01)diag = Q. The proof of Q = W(Q) then immediately carries over to show that the subintensity matrix of {mil) } is cp (Q(o)) = Q(l). Similarly by induction , the subintensity matrix of {min+i ) } is cp (Q(n)) = Q(n +l) which implies that qgj +1) = \!- - Qi + )%pij)

t pij and insert (2.4). p1^) Define a further kernel U by f

Now just note that

U(i,j; A) = f

Pi(mx = j) dx eie4xej dx

❑

(2.5)

-A

A

(note that we use -A = {x : -x E Al on the r. h.s. of the definition to make U be concentrated on (-co, 0)). Theorem 2 .5 R(i, j; A) = L' U(j, i; A). 7rE

Proof We shall show that

Fi(Jt=j, StEA ,T+>t)

_ îF 7ri

(JJ =i,St EA,St <S*,u< t), (2.6)


157

from which the result immediately follows by integrating from 0 to oo w.r.t. dt. To this end, consider stationary versions of {Jt}, {Jt }. We may then assume Ju=Jt-u, S;,=StSt-.,0 1 instead of (as up to now) p < 1. Let M+ denote the matrix with ijth entry M+(i,j) = xG+(i,j;dx). 0 Lemma 2 .10 Assume p > 1. Then IIG+II is stochastic with invariant probability vector C+ (say) proportional to -irK, S+ _ -7rK/(-7rKe). Furthermore, -irKM+e = p - 1. Proof From p > 1 it follows that St a4' oo and hence IIG+II is stochastic. That -7rK = -e'Q'0 is non-zero and has nonnegative components follows since -Qe has the same property for p > 1. Thus the formula for C+ follows immediately by multiplying (2.8) by --7r, which yields -irKIIG+II = -irK. Further M+ = fdzfeS(( x+z oo)) dx f 00 dy fy eKx dx S((y, oo)) 0 0 m K-' f (eKy - I) S((y, oo))dy, 0 00

-7rKM+e = 7r f d y(I - eKy) S((y, oo))e = lr(/3ipB;) diage -

irII G +Ile

=p-1

160

CHAPTER VI. MARKOVIAN ENVIRONMENT ❑

(since IIG+II being stochastic implies IIG+ IIe = e).

Notes and references The exposition follows Asmussen [17] closely (the proof of Proposition 2.4 is different). The problem of computing G+ may be viewed as a special case of Wiener-Hopf factorization for continuous-time random walks with Markov-dependent increments (Markov additive processes ); the discrete-time case is surveyed in Asmussen [15] and references given there.

3 Change of measure via exponential families We first recall some notation and some results which were given in Chapter II in a more general Markov additive process context. Define Ft as the measurevalued matrix with ijth entry Ft(i, j; x) = Pi[St < x; Jt = j], and Ft[s] as the matrix with ijth entry Ft[i, j; s] = Ei[e8St; Jt = j] (thus, F[s] may be viewed as the matrix m.g.f. of Ft defined by entrywise integration). Define further K[a] = A + ((3i(Bi[a] - 1))

diag

- aI

(the matrix function K[a] is of course not related to the matrix K of the preceding section]. Then (Proposition 11.5.2):

Proposition 3.1 Ft[a] = etK[a] It follows from II.5 that K[a] has a simple and unique eigenvalue x(a) with maximal real part, such that the corresponding left and right eigenvectors VW, h(a) may be taken with strictly positive components. We shall use the normalization v(a)e = v(a)hi') = 1. Note that since K[0] = A, we have vi°> = 7r, h(°) = e. The function x(a) plays the role of an appropriate generalization of the c.g.f., see Theorem 11.5.7. Now consider some 9 such that all Bi[9] and hence ic(9), v(8), h(e) etc. are well-defined. The aim is to define governing parameters f3e;i, Be;i, Ae = 0!^1)i,jEE for a risk process, such that one can obtain suitable generalizations of the likelihood ratio identitites of Chapter II and thereby of Lundberg's inequality, the Cramer-Lundberg approximation etc. According to Theorem 11.5.11, the appropriate choice is e9x

09;i =13ihi[9], Bo;i (dx) = Bt[B]Bi(dx),

Ae = AB 1K[9]De - r.(9)I oB 1 ADe + (i3i(Bi[9] -

1))diag - (#c(9) + 9)I

161


where AB is the diagonal matrix with h(e) as ith diagonal element . That is,

hie) DEB) _ ^Y' Me) iii

i#j

+ /i(Bi[9] -1) - r. (9) - 0

i=j

We recall that it was shown in II . 5 that Ae is an intensity matrix, that Eie°St h(o) = etK(e)hEe ) and that { eest - t(e)h(9 ) } is a martingale. t>o We let Pe;i be the governing probability measure for a risk process with parameters ,69;i, B9; i, A9 and initial environment Jo = i. Recall that if PBT) is ]p(T) the restriction of Pe ;i to YT = a {(St, Jt) : t < T} and PET) = PoT), then and PET) are equivalent for T < oo. More generally, allowing T to be a stopping time, Theorem II.2.3 takes the following form: Proposition 3.2 Let r be any stopping time and let G E Pr, G C {r < oo}. Then

PiG = Po;iG = hE°) Ee;i lh

1 j,)

exp {-BST + -rrc(0 ) }; G .

J

(3.1)

Let F9;t[s], ice ( s) and pe be defined the same way as Ft[s], c (s) and p, only with the original risk process replaced by the one with changed parameters. Lemma 3.3 Fe;t [s] = e-t"(B) 0 -1 Ft[s + O]0. Proof By II.( 5.8).

❑

Lemma 3.4 rte ( s) = rc(s+B ) - rc(O). In particular, pe > 1 whenever ic'(s) > 0. Proof The first formula follows by Lemma 3.3 and the second from Pe = rc'' (s). Notes and references The exposition here and in the next two subsections (on likelihood ratio identities and Lundberg conjugation) follows Asmussen [16] closely (but is somewhat more self-contained).

3a Lundberg conjugation Since the definition of c( s) is a direct extension of the definition for the classical Poisson model, the Lundberg equation is r. (-y) = 0. We assume that a solution


162

y > 0 exists and use notation like PL;i instead of P7;i; also, for brevity we write h = h(7) and v = v(7). Substituting 0 = y, T = T(u), G = {T(u) < oo} in Proposition 3.2, letting ^(u) = S7(u) - u be the overshoot and noting that PL;i(T(u) < oo) = 1 by Lemma 3.4, we obtain: Corollary 3.5 V)i(u,

T) =

ioi(u)

h ie -7uE L,i

= h ie -7u E

e -7{(u) h =(u) e -WO

hj,(„)

; T(u) < T ,

.

(3 . 2) (3.3)

Noting that 6(u) > 0, (3.3) yields Corollary 3.6 (LUNDBERG'S INEQUALITY) Oi(u) -
hj u

e(1tL)G+(i,j;dy).

0 G+(i,7; dy) = aj f Bj(dy - x ) R(i,j; dx).

00

Thus

C+ > hj f"o e7(Y-u)G +(i,

j; dy)

jEE u 0

00

C+ ijhj f R(i, j; dx) f e7( v-u)Bj (dy - x) jEE

00

u

0 //^^ C+E,3jhj

// f 00

jEE

R(i, j; dx)

100 ery(&-u+x)Bj (dy) C . Bj(u Bj (u - x)

0

R(i, j; dx ) Bj (u - x ) = Gi(u),

E Qj f jEE

°O

x)


167 ❑

proving the upper inequality, and the proof of the lower one is similar.

Proof of Theorem 3.13 Let first cp=°)(u) = C_ hie-"u in Lemma 3.13. We claim by induction that then cpin) (u) > C_ hie-7u for all n, from which the lower inequality follows by letting n -* oo. Indeed, this is obvious if n = 0, and assuming it shown for n, we get

Wo n +1) (u) = ? 7 i ( U ) + E J u gyp;n) ( u - y)G+(i, j; dy)

(3.10)

jEE o

C_ 1 f hje7(y- u)G+(i, j; dy) jEE u

U +C_ hje7( y-u)G

+(i, j; dy)

jEE""

C_e-7u 57 O+[i, j; y]hj = C_ e-7uhi,

(3.11)

jEE

estimating the first term in (3.10 ) by Lemma 3 . 13 and the second by the induction hypothesis , and using Lemma 3 . 9 for the last equality in (3.11). ❑ The proof of the upper inequality is similar , taking cps°) (u) = 0. Here is an estimate of the rate of convergence of the finite horizon ruin probabilities 'i (u, T) = Pi (7- (u) < T ) to 0i (u) which is different from Theorem 3.10: Theorem 3 . 14 Let yo > 0 be the solution of 'c'(yo ) = 0, let C+(yo) be as in (3.8) with -y replaced by yo and hi by h=7o ), and let 8 = e'(70). Then

0
u) = Pi(ST u), 0"(u) = P(M" > u)) Now consider the risk process in a Markovian environment and define i' (u) _ >iEE irioi(u). It was long conjectured that -0* Vi,,, where o*(u) is the ruin probability for the averaged compound Poisson model defined in Section 1 and ,0,, is the one for the Markov-modulated one in the stationary case (the distribution of J0 is 7r). The motivation that such a result should be true came in part from numerical studies, in part from the folklore principle that any added stochastic variation increases the risk, and finally in part from queueing theory, where it has been observed repeatedly that Markov-modulation increases waiting times and in fact some partial results had been obtained. The results to be presented show that quite often this is so, but that in general the picture is more diverse. The conditions which play a role in the following are: ,31:5)32 ... < ,3p.

(4.2)

Bl s=1

7r=

+)

b

f0 u

(u -

x)Bt (x) /pB,

dx (4.7)

7ri,3iBi(x)YPi(u - x)dx _ /3*B*(u) + f u / ^ t=1 ^j

B* ( ) > 3 *+ f

Tri/iBd(x) . E 7r i Wi(u - x) dx u o i =1 i=1

Q*B*(u)+,3* f uB(x) z/^,r(u -x)dx. 0

(4.8)

(4.9) (4.10)

Here (4.9) follows by considering the increasing functions 3iBi (x) and Oi (u - x) of i and using Lemma 4.2. Comparing (4.10) and (4.6), it follows by a standard


170

argument from renewal theory that tk.., dominates the solution 0* to the renewal equation (4.6). ❑ Here is a counterexample showing that the inequality tp* (u) < V),, (u) is not in general true: Proposition 4.4 Assume that ,3µi < 1 for all i, that P

P

/^2 /^ î/ji pBi < 1il3i i=1

i=1

/^ 1r1NiµBi

(4.11)

i=1`

and that A has the form eAo for some fixed intensity matrix A0. Then i/i*(u) < ,0,r (u ) fails for all sufficiently small e > 0. Proof Since 0..(0) = V,* (0), it is sufficient to show that 0',(0) < b *'(0) for e small enough. Using (4.6), (4.8) we get P

'*' (0) =

P

-3* + /3*1* (0) _ > lri'3qqi • E 7i/ipBi - /3*, i=1

i=1

7'r(0) _ EFioiwi(0) - 0*•

i=1

But it is intuitively clear (see Theorem 3.2.1 of [145] for a formal proof) that z/ii(u) converges to the ruin probability for the compound Poisson model with parameters ,3i, Bi as e J. 0. For u = 0, this ruin probability is /3iPBi, and from this the claim follows. ❑ To see that Proposition 4.4 is not vacuous, let = ( 1/2 1/2 ) , 01 = 10-3, Q2 = 1, µB, = 102, µB2 = 10-4. Then the l.h.s. of (4.11) is of order 10-4 and the r.h.s. of order 10-1. Notes and references The results are from Asmussen, Frey, Rolski & Schmidt [32]. As is seen, they are at present not quite complete. What is missing in relation to Theorem 4.1 and Proposition 4.4 is the understanding of whether the stochastic monotonicity condition (4.4) is essential (the present author conjectures it is).

4b Ordering of adjustment coefficients Despite the fact that V)* (u) < *,, (u) may fail for some u, it will hold for all sufficiently large u, except possibly for a very special situation . Recall that

4. COMPARISONS WITH THE COMPOUND POISSON MODEL

171

the adjustment coefficient for the Markov-modulated model is defined as the solution -y > 0 of ic(-y) = 0 where c(a) is the eigenvalue with maximal real part of the matrix A + (rci(a))diag where rci(a) = ai(Bi[a] - 1) - a. The adjustment coefficient -y* for the averaged compound Poisson model is the solution > 0 of rc*(ry*) = 0 where rc*(a) _ 13*(B*[a] - 1) - a = E irirci(a). (4.12) iEE

Theorem 4.5 y < ry*, with strict inequality unless rci (y*) does not depend on iEE. Lemma 4.6 Let (di)iEE be a given set of constants satisfying EiEE iribi = 0 and define A(a) as the eigenvalue with maximal real part of the matrix A + a(bi)diag• Then )t(a) > 0, with strict inequality unless a = 0 or bi = 0 for all i E E. Proof Define

X= f &ids. Then {(Jt, Xt)} is a Markov additive process (a so-called Markovian fluid model, cf. e.g. Asmussen [20]) as discussed in 11.5, and by Proposition II.5.2 we have (Ei[e"X'; Jt = i])' EE = vA+n(6.)a..g.

Further (see Corollary 11.5.7) )i is convex with

A'(0) = lim

EXt

t-ioo t

=

70i

= 0, (4.13)

iEE

A„(O) iioo varXt t t

(4.14)

By convexity, (4.13) implies A(a) > 0 for all a.

Now we can view {Xt} as a cumulative process (see A.ld) with generic cycle w = inf{t>0: Jt_54 k,Jt=kI A (the return time of k) where k E E is some arbitrary but fixed state. It is clear that the distribution of X,, is non-degenerate unless bi does not depend on i E E, which in view of EiEE 1ibi = 0 is only possible if Si = 0 for all i E E. Hence if 5i 54 0 for some i E E, it follows by Proposition A1.4(b) that the limit in (4.14) is non-zero so that A"(0) > 0. This implies that A is strictly convex, in particular .(a) > 0 for all a 0 0. 0

172


Proof of Theorem 4.5. Let bi = rci(y*), a = 1 in Lemma 4.6. Then > risi = 0 because of (4.12) and rc*(y*) = 0. Further a(1) = rc(y*) by definition of A(.) and rc (•). Hence rc (y*) > 0. Since ic is convex with rc'(0) < 0 , this implies that the solution y > 0 of K(y) = 0 must satisfy y < y*. If rci(y* ) is not a constant function of i E E, we get rc (y*) > 0 which in a similar manner implies that ❑ y < y*. Notes and references Theorem 4.5 is from Asmussen & O'Cinneide [40], improving upon more incomplete results from Asmussen, Frey, Rolski & Schmidt [32].

4c Sensitivity estimates for the adjustment coefficient Now assume that the intensity matrix for the environment is Ae = Ao/ e, whereas the ,Qi and Bi are fixed . The corresponding adjustment coefficient is denoted by ry(e). Thus -y(e) -* y* as e 10, and our aim is to compute the sensitivity

ay

ae

E=O

A dual result deals with the limit a -4 oo. Here we put a = 1/e, note that y(a) -+ mins=1,...,p yi and compute

8y 8a

a=0

In both cases, the basic equation is (A + (rci(y))diag)h = 0, where A, y, h depend on the parameter (e or a). In the case of e, multiply the basic equation by a to obtain 0 = (A0 + e(r£i(y))diag)h,

0 = ((ri(-Y))diag + ery (4{('Y))diag)h + (A0 + e(?i'Y))diag)h'. (4.15)

Normalizing h by 7rh = 0, we have 7rh' = 0, h(0) = e. Hence letting e = 0 in (4.15) yields

0 = (Ii(y*)) diage + Aoh'(0) = (rci('Y*)) diage + (Ao - eir)h'(0), h'(0) = -(Ao - e7r)-1 (Ici(Y*))diage. (4.16) Differentiating (4.15) once more and letting e = 0 we get

5. THE MARKOVIAN ARRIVAL PROCESS

173

0 = 27'(0)(r-i(`Y *)) diage + 2(ci('Y* )) diag h' (0) + Aoh" (0)

, (4.17)

0 = 27'(0)p+27r(rs;i(7' *))diagh'(0),

(4.18)

multiplying (4.17) by 7r to the left to get (4.18). Inserting (4.16) yields Proposition 4.7 8ry

aE

= 1 7r(ci ('Y*))diag ( Ao -e7r)-1(Xi(-Y*))diage *=0 P

Now turn to the case of a. We assume that 0 < -y < 7i,

i = 2, ... , p. (4.19)

Then 'y -^ ryl as a ^ 0 and we may take h(0) = el (the first unit vector). We get

0 = (aAo + ( lc&Y))diag)h, 0 = (Ao + ry'(ii(-Y)) diag )h + (aAo + (Ki(7'))diag)h'. (4.20) Letting a = 0 in (4.20) and multiplying by el to the left we get 0 = All + 7'(0)rci (0) + 0 (here we used icl (ry(0)) = 0 to infer that the first component of K[7(0)]h'( 0) is 0), and we have proved:

All

Proposition 4.8 If (4.19) holds, then 8a a=o

rci (0)

Notes and references The results are from Asmussen, Frey, Rolski & Schmidt [32]. The analogue of Proposition 4.8 when ryi < 0 for some i is open.

5 The Markovian arrival process We shall here briefly survey an extension of the model, which has recently received much attention in the queueing literature, and may have some relevance in risk theory as well (though this still remains to be implemented). The additional feature of the model is the following: • Certain transitions of {Jt} from state i to state j are accompanied by a claim with distribution Bid; the intensity for such a transition (referred to as marked in the following) is denoted by Aii l and the remaining intensity


174

f o r a transition i -+ j by A

. For i = j, we use the convention that

a1i = f3i where 3i is the Poisson rate in state i, that Bii = Bi , and that are determined by A = A(l ) +A(2) where A is the intensity matrix the governing {Jt}. Thus , the Markov-modulated compound Poisson model considered sofar corresponds to A(l) = (,6i ) diag, A(1) = A - (13i )diag, Bii = Bi ; the definition of Bij is redundant for i i4 j. Note that the case that 0 < qij < 1, where qij is the probability that a transition i -* j is accompanied by a claim with distribution, is neither 0 or 1 is covered by letting Bij have an atom of size qij at 0. Again , the claim surplus is a Markov additive process (cf. II.4). The extension of the model can also be motivated via Markov additive processes: if {Nt} is the counting process of a point process, then {Nt} is a Markov additive process if and only if it corresponds to an arrival mechanism of the type just considered. Here are some main examples: Example 5 .1 (PHASE-TYPE RENEWAL ARRIVALS) Consider a risk process where the claim sizes are i.i.d. with common distribution B, but the point process of arrivals is not Poisson but renewal with interclaim times having common distribution A of phase-type with representation (v, T). In the above setting, we may let {Jt} represent the phase processes of the individual interarrival times glued together (see further VIII.2 for details), and the marked transitions are then the ones corresponding to arrivals. This is the only way in which arrivals can occur, and thus

1i = 0, A(l) = T, A(l) = tv, Bij = B; the definition of Bi is redundant because of f3i = 0.

❑

Example 5 .2 (SUPERPOSITIONS) A nice feature of the set-up is that it is closed under superposition of independent arrival streams . Indeed, let { Jt 1) }, j(2)

} be two independent environmental processes and let E(k), A(1'k) A(2 k1),

B;^) etc. refer to

{ Jt k)

}. We then let (see the Appendix for the Kronecker

notation)

E = E(1) x E(2), Jt = (Jtl), Jt2)) (2;2) A(1) = A(' 1) ® A(1;2), A ( 2) = A (2`1 ) ® A,

5. THE MARKOVIAN ARRIVAL PROCESS

175

-

Bij,kj = Bik) B13 4k = Bak)

(the definition of the remaining Bij,kl is redundant). In this way we can model, e.g., superpositions of renewal processes. ❑ Example 5 .3 (AN INDIVIDUAL MODEL) In contrast to the collective assumptions (which underly most of the topics treated sofar in this book and lead to Poisson arrivals), assume that there is a finite number N of policies. Assume further that the ith policy leads to a claim having distribution Ci after a time which is exponential, with rate ai, say, and that the policy then expires. This means that the environmental states are of the form i1i2 • • • iN with il, i2i ... E 10, 11, where ik = 0 means that the kth policy has not yet expired and ik = 1 that it has expired. Thus, claims occur only at state transitions for the environment so that AN2... iN,1i2 ... iN = all BOi2... iN,1i2...iN C17 AilO...iN,iil...iN = a2, Bilo...iN,iil...iN = C27

All other off-diagonal elements of A are zero so that all other Bii are redundant. Similarly, all Al i2...iN are zero and all Bi are redundant. Easy modifications apply to allow for • the time until expiration of the kth policy is general phase-type rather than exponential;

• upon a claim, the kth policy enters a recovering state, possibly having a general phase-type sojourn time, after which it starts afresh.

Example 5 .4 (A SINGLE LIFE INSURANCE POLICY ) Consider the life insurance of a single policy holder which can be in one of several states, E = { WORKING, RETIRED, MARRIED, DIVORCED, WIDOWED, INVALIDIZED, DEAD etc.}. The individual pays at rate pi when in state i and receives an amount having distribution Bij when his/her state changes from i to j.

❑

Notes and references The point process of arrivals was studied in detail by Neuts [267] and is often referred to in the queueing literature as Neuts ' versatile point process , or, more recently, as the Markovian arrival process ( MAP). However , the idea of arrivals at transition epochs can be found in Hermann [193] and Rudemo [313]. The versatility of the set-up is even greater than for the Markov-modulated model. In fact , Hermann [193 ] and Asmussen & Koole [37] showed that in some appropriate


176

sense any arrival stream to a risk process can be approximated by a model of the type studied in this section : any marked point process is the weak limit of a sequence of such models . For the Markov-modulated model, one limitation for approximation purposes is the inequality Var Nt > ENt which needs not hold for all arrival streams. Some main queueing references using the MAP are Ramaswami [298], Sengupta [336], Lucantoni [248], Lucantoni et at. [248], Neuts [271] and Asmussen & Perry [42].

6 Risk theory in a periodic environment 6a The model We assume as in the previous part of the chapter that the arrival mechanism has a certain time-inhomogeneity, but now exhibiting (deterministic) periodic fluctuations rather than (random ) Markovian ones. Without loss of generality, let the period be 1; for s E E = [0, 1), we talk of s as the 'time of the year'. The basic assumptions are as follows: • The arrival intensity at time t of the year is 3(t) for a certain function /3(t), 0 < t < 1; • Claims arriving at time t of the year have distribution B(t); • The premium rate at time t of the year is p(t). By periodic extension, we may assume that the functions /3(t), p(t) and B(t) are defined also for t t [0, 1). Obviously, one needs to assume also (as a minimum) that they are measurable in t; from an application point of view, continuity would hold in presumably all reasonable examples. We denote throughout the initial season by s and by P(8) the corresponding governing probability measure for the risk process. Thus at time t the premium rate is p(s + t), a claim arrives with rate /3(s + t) and is distributed according to B(8+0 . Let 1

1

/3* _ f /3(t) dt, B* =

J

t

1

B(t) ((*) dt, p * = 0 p(t) dt. )3

J

(6.1)

Then the average arrival rate is /3* and the safety loading rt is 77 = (p* - p)/p, where

f

i f00 p = f /3(v) dv xB(°) (dx) _ ,3*µs • 0 0

(6.2)

Note that p is the average net claim amount per unit time and µ* = p//3* the average mean claim size.

6. RISK THEORY IN A PERIODIC ENVIRONMENT 177 In a similar manner as in Proposition 1.8, one may think of the standard compound Poisson model with parameters 3*, B*, p* as an averaged version of the periodic model, or, equivalently, of the periodic model as arising from the compound Poisson model by adding some extra variability. Many of the results given below indicate that the averaged and the periodic model share a number of main features. In particular, it turns out that they have the same adjustment coefficient. In contrast, for Markov-modulated model typically the adjustment coefficient is larger than for the averaged model (cf. Section 4b), in agreement with the general principle of added variation increasing the risk (cf. the discussion in 111.9). The behaviour of the periodic model needs not to be seen as a violation of this principle, since the added variation is deterministic, not random. Example 6 .1 As an example to be used for numerical illustration throughout this section, let ,3(t) = 3A(1 + sin 27rt), p(t) = A and let B(t) be a mixture of two exponential distributions with intensities 3 and 7 and weights w(t) _ (1 +cos27rt)/2 and 1 - w(t), respectively. It is easily seen that ,3* = 3A, p* = A whereas B* is a mixture of exponential distributions with intensities 3 and 7 and weights 1/2 for each (1/2 = ff w(t)dt = f o (1- w(t)) dt). Thus, the average compound Poisson model is the same as in III.(3.1) and Example 1.10, and we recall from there that the ruin probability is

*(u) _ 3245e-u1 + 35e-6u. (6.3) Note that A enters just as a scaling factor of the time axis, and thus the averaged standard compound Poisson models have the same risk for all A. In contrast, we shall see that for the periodic model increasing A increases the effect of the periodic fluctuations. ❑ Remark 6 .2 Define T

6(T) = p(t ) dt, St = Se-I(t). 0 Then (by standard operational time arguments )

{St}

is a periodic risk process

with unit premium rate and the same infinite horizon ruin probabilities. We ❑ assume in the rest of this section that p(t) - 1. The arrival process {Nt}t>0 is a time-inhomogeneous Poisson process with intensity function {/3(s + t)}t>0 . The claim surplus process {St } two is defined in the obvious way as St = ^N° Ui - t.

Thus , the conditional distribution

CHAPTER VL MARKOVIAN ENVIRONMENT

178

of U; given that the ith claim occurs at time t is B(8+t). As usual, r(u) _ inf It > 0 : St > u} is the time to ruin , and the ruin probabilities are

0(8) (U) = P(s )(r(u) < 00), 0 (5)(u,T) = P(8)(r(u) o = h(s; 9)

Lo

is a P ( 8)-martingale with mean one. Proof In the Markov additive sense of (6.4), we can write

Jt;9) east-t,t.(e) Let = h( h(Jo; 0) P(8)-a.s. so that obviously {Lo,t} is a multiplicative functional for the Markov process { (Jt, St)} . According to Remark 11.2.6 , it then suffices to note that E(8)Le,t = 1 by Theorem 6.3. ❑ Remark 6.5 The formula for h(s) = h(s; a) as well as the fact that rc = k` (a) is the correct exponential growth rate of Eeast can be derived via Remark 11.5.9 as follows. With g the infinitesimal generator of {Xt} = {(Jt, St)} and


180

ha(s,y) = eayh(s), the requirement is cha(i,0) = Kh(s). However, as above E (s) ha(Jdt, Sdt) = h(s + dt) e-adt (1 -,(3(s)dt) +,3(s)dt • B(s)[a]h(s) = gha(s, 0) =

h(s) + dt {-ah(s) -,3(s)h(s) + h'(s) +,3(s)ks)[a]h(s)} -ah(s) -13(s)h(s) + h'(s) +,3(s)B(s) [a]h(s).

Equating this to rch (s) and dividing by h(s) yields h(s ) = h(s)

=

a + ,6 ( s ) exp { -

0( s )&s) [a] + tc ,

J s [,3(v)( Bi"i [a] - 1) - a - tc] dv}

(normalizing by h(0) = 1). That rc = is*(a) then follows by noting that h(1) _ ❑ h(0) by periodicity. For each 0 satisfying the conditions of Corollary 6.4, it follows by Theorem II.2.5 that we can define a new Markov process {(Jt, St)} with governing probability measures Fes), say, such that for any s and T < oo, the restrictions of Plsi and Pest to Ft are equivalent with likelihood ratio Le,T. Proposition 6.6 The P(s), 0 < s < 1, correspond to a new periodic risk model with parameters

ex ,60(t) = a(t)B(t)[0], Bet)(dx) = ^ B(t ) (dx). Proof (i) Check that m.g.f. of St is as for the asserted periodic risk model, cf. Proposition 6.3; (ii) use Markov-modulated approximations (Section 6c); ( iii) use approximations with piecewiese constant /3(s), B(s); (iv) finally, see [44] for 11 a formal proof. Now define 'y as the positive solution of the Lundberg equation for the averaged model. That is, -y solves n* (-y) = 0. When a = y, we put for short h(s) = h(s;'y). A further important constant is the value -yo (located in (0, ry)) at which n* (a) attains its minimum. That is, -yo is determined by 0 = k* (70) = QB*, [70] - 1. Lemma 6 .7 When a > -yo, P(s) (T(u) < oo) = 1 for all u > 0.

6. RISK THEORY IN A PERIODIC ENVIRONMENT

181

Proof According to (6.2), the mean number of claims per unit time is p«

dv J ' xe«xB (°) (dx) Jo 1,6(v) ✓✓ o r^ xe«xB'(dx) = Q'B' [ a] = ^' J 0

=

= ^c"'(a) + 1, ❑

which is > 1 by convexity.

The relevant likelihood ratio representation of the ruin probabilities now follows immediately from Corollary 11.2.4. Here and in the following, ^(u) = ST(u) - u is the overshoot and 9(u) = (T(u) + s) mod 1 the season at the time of ruin.

Corollary 6.8 The ruin probabilities can be computed as (u)+T(u)k'(a)

^/i(8) (u, T)

= h(s; a) e-«uE(8 ) e «^ ; T(u) < (6.7) h(B(u); a) TI

0(')(u) = h(s; a)e-«uE (a

h(9(u); a) a > ry0 (6.8)

iP(s) (u) = h( s)e-7uE(` ) h(O(u))

(6.9)

To obtain the Cramer-Lundberg approximation from Corollary 3.1, we need the following auxiliary result . The proof involves machinery from the ergodic theory of Markov chains on a general state space, which is not used elsewhere in the book, and we refer to [44]. Lemma 6 .9 Assume that there exist open intervals I C [0, 1), J C R+ such that the B(8), s E I, have components with densities b(8)(x) satisfying

inf sEI, xEJ

0 (s)b(8)(x) > 0.

(6.10)

Then for each a, the Markov process {(^(u),9(u))} u>0, considered with governing probability measures { E(8) }E[ , has a unique stationary distribution, say s0,1)

the distribution of (l: (oo), B(oo)), and no matter what is the initial season s, Wu), 0(u)) -* (b(oo), e(cc)) Letting u --> oo in (6.9) and noting that weak convergence entails convergence of E f (^(u), 9(u)) for any bounded continuous function (e.g. f (x, q) = e-ryx/h(q)), we get:


182

Theorem 6.10 Under the condition (6.10) of Lemma 3.1, Vi(8) (u) - Ch(s)e-ry", where

u -+ oo, (6.11)

e- -W- ) C = E1 h(B(oo))

Note that ( 6.11) gives an interpretation of h(s ) as a measure of how the risks of different initial seasons s vary. For our basic Example 6 . 1, elementary calculus yields h(s) = exp

{ A C 2^

cos 2irs -

4^ sin 21rs + 11 cos 41rs - 16,ir) }

Plots of h for different values of A are given in Fig. 6.1, illustrating that the effect of seasonality increases with A.

A=1/4 A=1

A=4

0

Figure 6.1 In contrast to h, it does not seem within the range of our methods to compute C explicitly, which may provide one among many motivations for the Markovmodulated approximation procedure to be considered in Section 6c. Among other things, this provides an algorithm for computing C as a limit. At this stage , Theorem 6 . 10 shows that certainly ry is the correct Lundberg exponent. Noting that ^(u) > 0 in ( 6.9), we obtain immediately the following version of Lundberg ' s inequality which is a direct parallel of the result given in Corollary 3.6 for the Markov-modulated model: Theorem 6 . 11 7/'O (u) < C+°)h(s) e-ry", where C(o) = 1

+ info < t -Y and ay > ry, r.(ay) > 0 when y < 1/ic' (7), whereas ay < -y, #c( ay) < 0 when y > 1/tc'('y). Theorem 6 .12 Let 00)(y)

1 Then info < t 0, letting x0 = 0 (i.e. xk = kh) and writing 9k = 9(xk ), Kk,e = K(xk, xe), this leads to h 9N = hN + 2 {KN,09o+KN,N9N}+h{KN,191+'''+KN,N-19N-1},

i.e.

hN+ ZKN ,ogo +h{KN,lgl+•••+KN,N-19N-1} 1 - ZKNN

9 N=

(

1.11

)

In the case of (1.8), the unknown yo is involved. However, (1.11) is easily seen to be linear in yo. One therefore first makes a trial solution g*(x) corresponding to yo = 1, i.e. h(x) = h*(x) = (3B(x)/p(x), and computes f o' g*(x)dx numerically (by truncation and using the gk). Then g(x) = yog*(x), and IIGII = 1 then yields f 00 g*(x)dx (1.12) 1= 1+ 'Yo from which yo and hence g(x) and z/'(u) can be computed. ❑

la Two-step premium functions We now assume the premium function to be constant in two levels as in Example 1.1, p(r) _ J 1'1 r < v P2 r > v.

(1.13)

We may think of the risk reserve process Rt as pieced together of two risk reserve processes R' and Rt with constant premiums p1, P2, such that Rt coincide with Rt under level v and with above level v. For an example of a sample path, Rt see Fig. 1.1.

Rt

V

Figure 1.1

195

1. INTRODUCTION

Proposition 1.10 Let V)' (u) denote the ruin probability of {Rt}, define a = inf It > 0 : Rt < v}, let pi ( u) be the probability of ruin between a and the next upcrossing of v (including ruin possibly at a), and let q(u) = 1 - V" (u)

0 < u < v. (1.14)

Then 0 v then the probability of ruin is the sum of being ruined between a and the next upcrossing of v which is pl (u), and the probability of ruin given the process hits v before (- oo, 0) again after a, (Pu(a < oo ) - p1(u))''(v) = (Vi2(u - v) - p1 (u))''(v)• This yields the expression for u > v, and the one for u = v then immediately follows. ❑ Example 1 .11 Assume that B is exponential, B(x) = e-62. Then 01 (u)

_

0 e -.yiu ,,2 (u) = )3 e -72u p1S P2S

where ry; = S - ,Q/p;, so that

q

-

1 - ~ e-ry1u p1S 1 - Q e-ryly P1S

Furthermore , for u > v P(a < oo ) = 02(u - v) and the conditional distribution of v - Ro given a < oo is exponential with rate S . If v - Ro < 0, ruin occurs at time a . If v - R, = x E [0, v], the probability of ruin before the next upcrossing of v is 1 - q(v - x). Hence

196

CHAPTER VII. RESERVE-DEPENDENT PREMIUMS

( pi(u) _ 02 ( u - v){ a-av + J (1 - q(v - x))be-dxdx 0 I 1- a e- 7i(v -x)

P2,ee- 7z(u-v)

1

_

P16 0 1 - a e-7iv P16

1 - e -6V Qbe-72(u-v) P2

Se-6xdx

a

e -71v (e(71 -6)v - 1)

1 - p1(71 - b)

1

p2be- 7z(u-v) 1 _

-

Ie-71v P16

1 - e-71v a

1 - -e -7iv P '6

0 Also for general phase-type distributions, all quantities in Proposition 1.10 can be found explicitly, see VIII.7. Notes and references Some early references drawing attention to the model are Dawidson [100] and Segerdahl [332]. For the absolute ruin problem, see Gerber [155] and Dassios & Embrechts [98]. Equation (1.6) was derived by Harrison & Resnick [186] by a different approach, whereas (1.5) is from Asmussen & Schock Petersen [50]; see further the notes to II.3. One would think that it should be possible to derive the representations (1.7), (1.8) of the ruin probabilities without reference to storage processes. No such direct derivation is, however, known to the author. For some explicit solutions beyond Corollary 1.8, see the notes to Section 2 Remark 1.9 is based upon Schock Petersen [288]; for complexity- and accuracy aspects, see the Notes to VIII.7. Extensive discussion of the numerical solution of Volterra equations can be found in Baker [57]; see also Jagerman [209], [210].

2 The model with interest In this section, we assume that p(x) = p + Ex. This example is of particular application relevance because of the interpretation of f as interest rate. However, it also turns out to have nice mathematical features.

197

2. THE MODEL WITH INTEREST

A basic tool is a representation of the ruin probability in terms of a discounted stochastic integral (2.1)

Z = - f e-EtdSt 0

w.r.t. the claim surplus process St = At - pt = EN` U; - pt of the associated compound Poisson model without interest . Write Rt") when Ro = u. We first note that: Proposition 2.1 Rt") = eetu + Rt°) Proof The result is obvious if one thinks in economic terms and represents the reserve at time t as the initial reserve u with added interest plus the gains/deficit from the claims and incoming premiums. For a more formal mathematical proof, note that

dR(u) = p + eR(u) - dAt, d [R(") - eetu] = p + e [R(u) - eEtu] - dAt . Since R( ;u) - eE'0u = 0 for all u, Rt") - eEtu must therefore be independent of u which yields the result. 0 Let

Zt = e-etR(0) = e-et (ft (p + eR(°)) ds - At I Then dZt

= e -Et

(_edt

f t (p + eR°) ds + (p + eR°)) dt + e dt A- dA

= e_et (pdt - dAt) = -e-EtdSt. / Thus v Z,, = - e-etdSt,

0 where the last integral exists pathwise because {St} is of locally bounded variation. Proposition 2.2 The r.v. Z in (2.1) is well-defined and finite, with distribution H(z) = P(Z < z) given by the m.g.f.

H[a] = Ee" = exp where k(a) _

(-ae-Et) dt} = exp {f °° k

13(B[a] - 1) - pa. Further Zt a ' Z

k

{fa

as t --+ oo.

(-y) dy}

198


Proof Let Mt =At -tAUB. Then St = Mt+t(/3pB-p) and {M„} is a martingale. e-EtdMt} From this it follows immediately that {fo is again a martingale. The mean is 0 and (since Var(dMt) = /3PB2)dt)

Var (

Z

'

e-'tdMt )

/' v

(2)

J e- eft/3p(B)dt = a2B (1 - e-2ev). o

Hence the limit as v -3 oo exists by the convergence theorem for L2-bounded martingales, and we have

v

v -

Zv =

e-EtdSt = -f e-t(dMt + (,3pB - p)dt)

J

0 - f0"

e-Et

a' 0 - f 0

oo

o o

(dMt + (3p$ -

p)dt)

e-EtdSt = Z.

Now if X1i X2, ... are i.i.d. with c.g.f. 0 and p < 1, we obtain the c .g.f. of E0° p'Xn at c as 00

00

log E fl ea°n X„ n=1

00

= log 11 e0(av ") _ n=1

E 0(apn). n=1

Letting p = e-Eh, Xn = Snh - S( n+1)h, we have q5(a) = hic(- a), and obtain the c.g.f. of Z = - f0,30 e-'tdSt as 00 00 00 lim E 0(apn ) = li h E rc(-ae -Fnh) = f tc (-ae-t) dt; n=1

1

n=1

0

the last expression for H[a] follows by the substitution y = ae-Et Theorem 2.3 z/'(u) =

H(-u) E [H(-RT(u)) I r(u) < oo] .

Proof Write r = r(u) for brevity. On {r < oo }, we have

u + Z =

(u + Zr ) + ( Z - Zr) = e

-

ET {e

(u + Zr) - f '* e-E(t-T )dSt] T

e-

ET [

R( u)

+ Z`],

❑

199

2. THE MODEL WITH INTEREST

where Z* = - K* e-E(t-T)dSt is independent of F, and distributed as Z. The last equality followed from Rt") = eEt(Zt + u), cf. Proposition 2.1, which also yields r < oo on {Z < -u}. Hence H(-u)

= P(u + Z < 0) = P(RT + Z* < 0; r < oo) zb(u)E [P(RT + Z* < 0 I)7T, r < oo)] _ O(u)E [H(-RT(")) I r(u) < oo] .

Corollary 2.4 Assume that B is exponential, B(x) = e-6', and that p(x) _ p + Ex with p > 0. Then o€Q/E -Ir, (8(p + cu);.

V) (u)

1\ E E

aA/Epal Ee -6n1 E +^3E1 / E

1r

Cbp;

E El al

where 1'(x; i) = f 2°° tn-le-tdt is the incomplete Gamma function. Proof 1 We use Corollary 1.8 and get

w(x) fo P + Etdt = g(x) = p +0x

e log(p + Ex) - e loge,

exp { - log(p + Ex) - - log p - 6x }

pal(p ryo)3 + ex)plE-1e-6^ J 70 = 1 + J p) exp {Ow(x) - Sx} dx x r^ = 1+ + ' /E (p + Ex)01'-le-ax dx 0

fJ

= 1+

a Epo/ E

f yI/ E- 1e- 6(Y -P)/E dy P

1+ OEA/E- 1e6 P /Er 60/e po/ e

(

,;,3 ) E E

lp(u) = -to foo a exp {w(x) - bx} AX) acO/E" 1 ePE l

Yo

50 1epolE

(

+ cu); 0)

5(p

E E

200

CHAPTER VII. RESERVE-DEPENDENT PREMIUMS ❑

from which (2.2) follows by elementary algebra. Proof 2 We use Theorem 2.3. From ic(a) = ,3a/ (5 - a) - pa, it follows that = f 1 c(-y)dy = 1 f '(p-a/(a +y))dy f 0 0 Ey

logH[a]

R/E 1 [pa + )3log 8 - /3 log(b + a)] = log ePa/f (a + a )

e which shows that Z is distributed as p/E - V, where V is Gamma(b, 13/E), i.e. with density x(3/e-1aQ/e fV (x) _

e

-6X

' x > 0.

r (j3/E)

In particular, H(-u)

=

P(Z

r < -u) = P(V

>

u + p/E) =

(8(p + Eu)/E; ,13 /E) r (,3/E)

By the memoryless property of the exponential distribution, -RT(u) has an exponential distribution with rate (S) and hence E [H(-RT(u))I r(u) < oo]

L Pe-6'r (P/C - V < x)]0 + f P(V > p/E ) +

/' P/ ' (p/

-

e-by fv (p/E - x) dx

x)p/e -150/f

e- b P/E dx

I' (/3/E) (6P1'E;01'E) + (p/E)al aO l fe-bP/E } IF (0 /0 jF

From this (2.2) follows by elementary algebra.

/^

❑

Example 2 .5 The analysis leading to Theorem 2.3 is also valid if {Rt} is obtained by adding interest to a more general process {Wt} with stationary independent increments. As an example, assume that {Wt} is Brownian motion with drift µ and variance v2; then {Rt} is the diffusion with drift function p+Ex and constant variance a2. The process {St} corresponds to {-Wt} so that c(a) or2a2/2 - pa, and the c.g.f. of Z is IogH[a]

= f ytc(-y)dy = e fa (0,2y +µ ) dy

3. THE LOCAL ADJUSTMENT COEFFICIENT

201

_ Q2a2 pa 4e E

I.e., Z is normal (p/E, Q2/2E), and since RT = 0 by the continuity of Brownian motion, it follows that the ruin probability is

Cu)

H(-u) H(0)

11 Notes and references Theorem 2.3 is from Harrison [185]; for a martingale proof, se e.g. Gerber [157] p. 134 (the time scale there is discrete but the argument is easily adapted to the continuous case). Corollary 2.4 is classical. The formula (2.3) was derived by Emanuel et at. [129] and Harrison [185]; it is also used as basis for a diffusion approximation by these authors. Paulsen & Gjessing [286] found some remarkable explicit formulas for 0(u) beyond the exponential case in Corollary 1.8. The solution is in terms of Bessel functions for an Erlang(2) B and in terms of confluent hypergeometric functions for a H2 B (a mixture of two exponentials). It must be noted, however, that the analysis does not seem to carry over to general phase-type distributions, not even Erlang(3) or H3, or to non-linear premium rules p(•). A r.v. of the form Ei° p"X" with the X„ i.i.d. as in the proof of Proposition 2.2 is a special case of a perpetuity; see e.g. Goldie & Griibel [167]. Further studies of the model with interest can be found in Boogaert & Crijns [71], Gerber [155], Delbaen & Haezendonck [104], Emanuel et at. [129], Paulsen [281], [282], [283], Paulsen & Gjessing [286] and Sundt & Teugels [356], [357]. Some of these references also go into a stochastic interest rate.

3 The local adjustment coefficient. Logarithmic asymptotics For the classical risk model with constant premium rule p(x) - p*, write y* for the solution of the Lundberg equation

f3(B[ry *] - 1) - -Y*p*

= 0,

write Vi* (u) for the ruin probability etc., and recall Lundberg 's inequality W*(u) < e-ry*u

202


and the Cramer-Lundberg approximation (3.3)

V,*(u) - C*e--f*".

When trying to extend these results to the model of this chapter where p(x) depends on x, a first step is the following: Theorem 3 .1 Assume that for some 0 < 5o < oo, it holds that f3[s] T oo, log ?i(u) < < 00 -JO . If 60 s f 6o, and that p(x) -* oo, x -* oo. Then lim sup u->oo u

and e -E''p(r) -+ 0, e(1o+e)2 (x ) u -> 00.

oo for all E > 0, then

log u (u)

In the proof as well as in the remaining part of the section , we will use the local adjustment coefficient 'y(x), i.e. the function -y(x) of the reserve x obtained by for a fixed x to define -y(x) as the adjustment coefficient of the classical risk model with p* = p(x), i.e. as solution of the equation n(x,'y ( x)) = 0

where

r. (x, a) = f3(B[a] - 1) - ap(x);

(3.4)

we assume existence of -y(x) for all x, as will hold under the steepness assumption of Theorem 3.1, and (for simplicity) that

inf p(x) > (3µs ,

(3.5)

x>0

which implies inf.,>o 7(x) > 0. The intuitive idea behind introducing local adjustment coefficients is that the classical risk model with premium rate p* = p(x) serves as a 'local approximation ' at level x for the general model when the reserve is close to x. Proof of Theorem 3.1. The steepness assumption and p(x) -+ oo ensure 'y(x) -* So. Let y* < So, let p* be a in (3. 1) and for a given E > 0, choose uo such that p( x) > p* when x > u0E. When u > uo, obviously O(u) can be bounded with the probability that the Cramer -Lundberg compound Poisson model with premium rate p* downcrosses level uE starting from u , which in turn by Lundberg's inequality can be bounded by e-ry*(1-E)" Hence limsup„,.log '(u)/u < -ry*(1 - E). Letting first E -* 0 and next ry * T 5o yields the first statement of the theorem. For the last asssertion , choose c(,1 ), c(,2) such that p(x) < c(.i)eex, B(x) > C(2)e-(ao+f)x for all x. Then we have the following lower bound for the time for the reserve to go from level u to level u + v without a claim: w(u + v) - w (u) J

dt > c(3)e-eu v 1 p(u+ t)


203

where c,(3) = (1 - e-a°/(ecf1)). Therefore the probability that a claim arrives before the reserve has reached level u + v is at least c(,4)e-E" Given such an arrival, ruin will occur if the claim is at least u + v, and hence '(u) > c(4)e-euc( 2)e-(do+e)u

The truth of this for all e > 0 implies lim inf log V,(u) > -so.

❑

Obviously, Theorem 3.1 only presents a first step, and in particular, the result is not very informative if bo = oo. The rest of this section deals with tail estimates involving the local adjustment coefficient. The first main result in this direction is the following version of Lundberg's inequality: Theorem 3 . 2 Assume that p(x) is a non-decreasing function of x and let I(u) = fo ry(x)dx. Then ,' (u) < e-I("). (3.6) The second main result to be derived states that the bound in Theorem 3.2 is also an approximation under appropriate conditions. The form of the result is superficially similar to the Cramer-Lundberg approximation, noting that in many cases the constant C is close to 1. However, the limit is not u -+ oo but the slow Markov walk limit in large deviations theory (see e.g. Bucklew [81]).

For e > 0, let 0e (u) be evaluated for the process only with 3 replaced by /0/e and U; by cU2.

{Rte)}

defined as in (1.2),

Theorem 3 .3 Assume that either (a) p(r) is a non -decreasing function of r, or (b) Condition 3.13 below holds. Then lim-elog l/ie (u) = I(u). (3.7) CIO

Remarks: 1. Condition 3.13 is a technical condition on the claim size distribution B, which essentially says that an overshoot r.v. UJU > x cannot have a much heavier tail than the claim U itself. 2. If p(x) = pis constant , then Rte) = CRtie for all t so that V), (u) = O(u/e), I.e., the asymptotics u -* oo and c -- 0 are the same. 3. The slow Markov walk limit is appropriate if p(x) does not vary too much compared to the given mean interarrival time 1/0 and the size U of the claims; one can then assume that e = 1 is small enough for Theorem 3.3 to be reasonably precise and use e` (u) as approximation to 0 (u).


204

4. One would expect the behaviour in 2) to be important for the quantitative performance of the Lundberg inequality (3.6). However, it is formally needed only for Theorem 3.3. 5. As typical in large deviations theory, the logaritmic form of (3.7) is only captures 'the main term in the exponent' but is not precise to describe the asymptotic form of O(u) in terms of ratio limit theorems (the precise asymptotics could be logI(u)e-1(U) or I(u)"e_I(u), say, rather than e-I(u)).

3a Examples Before giving the proofs of Theorems 3.2, 3.3, we consider some simple examples. First, we show how to rewrite the explicit solution for ti(u) in Corollary 1.8 in terms of I(u) when the claims are exponential: Example 3 .4 Consider again the exponential case B(x) = e-ax as in Corollary 1.8. Then y(x) = b - (3/p(x), and r

j

v(x)dx = bu -

a J0

p(x)-ldx =

Integrating by parts, we get 1 'Yo

/' oo

exp {(3w (x) - bx} dx J" AX) = 1 + J0 dodx(x) exp {,(iw(x) - bx} dx fo

= 1+

00

1 + [exp {/(3w(x) - bx}]o + b

exp low (x) - bx} dx

1+0- 1 + b f e-,(x) dx, J0

1

ôo

70 Ju

exp low(x) - bx} dx

g(x ) dx f AX)

lexp IOW (X )

bx + b

oo exp low(x) bx dx u

r oo = b

J

u

exp low (x) - bx} dx - exp {/33w(u) - bu},


205

and hence

f°° e-I(v )dy - e- I ( u) fool,

ry(x

/b -I u

e- I ( v )dy

o e -f0 °° e -

+u) dxdy - 1/8 . (3.8)

fo

7(x)dxdy

1-1 We next give a direct derivations of Theorems 3.2, 3.3 in the particularly simple case of diffusions: Example 3.5 Assume that {Rt} is a diffusion on [0, oo) with drift µ(x) and variance a2 (x) > 0 at x. The appropriate definition of the local adjustment coefficient 7(x) is then as the one 2p(x)la2(x) for the locally approximating Brownian motion. It is well known that (see Theorem XI.1.10 or Karlin & Taylor [222] pp. 191-195) that

1P (U) = fu0 e-I(v)dy = e-I(u) follo e- fory(x+u)dxdy ( 3.9 ) 11000 e-I(v)dy f000 e- f y(x)dxd y

If 7(x) is increasing , applying the inequality 7(x + u) > 7(x) yields immediately the conclusion of Theorem 3.2. For Theorem 3.3, note first that the appropriate slow Markov walk assumption amounts to u, (X) = µ(x), 0,2(X) = ev2(x) so that 7e(x) = 7(x)/e, IE(u) = I(u)/e, and (3.9) yields

-e log ,0, (u) = I(u) + AE - BE, (3.10) where AE = e log

000 e- fo 7(x)dx/Edy f ,

Be = e log U000 e- fa

7(x+u)dx/Edy

o The analogue of (3.5) is infx>o 7(x) > 0 which implies that f °O ... in the definition of AE converges to 0. In particular, the integral is bounded by 1 eventually and hence lim sup AE < lim sup a log 1 = 0. Choosing yo, 70 > 0 such that 7(x) < 7o for y < yo, we get

r

00 e- fo 7(x) dx /E dy >

Yo

a-v 'yo /Edy = E (1 - e-v 0 O /E) J0 70 70

This implies lim inf A, > lime log e = 0 and AE -* 0. Similarly, BE -* 0, and (3.7) follows. ❑

206


The analogue of Example 3.5 for risk processes with exponential claims is as follows: Example 3 .6 Assume that B is exponential with rate S. Then the solution of the Lundberg equation is -y* = b - ,6/p* so that u

1 dx. I (U) = bu - /3 1 AX ) Note that this expression shows up also in the explicit formula for lk(u) in the form given in Example 3.4. Ignoring 1/5 in the formula there, this leads to (3.6) exactly as in Example 3.5. Further, the slow Markov walk assumption means 5E = b/c, 0, _ ,0/e. Thus 7e(x) _7(x)/e and (3.10) holds if we redefine AE as AE = flog (j °° efo 7(x)dx/edy _ E/5 I

and similarly for B. As in Example 3.5, lim sup Af < lim sup c log(1 - 0) = 0. E-+o

e-*O

By (3.5) and 7* = 5 -,Q/p*, we have 5 > 7o and get

C

lim inf AE > lime log e - 7o

15 - 1 I I > 0. -

Now (3.7) follows just as in Example

3.5.

0

We next investigate what the upper bound / approximation a-I (°) looks like in the case p(x) = a + bx (interest) subject to various forms of the tail B(x) of B. Of course, 7(x) is typically not explicit, so our approach is to determine standard functions Gl (u), . . . , G. (u) representing the first few terms in the asymptotic expansion of I(u) as u -+ oo. I.e.,

G,(u) oo,

G;+1 (u) = o( 1), I(u ) = G1(u) + ... + Gq(u) + o(G9(u))• Gi (u)

It should be noted , however , that the interchange of the slow Markov walk oo is not justified and in fact, the slow Markov limit a -* 0 and the limit u walk approximation deteriorates as x becomes large. Nevertheless , the results are suggestive in their form and much more explicit than anything else in the literature.


207

Example 3 .7 Assume that B(x) - clxa-le-5x

(3.11)

with a > 0. This covers mixtures or convolutions of exponentials or, more generally, phase-type distributions (Example 1. 2.4) or gamma distributions; in the phase-type case , the typical case is a = 1 which holds , e.g., if the phase generator is irreducible ( Proposition VIII. 1.8). It follows from (3.11) that b[s] -* co as s f S and hence 7* T S as p* -+ oo. More precisely, B[s] = 1 + s exB(x)dx = 1 +c1SF(a) ('+o(')) (S - s)C' f "o

o

as s T S, and hence (3.1) leads to (S-7T N Ocp

a,

,Y ,:;C2p*

fu I(u) Su - c2

J

Su

a + bx 1/

0

(

C2 = (3clr( a))11',

)

Su

a 1

❑

where c3 = c2 /b, c4 = c2b -1/'/(1 - 1/a).

Example 3 .8 Assume next that B has bounded support, say 1 is the upper limit and B(x) - cs(1 - x)n-1, x T 1,

(3.12)

with y > 1. For example, 77 = 1 if B is degenerate at 1, y = 2 if B is uniform on (0, 1) and 17 = k + 1 if B is the convolution of k uniforms on (0,1/k). Here B[s] is defined for all s and B[s] - 1 =$

e"B(x)dx = e8 f

cse8 Sn

-1

f

f

Jo s e-IB ( 1 - y/s)dy

' e-vy'7-ldy = cse8r(T7)

sn -1

as s T oc. Hence (3.1) leads to ,3cse7*I7(77) - ry*°p*, ry* loge*+ g7loglogp*, I(u) Pt; u(logu + r7loglogu).


208

Example 3 .9 As a case intermediate between (3.11) and (3.12), assume that B(x) CO -x2/2c7, x f oo . (3.13)

We get b[s] - 1

Cgs o"O 0

e-c78)2/2c7

esxe-x2/2c7 dx = cgsec782/2

dx

f - css 2%rc7eC782/2,

C7

- log p*, 7 * - c8

where c8 =

,

log

I (u) c8u

log u

2/c7.

0

3b Proof of Theorem 3.2 We first remark that the definition (3.4) of the local adjustment coefficient is not the only possible one: whereas the motivation for (3.4) is the formula

h

logEues ( Rh-u) ,•, ,3 (B[s] - 1) - sp(u), h 10, (3.14)

for the m .g.f. of the increment in a small time interval [0, h], one could also have considered the increment ru (T1) - u - Ul up to the first claim (here ru (•) denotes the solution of i = p (r) starting from ru(0) = u). This leads to an alternative local adjustment coefficient 7o(u) defined as solution of

1 = Ee''o(u)(vi+u - ru(TI)) - B[7o (u)] .

1

3e- Ote7o( u)(u- r^.(t))dt.

(3.15)

0

Proposition 3.10 Assume that p(x) is a non-decreasing function of x. Then: (a) -y(x) and 7o(x) are also non-decreasing functions of x; (b) 'y(x) 0 and x < 0 separately, it is easily seen that fu x7o(y)dy < x-yo (u). Also, fa 7o(y)dy < u7o(u) < x-yo (u) for x > u. Hence „/,(n+1) (u)

e-fo Yo(x)dxI^"Q exyo( I

u

fo -yo( x)dx j,u[70(u)] fo e-

e-

-yo(x)dx

u)Fu(dx )+ J - es'Yo(u)Fu(dx)} o0


210

where the last identity immediately follows from (3.15); we used also Proposition 3.10(a) for some of the inequalities. 0 It follows from Proposition 3.10(b ) that the bound provided by Theorem 3.11 is sharper than the one given by Theorem 3.2. However, yo(u) appears more difficult to evaluate than y(u). Also, for either of Theorems 3.2, 3.11 be reasonably tight something like the slow Markov walk conditions in Theorem 3.3 is required, and here it is easily seen that yo(u) ,: y(u). For these reasons, we have chosen to work with -y(u) as the fundamental local adjustment coefficient.

3c Proof of Theorem 3.3 The idea of the proof is to bound { R( f) } above and below in a small interval [x - x/n, x + x/n] by two classical risk processes with a constant p and appeal to the classical results (3.2), (3.3). To this end, define

uk,n =

ku,

P k,n

=

p(x),

sup

inf

pk n =

n uk-1,n 0 and the root of q(s) with the smallest real part should be unique (not necessarily simple, cf. the Erlang case). No satisfying

223

2. RENEWAL THEORY

algorithm for finding a phase representation of a distribution B (which is known to be phase-type and for which the m.g.f. or the density is available ) is, however, known. A related important unsolved problem deals with minimal representations: given a phase-type distribution , what is the smallest possible dimension of the phase space E?

2 Renewal theory A summary of the renewal theory in general is given in A.1 of the Appendix, but is in part repeated below. Let U1, U2, ... be i.i.d. with common distribution B and define4

U(A)

= E# {n = 0,1, ...: U1 + ... +UnEA} 00 = EEI(U1 +...+UnEA). n=O

We may think of the U; as the lifetimes of items (say electrical bulbs) which are replaced upon failure, and U(A) is then the expected number of replacements (renewals) in A. For this reason, we refer to U as the renewal measure; if U is absolutely continuous on (0, oo) w.r.t. Lebesgue measure, we denote the density by u(x) and refer to u as the renewal density. If B is exponential with rate 0, the renewals form a Poisson process and we have u(x) = 0. The explicit calculation of the renewal density (or the renewal measure) is often thought of as infeasible for other distributions, but nevertheless, the problem has an algorithmically tractable solution if B is phase-type: Theorem 2.1 Consider a renewal process with interarrivals which are phasetype with representation (cr,T). Then the renewal density exists and is given

by u(x) = ae(T+ta)xt.

(2.1)

Proof Let {Jtk)} be the governing phase process for Uk and define {Jt} by piecing the { J(k) } together, JtJt1)

0 0, -s ¢ sp(T). Then -s is an eigenvalue of Q = T + ta+ if and only if 1 =,P[s] = A[-s]B[s], with B[s], F[s] being interpreted in the sense of the analytical continuation of the m.g.f. In that case, the corresponding right eigenvector may be taken as (-sI - T)-It.

Proof Suppose first Qh = -sh. Then e4'h = e-82h and hence -sh = Qh = (T + taA[Q])h = Th + A[-s]tah. (4.5) Since -s $ sp(T), this implies that ahA[-s] # 0, and hence we may assume that h has been normalized such that ahA[-s] = 1. Then (4.5) yields h = (-sI - T)-1t. Thus by (4.4), the normalization is equivalent to F(s) = 1.

4. THE RENEWAL MODEL

233

Suppose next F(s) = 1. Since R(s) > 0 and G _ is concentrated on (-oo, 0), we have IG_ [s] I < 1 , and hence by the Wiener-Hopf factorization identity (A.9) we have G+[s] = 1 which according to Theorem 1.5(c) means that a+(-sI T)-1t = 1. Hence with h = (-sI -T)- lt we get

Qh = (T + to+)h = T(-sI - T)-lt + t = -s(-sI - T)-lt = -sh.

Let d denote the number of phases. Corollary 4.6 Suppose u < 0,' that the equation F(s) = 1 has d distinct roots p1, ... , Pd in the domain ER(s) > 0 , and define hi = (-piI - T)-It, Q = CD-1 where C is the matrix with columns hl,..., hd, D that with columns -p1 hl, ... , -pdhd. Then G+ is phase- type with representation (a+, T) with a+ = a(Q-T)/at. Further, letting vi be the left eigenvector of Q corresponding to -pi and normalised by vihi = 1 , Q has diagonal form d

d

Q = -dpivi®hi = -dpihivi. (4.6) i=1

i=1

Proof Appealing to Theorem 4.5, the matrix Q in Theorem 2.1 has the d distinct eigenvalues - p1i ... , -Pd with corresponding eigenvectors hl,..., hd. This immediately implies that Q has the form CD-1 and the last assertion on the diagonal form . Given T has been computed, we get at a(Q - T) = 1 ata+ = a+.

Notes and references Results like those of the present section have a long history, and the topic is classic both in risk theory and queueing theory (recall that we can identify 0(u) with the tail P(W > u) of the GI/PH /1 waiting time W; in turn, W v M(d) in the notation of Chapter V). In older literature , explicit expressions for the ruin/ queueing probabilities are most often derived under the slightly more general assumption that b is rational (say with degree d of the polynomial in the denominator) as discussed in Section 6. As in Corollary 4.6, the classical algorithm starts by looking for roots in the complex plane of the equation f3[y]A[-ry] = 1, t(ry) > 0. The roots are counted and located by Rouche' s theorem (a classical result from complex analysis giving a criterion for two complex functions to have the same number of zeros within the unit circle ). This gives d roots 'y,,. .. , -yd satisfying R(ryi) > 0, and the solution is

CHAPTER VIII. MATRIX-ANALYTIC METHODS

234 then in transform terms

d

F 1 + a J e°" ip(u) du = Ee°w =

11(--t,) d

(see, e.g., Asmussen & O'Cinneide [ 41] for a short self- contained derivation). In risk theory, a pioneering paper in this direction is Tacklind [373], whereas the approach was introduced in queueing theory by Smith [350]; similar discussion appears in Kemperman [227] and much of the queueing literature like Cohen [88]. This complex plane approach has been met with substantial criticism for a number of reasons like being lacking probabilistic interpretation and not giving the waiting time distribution / ruin probability itself but only the transform. In queueing theory, an alternative approach (the matrix-geometric method ) has been developed largely by M.F. Neuts and his students, starting around in 1975. For surveys , see Neuts [269], [270] and Latouche & Ramaswami [241]. Here phase- type assumptions are basic, but the models solved are basically Markov chains and -processes with countably many states ( for example queue length processes ). The solutions are based upon iterations schemes like in Theorem 4.4; the fixpoint problems look like

R=Ao+RAI+R2A2+ , where R is an unknown matrix, and appears already in some early work by Wallace [377]. The distribution of W comes out from the approach but in a rather complicated form . The matrix- exponential form of the distribution was found by Sengupta [335] and the phase-type form by the author [18]. The exposition here is based upon [18], which contains somewhat stronger results concerning the fixpoint problem and the iteration scheme. Numerical examples appear in Asmussen & Rolski [43]. For further explicit computations of ruin probabilities in the phase-type renewal case , see Dickson & Hipp [118], [119].

5 Markov-modulated input We consider a risk process {St } in a Markovian environment in the notation of Chapter VI. That is , the background Markov process with p states is {Jt}, the intensity matrix is A and the stationary row vector is ir . The arrival rate in background state i is a; and the distribution of an arrival claim is B;. We assume that each B; is phase-type, with representation say (a(' ), T('), E(t)). The number of elements of El=> is denoted by q;. It turns out that subject to the phase- type assumption , the ruin probability can be found in matrix-exponential form just as for the renewal model, involving

5. MARKOV-MODULATED INPUT

235

some parameters like the ones T or a+ for the renewal model which need to be determined by similar algorithms. We start in Section 5a with an algorithm involving roots in a similar manner as Corollary 4.6. However, the analysis involves new features like an equivalence with first passage problems for Markovian fluids and the use of martingales (these ideas also apply to phase-type renewal models though we have not given the details). Section 5b then gives a representation along the lines of Theorem 4.4. The key unknown is the matrix K, for which the relevant fixpoint problem and iteration scheme has already been studied in VI.2.

5a Calculations via fluid models. Diagonalization Consider a process {(It, Vt)}t>o such that {It} is a Markov process with a finite state space F and {Vt} has piecewiese linear paths, say with slope r(i) on intervals where It = i. The version of the process obtained by imposing reflection on the V component is denoted a Markovian fluid and is of considerable interest in telecommunications engineering as model for an ATM (Asynchronuous Transfer Mode) switch. The stationary distribution is obtained by finding the maximum of the V-component of the version of {(It,Vt)} obtained by time reversing the I component. This calculation in a special case gives also the ruin probabilities for the Markov-modulated risk process with phase-type claims. The connection between the two models is a fluid representation of the Markov-modulated risk process given in Fig. 5.1. (a) 0

o

0 ♦ o ° tl ♦ • 0 0 o } o

(b)

0

}

o

♦ •

f

0

o

Figure 5.1 In Fig. 5.1, p = ql = Q2 = 2. The two environmental states are denoted o, •, the phase space E(°) for B. has states o, O, and the one E(•) for B. states


236

4, 4. A claim in state i can then be represented by an E()-valued Markov process as on Fig. 5.1(a). The fluid model on Fig . 5.1(b) {(It ,Vt)} is then obtained by changing the vertical jumps to segments with slope 1. Thus F = {o, o, V, •, 4, 4}. In the general formulation , F is the disjoint union of E and the Eli), r(i, a) = 1.

F = E U { (i, a) : i E E, a E E(i) } , r(i) _ -1, i E E, The intensity matrix for { It} is (taking p = 3 for simplicity)

AI =

0

0

A - (Ni)diag

'31a(1) 0

f32a(2)

0

0 t(2) 0

0 T1 0 0

0 0 T(2) 0

'33a(3) 0 0 T(3)

I

t(1) 0 0

0 0 t(3)

The reasons for using the fluid representation are twofold. First, the probability in the Markov-modulated model of upcrossing level u in state i of {Jt} and phase a E Eli) is the same as the probability that the fluid model upcrosses level u in state (i, a) of {It}. Second, in the fluid model Eel', < oo for all s, t, whereas Ee8s' = oo for all t and all s > so where so < oo. This implies that in the fluid context, we have more martingales at our disposal. Recall that in the phase-type case, Bi[s] = -a(i)(T(i) + sI)-it('). Let E denote the matrix

-,31a(l) 0

(/3i)diag - A

Or 1A/ _

t(i)

0

0 0

t(2)

0

0 0 t(3)

0 T1 0 0

0 - 92a(2) 0 0 T(2) 0

0 0

-f33a(3) 0 0 T(3)

with the four blocks denoted by Ei„ i, j = 1, 2, corresponding to the partitioning + Epp). of E into components indexed by E, resp. Eli) + Proposition 5.1 A complex number s satisfies

'A+

(f3i(Bi[-s] - 1))diag + sII = 0 (5.1)

if and only if s is an eigenvalue of E. If s is such a number, consider the vector a satisfying (A + (13i(Bi[ -s] - 1))diag ) a = -sa and the eigenvector b =


(a>

237

of 0* 1 AI, where c, d correspond to the partitioning of b into components

indexed by E, resp . E(1) + + E(P). Then (up to a constant) c = a, d = (sI -

E22)-1E21a

= E ai(sI - T('))-1t(i) . iEE

Proof Using the well-known determinant identity Ell E12 E21 E22

E22 I ' I Ell - E12E22 E21 I ,

with Eii replaced by Eii - sI, it follows that if

(/3i)diag

t(1)

0 0

0 t(2) 0

0 0

-Qla(1) 0 0 -,32a(2)

- A - sI 0 0 t(3)

- Nla(1) 0 0 T 1- sI 0 0 0 T(2) - sI 0 0 0 T(3) - sI

= 0,

then also ()3i)diag - A - sI+ ((3ia(i)(T(i) - sI)-1t)) iag

I = 0

which is the same as (5.1). For the assertions on the eigenvectors, assume that a is chosen as asserted which means (Ell - sI + E12 (sI - E22)-1 E21) a = 0, and let d = (sI - E22)-1 E21a, c = a. Then E21c+E22d =

E21a - (sI - E22 - sI) (sI - E22)-1 E21a E21a - E21a + sd = sd.

Noting that E11c + E12d = se by definition, it follows that Ell E12

( E 21 E22) (d) = s 1 d I . 0


238

Theorem 5.2 Assume that E = Or 'Al has q = ql + + qp distinct eigenvalues si, ... , sq with $2s,, < 0 and let b(v) = I d(„)) be the right eigenvector corresponding to s,,, v = 1, . . . , q. Then ,,/'

u = e'

(esiuc ( 1)

... e89uc(e)) (d(1) ... d("))-1 e.

Proof Writing Or-'Alb( v) = svb( v) as (AI - O,.sv)b(v) = 0, it follows by Proposition II.5.4 that {e--"1b(v) is a martingale . For u, v > 0, define w(u,v)=inf{t >0:Vtu orVt=- v}, w(u)=inf{t >O:Vt-u}, pi(u, v;

p i( u , v; j) pi( u ; j, a)

(j, a)), I' i( Vw(u,v) = -v) I,( u,v) = j), P2 (w (u) < oo, Iw(u,v) = (j, a)).

= Pi (Vw(u,v) =

j, a)

=

=

u)

Iw(u,v) =

Optional stopping at time w (u, v) yields C{V) = e8 ,upi(u, v;

Letting v -^ oo and using

Rsv

j, a )d(a

+ e8 °vpi (u ,v;j)c

v

.

< 0 yields

e8'uc = Epi(u;j,a)d^ ). j,a

Solving for the pi(u; j, a) and noting that i1 (u) = >I j,,,,pi(u; j, a), the result ❑ follows. Example 5 .3 Consider the Poisson model with exponential claims with rate 5. Here E has one state only. To determine 0 (u), we first look for the negative eigenvalue s of E = I -0 I which is s = -ry with yy = b -,Q. We can take a = c = 1 and get d = (s + b)-16 = 5/(3 = 1/p. Thus 0(u) = esu/d = pe-7 ° as ❑ should be. Example 5 .4 Assume that E has two states and that B1, B2 are both exponential with rates 51 i b2. Then we get V)i (u) as sum of two exponential terms where the rates s1, s2 are the negative eigenvalues of Al +01 -A1

E _

-A 2 b1

A2 +32 0

0 52

239


5b Computations via K Recall the definition of the matrix K from VI.2. In terms of K, we get the following phase-type representation for the ladder heights (see the Appendix for the definition of the Kronecker product 0 and the Kronecker sum ®): Proposition 5.5 G+(i, j; •) is phase-type with representation (E(i), 8^')IT(j)) where e 3^') =,33(e = 0 a(j))(-K ®T ( j))(ej (9 I). Proof We must show that G+ (i, j; (y, oo))

j)ye. (') a T(

However , according to VI.( 2.2) the l.h.s. is

0 /3 f R(i , j; dx)Bj(y - x) 00 f ° (') (j) eT (y-y)edx ,3j eye- xxej • a 00 oo f

el ,Qj eie

x T(j)y eej (j)a T(') e dx e e

0

00

eKx

®

e T(')'

dx (ej (& I)e T(')ye

00 eKa®T(')x dx (ej (9 I)eT(') Ye

e(i)eT(')ye. 0

Theorem 5 .6 For i E E, the Pi-distribution of M is phase-type with representation (E(1) + + E(P), 9('), U) where t(j) + t(j)O(j j = k

uja,k.y = to

B k7

j # k

In particular,

i,b (u) = Pi(M > u) = 9(i)euue. (5.3)

240


Proof We decompose M in the familiar way as sum of ladder steps . Associated with each ladder step is a phase process, with phase space EU> whenever the corresponding arrival occurs in environmental state j (the ladder step is of type j). Piecing together these phase processes yields a terminating Markov process with state space EiEE E('), intensity matrix U, say, and lifelength M, and it just remains to check that U has the asserted form. Starting from Jo = i, the initial value of (i, a) is obviously chosen according to e(`). For a transition from (j, a) to (k, ,y) to occur when j # k, the current ladder step of type j must terminate, which occurs at rate t(i), and a new ladder step of type k must start in phase y, which occurs w.p. Bk7 . This yields the asserted form of uja,k y. For j = k, we have the additional possibility of a phase change from a to ry within the ladder step, which occurs at rate t^^7.

❑

Notes and references Section 5a is based upon Asmussen [21] and Section 5b upon Asmussen [17]. Numerical illustrations are given in Asmussen & Rolski [43].

6 Matrix-exponential distributions When deriving explicit or algorithmically tractable expressions for the ruin probability, we have sofar concentrated on a claim size distribution B of phase-type. However, in many cases where such expressions are available there are classical results from the pre-phase-type-era which give alternative solutions under the slightly more general assumption that B has a Laplace transform (or, equivalently, a m.g.f.) which is rational, i.e. the ratio between two polynomials (for the form of the density, see Example 1.2.5). An alternative characterization is that such a distribution is matrix-exponential, i.e. that the density b(x) can be written as aeTxt for some row vector a, some square matrix T and some column vector t (the triple (a, T, t) is the representation of the matrix-exponential distribution/density): Proposition 6.1 Let b(x) be an integrable function on [0, oo) and b* [0] = f °O e-Bxb(x) dx the Laplace transform. Then b*[0] is rational if and only b(x) is matrix-exponential. Furthermore, if

b* [0] =

b1 +b20+b302 +... +bn0i-1 0n +a10n-1 +... +aii-10+anI

then a matrix-exponential representation is given by b(x) = aeTxt where

a = (b1 b2 ... bn-1 bn), t = (0 0 ... 0 1)', (6.2)

6. MATRIX-EXPONENTIAL DISTRIBUTIONS

T =

241

0 1 0 0 0 ... 0 0 0 0 1 0 0 ... 0 0 .. .(6.3) 0 0 0 0 0 ... 0 1 -an -an-1 -an _2 - an_3 -an _ 4 ... -a2 -a1

Proof If b(x) = aeTxt, then b*[0] = a(0I -T)-1t which is rational since each element of (01 - T)-1 is so. Thus, matrix-exponentiality implies a rational transform. The converse follows from the last statement of the theorem. For a proof, see Asmussen & Bladt [29] (the representation (6.2), (6.3) was suggested by Colm O'Cinneide, personal communication). ❑ Remark 6.2 A remarkable feature of Proposition 6.1 is that it gives an explicit Laplace tranform inversion which may appear more appealing than the first attempt to invert b* [0] one would do, namely to asssume the roots 6l, . . . , bn of the denominator to be distinct and expand the r.h.s. of (6.1) as E 1 c;/(0 + bi), ❑ giving b(x) = E 1 cie-biz/bY. Example 6 .3 A set of necessary and sufficient conditions for a distribution to be phase-type are given in O'Cinneide [276]. One of his elementary criteria, b(x) > 0 for x > 0, shows that the distribution B with density b(x) = c(1 cos(21r x))e-x, where c = 1 + 1/47r 2, cannot be phase-type. Writing b(x) = c(-e( 2ni-1 ) y/2 - e(-tai-1)x/2 + e-'T) it follows that a matrix-exponential representation ()3, S, s) is given by

27r i - 1 0 0 )3 = (111), S =

f -c/2

0 -21ri - 1 0 , s = -c/ 2 . (6.4) 0 0 -1 c

This representation is complex, but as follows from Proposition 6.1, we can always obtain a real one (a, T, t). Namely, since 1 + 4ir2 03 + 302 + (3 + 47x2)0 + 1 + 47r2 it follows by (6.2), (6.3) that we can take

0 1 0 0 a= (1 + 47r2 0 0), T= 0 0 1 , t= 0 . -1 - 47r2 -3 - 47x2 -3 1

0


242

Example 6 .4 This example shows why it is sometimes useful to work with matrix-exponential distributions instead of phase-type distributions: for dimension reasons . Consider the distribution with density

b(x)

=

15 ((2e-2x - 1)2 + 6). 7 + 155e-x

Then it is known from O'Cinneide [276] that b is phase-type when 6 > 0, and that the minimal number of phases in a phase-type representation increases to 0o as 5 , 0, leading to matrix calculus in high dimensions when b is small. But since

15(1 +6)02 + 1205 0 + 2255 + 105 b* [9] _ (7 + 155)03 + (1355 + 63)92 + (161 + 3455)9 + 2256 + 105 Proposition 6.1 shows that a matrix-exponential representation can always be ❑ obtained in dimension only 3 independently of J. As for the role of matrix-exponential distributions in ruin probability calculations, we shall only consider the compound Poisson model with arrival rate 0 and a matrix-exponential claim size distribution B, and present two algorithms for calculating '(u) in that setting. For the first, we take as starting point a representation of b* [0] as p( O)/q(9) where p, q are polynomials without common roots. Then (cf. Corollary 111.3.4) the Laplace transform of the ruin probability is

/g(e)-PO 0*[e] _ /' e-eu^G(u)dU = 0 9(/3--a0p(-9)ap (9)/q(9)) .

(6.5)

Thus, we have represented ti* [0] as ratio between polynomials (note that 0 must necessarily be a root of the numerator and cancels), and can use this to invert by the method of Proposition 6.1 to get i (u) = f3esus. For the second algorithm, we use a representation (a, T, t) of b(x). We recall (see Section 3; recall that t = -Te) that if B is phase-type and (a, T, t) a phase-type representation with a the initial vector, T the phase generator and t = -Te, then 5(u) = -a+e(T+t-+)uT-le

where a+ = -/3aT-1. (6.6)

The remarkable fact is, that despite that the proof of (6.6) in Section 3 seems to use the probabilistic interpretation of phase-type distribution in an essential way, then: Proposition 6.5 (6.6) holds true also in the matrix-exponential case.

6. MATRIX-EXPONENTIAL DISTRIBUTIONS 243 Proof Write b* = a(9I - T)-1t, b+ = a+(9I - T)- 't, b+ = a +(BI - T)-1T-1t. Then in Laplace transform formulation , the assertion is equivalent to -a+(BI - T - to+)-1T - 1t du =

9(

, - 6b* - b* (6.7)

cf. (6.5 ), (6.6). Presumably, this can be verified by analytic continuation from the phase-type domain to the matrix-exponential domain , but we shall give an algebraic proof. From the general matrix identity ([331] p. 519)

(A + UBV )- 1 = A-1 - A - 1UB(B + BVA-1UB)- 1BVA-1, with A = 91-T, U =- t,B=land V=a+, we get (91- T - to+)-1 = (BI - T)-1 + (6I - T)-1t ( l - a+(9I - T)-1t)-1a +(9I - T)-1 (91- T)-1 + 1 ib* (91- T)-1ta+(OI - T)-1 so that b* b** b** -a+(9I - T - to+)-1T - 1t = -b* - 1 + b+ = b++ 1 . Now, since (91-T)-1T - 1 = ^(T-1 + ( 91-T)-1),

(91- T)-1T -2 =

IT-2 + 82T - 1 + 82 (9I - T)-1

and 1 =

J0 00 b(x) dx =

-aT-1t,

xb(x) dx = aT2t,

AB

f

we get b+ = -0aT-1(9I -T)- 1t = -f3a (0I -T)-1T-1t


244

- 8 a(T-1 + (01- T)-1)t = 8 (1 - b*), -/3aT-1(0I - T)-1T- 1t = -/3a (9I - T)-1T-2t -,3a

(1 0

T -2

+

1

1 T -1 + (9I 02 02

-T)-1) t

-P + 7- 82b*.

From this it is straightforward to check that b**/(b+ - 1) is the same as the r.h.s. of (6.7). 0 Notes and references As noted in the references to section 4, some key early references using distributions with a rational transform for applied probability calculations are Tacklind [373] (ruin probabilities) and Smith [350] (queueing theory). A key tool is identifying poles and zeroes of transforms via Wiener-Hopf factorization. Much of the flavor of this classical approach and many examples are in Cohen [88]. For expositions on the general theory of matrix-exponential distributions, see Asmussen & Bladt [29], Lipsky [247] and Asmussen & O'Cinneide [41]; a key early paper is Cox [90] (from where the distribution in Example 6.3 is taken). The proof of Proposition 6.5 is similar to arguments used in [29] for formulas in renewal theory.

7 Reserve-dependent premiums We consider the model of Chapter VII with Poisson arrivals at rate 0, premium rate p(r) at level r of the reserve {Rt} and claim size distribution B which we assume to be of phase-type with representation (E, a, T). In Corollary VII.1.8, the ruin probability(u) was found in explicit form for the case of B being exponential. (for some remarkable explicit formulas due to Paulsen & Gjessing [286], see the Notes to VII.1, but the argument of [286] does not apply in any reasonable generality). We present here first a computational approach for the general phase-type case (Section 7a) and next (Section 7b) a set of formulas covering the case of a two-step premium rule, cf. VII.la.

7a Computing O(u) via differential equations The representation we use is essentially the same as the ones used in Sections 3 and 4, to piece together the phases at downcrossing times of {Rt} (upcrossing times of {St}) to a Markov process {mx} with state space E. See Fig. 7.1, which is self-explanatory given Fig. 3.1.

7. RESERVE-DEPENDENT PREMIUMS

245

Rt l0 -u

--------------------- 1z I.

Figure 7.1 The difference from the case p(r) = p is that {m2}, though still Markov, is no longer time-homogeneous. Let P(tl,t2) be the matrix with ijth element P (mt2 =j I mtl = i), OiEE Vi (U) < 1. In fact, >iEE Vi (U) is the ruin probability for a risk process with initial reserve 0 and premium function p(u + •). Also, in contrast to Section 3, the definition of {m8} depends on the initial reserve u = Ro. Since v(u) = (vi(u))iEE is the (defective) initial probability vector for {m8}, we obtain V)(u) = P(m„ E E) = v(u)P(0,u)e = A(u)e (7.1) where A(t) = v(u)P(0, t) is the vector of state probabilities for mt, i.e. Ai(t) = P(mt = i). Given the v(t) have been computed, the A(t) and hence Vi(u) is available by solving differential equations: Proposition 7.1 A(0) = v(u) and A'(t) = A(t)(T + tv(u - t)), 0 < t < u. Proof The first statement is clear by definition. By general results on timeinhomogeneous Markov processes, P(tl, t2) = exp

{

tq

f Q(v) dvl

t1 1

where Q(t) = ds [P(t, t + s) - I] I 8-0


246

However, the interpretation of Q(t) as the intensity matrix of {my} at time t shows that Q(t) is made up of two terms: obviously, {mx} has jumps of two types, those corresponding to state changes in the underlying phase process and those corresponding to the present jump of {Rt} being terminated at level u - t and being followed by a downcrossing. The intensity of a jump from i to j is tij for jumps of the first type and tivj(u - t) for the second. Hence Q(t) _ T + tv(u - t), A'(t) = A(t)Q(t) = A(t)(T + tv(u - t)).

0 Thus, from a computational point of view the remaining problem is to evaluate the v(t), 0 < t < u. Proposition 7.2 For i E E,

-vi,(u)

p ( u)

=

,(tai

+ vi(u)

E

vj(u)tjp (u) -

jEE

Q

+ vj (u)tjip ( u). (7.4)

jEE

Proof Consider the event A that there are no arrivals in the interval [0, dt], the probability of which is 1 -,3dt. Given A', the probability that level u is downcrossed for the first time in phase i is ai. Given A, the probability that level u + p(u)dt is downcrossed for the first time in phase j is vj (u + p(u)dt). Given this occurs, two things can happen: either the current jump continues from u + p(u)dt to u, or it stops between level u + p(u)dt and u. In the first case, the probability of downcrossing level u in phase i is

8ji(1 + p(u)dt • tii) + (1 - Sj i)p(u)dt • tji = Sji

+ p(u)tji dt,

whereas in the second case the probability is p(u)dt • tjvi(u). Thus, given A, the probability of downcrossing level u in phase i for the first time is E vj (u + p(u)dt) (Sji + p( u)dt • tji + p(u)dt • tjvi(u)) jEE

vi(u) + vi' (u)p(u)dt + p(u) dt E {tji + tjvi(u)} jEE

Collecting terms, we get vi(u) = aidt + (1 -,Qdt) vi(u) + vi'(u)p(u)dt + p(u) dt E{tji+tjvi(u)}. jEE

7. RESERVE-DEPENDENT PREMIUMS 247 Subtracting v; (u) on both side and dividing by dt yields the asserted differential ❑ equation. When solving the differential equation in Proposition 7.2, we face the difficulty that no boundary conditions is immediately available. To deal with this, consider a modification of the original process {Rt} by linearizing the process with some rate p, say, after a certain level v, say. Let p" (t), Rt , F" etc. refer to the modified process. Then pv(r)

p(r) r < v p r>v '

and (no matter how p is chosen) we have: Lemma 7.3 For any fixed u > 0, vi (U) = lim v= (u). V - 00

Proof Let A be the event that the process downcrosses level u in phase i given that it starts at u and let B" be the event By={o, v l t 0 : Rt < v}. To evaluate p1(u), let v(u) = a+2ieiT +ta+>)(u-v), assuming u > v for the moment. Then v(u) is the initial distribution of the undershoot when downcrossing level v given that the process starts at u, i.e. for u > v the distribution of v - RQ (defined for or < oo only) is defective phase-type with representation (v(u), T). Recall that q(w) is the probability of upcrossing level v before ruin given the process starts at w < v. Therefore u vvueTa t 1vueTva (7.7) pl( ) = ( ) ( q(v q( dx ))+( ) f o (the integral is the contribution from {R, > 0} and the last term the contribu-

tion from {R, < 0}). The f iin

in (7.7) equals

-01 (v - x) dx f v(u)eT xt dx - v v(u)eTat 1 1 - V" M 0

7. RESERVE-DEPENDENT PREMIUMS 1

1 - v(u)eTVe - 1

249

1 - v(u ) eTV e

-

- ^1(v)

J

v(u)eTxtz/)l (v - x) dx} V

from which we see that

1

pl (u) = 1 + 1

-1(v) f V

v(u) eTxt,01 (v - x) dx -

1 - v(u)eTve). 1 -^(v) ( (7.8)

The integral in (7.8) equals v v(u)eTxta+2) e(T+ta +))( v-x)edx which using Kronecker calculus (see A.4) can be written as (Y(u)

®a+)e(T+t°+>)°1

(T ® (-T - to+))1-1

{e{T®(-T-toy+ ))}„ - jl (t ®e)

Thus, all quantities involved in the computation of b(u) have been found in matrix form.

Example 7.4 Let {Rt } be as in Example 3.2. I.e., B is hyperexponential corresponding to

-3 0 3 a-(2 2)' T= ( 0 7 t- (7 The arrival rate is (i = 3. Since µB = 5/21, p2 < 3.21 = ? yields 0(u) = 1, so we consider the non-trivial case example p2 = 4 and p1 = 1. From Example 3.2, 01(u)

_ 24 -u + 35 1 e-6u 35 e

4(u)

_ 35 - 24e- u - e-6u 35 - 24e-v - e-6v

Let Al = -3 + 2V'2- and A2 = -3 - 2V"2- be the eigenvalues of T + to( 2 ). Then one gets

f

X20 20 21

1ea1(u -v) + 1

3

3 ^ A 2(u e

- v)

1eai(u -v) +

7

31 ^') eA2 (u- v) + (2^ + 3v2 ea'(u "

1

7

e\2(u

-v)


250

From (7.7) we see that we can write pi (u) = v(u)V2 where V2 depends only on v, and one gets 12e5" - 2 35e6v - 24e5v - 1 V2 = 4e5"+6 35e6v - 24es" - 1 Thus, pi (u) = p12(u)/p1 l(u) where p1i(u)

35e6v - 24es" - 1,

p12(u)

) e sv + ( 2v/2- + it (3 4'I 1 ea2(u-v

e1\2(u-")

7

+

(

32 +4,/-2-) ea 1(u - v)esv + 7

4_

2,,/2-

ea1(u-")

.

21 3

In particular, 192esv + 8 P1 - 21(35e6v - 24e5v - 1)' ?,b(v) =

192esv +8 35e6v + 168esv + 7*

Thus all terms involved in the formulae for the ruin probability have been ex❑ plicitly derived. Notes and references [30].

The analysis and the example are from Asmussen & Bladt

Chapter IX

Ruin probabilities in the presence of heavy tails 1 Subexponential distributions We are concerned with distributions B with a heavy right tail B(x) = 1- B(x). A rough distinction between light and heavy tails is that the m.g.f. B[s] = f e8x B(dx) is finite for some s > 0 in the light-tailed case and infinite for all s > 0 in the heavy-tailed case. For example, the exponential change of measure techniques discussed in II.4, III.4-6 and at numerous later occasions require a light tail. Some main cases where this light-tail criterion are violated are (a) distributions with a regularly varying tail, B(x) = L(x)/x" where a > 0 and L(x) is slowly varying, L(tx)/L(x) -4 1, x -4 oo, for all t > 0; (b) the lognormal distribution (the distribution of eu where U - N(µ, a2)) with density 1 e-(logy-Fh) 2/2az .

x 2iror2 (c) the Weibull distribution with decreasing failure rate , B(x) = e-x0 with 0 x) 2P(Xi > x). To capture the intuition behind this definition, note first the following fact: Proposition 1.1 Let B be any distribution on (0, oo). Then: (a) P(max(Xi, X2) > x) ^' 2B(x), x -3 00.

(b) liminf

BB(()

) > 2.

Proof By the inclusion-exclusion formula, P(max(Xi, X2) > x) is P(X1 > x) + P(X2 > x) - F(X1 > x, X2 > x) = 2B(x) - B(x)2 - 2B(x), proving (a). Since B is concentrated on (0, oo), we have {max(Xi, X2) > x} C {X1 + X2 > x}, and thus the lim inf in (b) is at least lim inf P(max(Xi, X2) > ❑ x)/B(x) = 2. The proof shows that the condition for B E S is that the probability of the set {X1 + X2 > x} is asymptotically the same as the probability of its subset {max(Xi, X2) > x}. That is, in the subexponential case the only way X1 + X2 can get large is by one of the Xi becoming large. We later show: Proposition 1.2 If B E S, then P(X1>xI X1+X2>x)--* 2, P(Xi x) 1B(y). That is, given X1 + X2 > x, the r.v. X1 is w.p. 1/2 'typical' (with distribution B) and w.p. 1/2 it has the distribution of X1I X1 > x. In contrast, the behaviour in the light-tailed case is illustrated in the following example: Example 1.3 Consider the standard exponential distribution, B(x) a-x. Then X1 +X2 has an Erlang(2) distribution with density ye-Y so that B*2(x) xe-x. Thus the liminf in Proposition 1.1(b) is oo. As contrast to Proposition 1.2, one can check that

x x where U is uniform on (0, 1). Thus , if X1 + X2 is large , then (with high proba❑ bility) so are both of X1, X2 but none of them exceeds x.

253

1. SUBEXPONENTIAL DISTRIBUTIONS Here is the simplest example of subexponentiality: Proposition 1.4 Any B with a regularly varying tail is subexponential.

Proof Assume B(x) = L(x)/xa with L slowly varying and a > 0. Let 0 < 5 < 1/2. If X1 + X2 > x, then either one of the Xi exceeds (1 - S)x, or they both exceed Sx. Hence lim sup a--+oo

B*2(x) 2B((1 - S)x + B(Sx)2 < lim sup x-aoo

B(x)

lim sup 2L((1 x-ôo

B(x)

- 6)x)/((1 - 5)x)' + 0 _ 2 L(x)l xa (1-6)-

Letting S 10, we get limsupB*2(x)/B(x) < 2, and combining with Proposition ❑ 1.1(b) we get B*2(x)/B(x) -* 2. We now turn to the mathematical theory of subexponential distributions. Proposition 1.5 If B E S, then B(B(x)y) -* 1 uniformly in y E [0, yo] as X -+ 00. [In terms of r.v.'s: if X - B E S, then the overshoot X - xIX > x converges in distribution tooo. This follows since the probability of the overshoot to exceed y is B (x + y)/B(x ) which has limit 1.] Proof Consider first a fixed y. Using the identity + 1)(x) 1+ 2 1 - B*n(x - z B(x) - B*(n ) B(dz) (1.2) B(x) B(x ) B(x) Jo

B*(n+1)(x) = 1+

with n = 1 and splitting the integral into two corresponding to the intervals [0, y] and (y, x], we get

BZ(x)) > 1 + B(y) + B(B(-)y) (B(x) - B(y)) . If lim sup B(x - y)/B(x) > 1, we therefore get lim sup B*2(x)/B(x) > 1+B(y)+ 1 - B(y) = 2, a contradiction. Finally lim inf B(x - y)/B(x) > 1 since y > 0. The uniformity now follows from what has been shown for y = yo and the obvious inequality

1 < B(x ) Y) B( < B( x) 0), B(

y E [0,yo].

0

254 CHAPTER IX. HEAVY TAILS Corollary 1.6 If B E 8, then e"R(x) -* oo,

b[c] = oo for all e > 0.

Proof For 0 < 5 < e, we have by Proposition 1.5 that B(n) > e-6B(n - 1) for all large n so that B(n) > cle-6n for all n. This implies B(x) > c2e-5x for all x, and this immediately yields the desired conclusions. 0 Proof of Proposition 1.2. P(X1 > xIX1 + X2 > x)

_ P(Xi > x) _ B(x) 1 P(X1 + X2 > x) B2(x) 2 1

y

P(X1 x) B(x - z) B(dz) 2B(x) o rv 2 0

2

using Proposition 1.5 and dominated convergence. O The following result is extremely important and is often taken as definition of the class S; its intuitive content is the same as discussed in the case n = 2 above. Proposition 1.7 If B E S, then for any n B*n(x)/B(x) -* n, x

oo.

Proof We use induction. The case n = 2 is just the definition, so assume the proposition has been shown for n. Given e > 0, choose y such that IB*n(x)/B(x) - nI < e for x > y. Then by (1.2), B*(n+1) (x

I x-y + Jxx y) W-(x - z ) ) = 1 + (^ B(x - z) B(dz). B(x) \Jo _ B(x - z) B(x)

Here the second integral can be bounded by B*n(y) B(x) - B(x - y) sup v>o B(v) B(x) which converges to 0 by Proposition 1.5 and the induction hypothesis. The first integral is y B(x - z) B(dz) (n + O(e)) ^x JO B(x) B (x) - B*2 (x) -

(n + 0(0) I

B(x)

(x - z) B(dz) _yBB(x) 111 Lx

255

1. SUBEXPONENTIAL DISTRIBUTIONS

Here the first term in {•} converges to 1 (by the definition of B E S) and the second to 0 since it is bounded by (B(x) - B(x - y))/B(x). Combining these estimates and letting a 4.0 completes the proof. 0 Lemma 1.8 If B E S, e > 0, then there exists a constant K = KE such that B*n(x) < K(1 + e)nB(x) for all n and x. Proof Define 5 > 0 by (1+5)2 = 1+e, choose T such that (B(x)-B*2(x))/B(x) < 1 + b for x > T and let A = 1/B(T), an = supx>o B*n(x)/B(x). Then by (1.2), an+1

fX B*n( *n(x - z) B(x - z) B(dz) x - z) B(dz ) + sup < 1 + sup - z) B(x) f xT 0 B(x < 1 + A + an sup f x B(x - z) B(dz) < 1 + A + an(1 + d) . x>T o B(x) The truth of this for all n together with al = 1 implies an < K(1 + 5)2n where K

=

(1

+

A)/e.

0

Proposition 1.9 Let A1, A2 be distributions on (0, oo) such that Ai (x) _ aiB(x) for some B E S and some constants al, a2 with a1 + a2 > 0. Then Al * A2 (x) - (al + a2)B(x).

Proof Let X1, X2 be independent r.v.'s such that Xi has distribution Ai. Then Al * A2(x) = P(X1 + X2 > x). For any fixed v, Proposition 1.5 easily yields P(X1 + X2 > x, Xi x,X1 > x-v,X2 > x-v) < A1(x-v)A2(x -v)

- ala2B(x)2

which can be neglected, it follows that it is necessary and sufficient for the assertion to be true that JX_VA

(x - y)Ai(dy) = (x)o(1)

(1.3)

Using the necessity part in the case Al = A2 = B yields

f x-v B(x - y)B(dy) = B(x)ov (1)• v

(1.4)

256

CHAPTER LX. HEAVY TAILS

Now (1.3) follows if 'V-V B(x - y)Ai(dy) = B(x)o„(1).

(1.5)

f" By a change of variables, the l.h.s. of (1.5) becomes x B(x - v)Ai(v) - Ai(x - v)B(v) + -_'U Aq(x - y)B(dy). V

Here approximately the last term is B(x)o„(1) by ( 1.4), whereas the two first yield B(x)(Ai(v) - aiB(v)) = B(x)o„(1). ❑ Corollary 1.10 The class S is closed under tail-equivalence. That is, if q(x) aB(x) for some B E S and some constant a > 0, then A E S. Proof Taking Al = A2 = A, a1 = a2 = a yields A*2(x) - 2aB(x) - 2A(x).

❑

Corollary 1.11 Let B E S and let A be any distribution with a ligther tail, A(x) = o(B(x)). Then A * B E S and A * B(x) - B(x) Proof Take Al = A, A2 = B so that a1 = 0, a2 = 1.

❑

It is tempting to conjecture that S is closed under convolution. That is, it should hold that B1 * B2 E S and B1 * B2 (x) - Bl (x) + B2 (x) when B1, B2 E S. However, B1 * B2 E S does not hold in full generality (but once B1 * B2 E S has been shown, B1 * B2 (x) - Bl (x) + B2 (x) follows precisely as in the proof of Proposition 1.9). In the regularly varying case, it is easy to see that if L1, L2 are slowly varying, then so is L = L1 + L2. Hence Corollary 1.12 Assume that Bi(x) = Li(x)lxa, i = 1,2, with a > 0 and L1, L2 slowly varying. Then L = L1 + L2 is slowly varying and B1 * B2(x) sim L(x)/x«. We next give a classical sufficient (and close to necessary) condition for subexponentiality due to Pitman [290]. Recall that the failure rate A(x) of a distribution B with density b is A(x) = b(x)/B(x) Proposition 1.13 Let B have density b and failure rate A(x) such that .(x) is decreasing for x > x0 with limit 0 at oo. Then B E S provided

fo

"O

exA(x) b(x)

dx < oo.

257

1. SUBEXPONENTIAL DISTRIBUTIONS

Proof We may assume that A(x) is everywhere decreasing (otherwise, replace B by a tail equivalent distribution with a failure rate which is everywhere decreasing). Define A(x) = fo .(y) dy. Then B(x) = e-A(x). By (1.2), B*2(x) - 1 B(x) eA( x)-A(x-v )-A(y)A(y) dy f B(x - y ) b(y)dy = B (x) o ox _

J

= ox/2 eA( x)-A(x-y )- A(y)\(y)

Jo

dy + fox/ 2 eA(x

)- A(x-y)-A ( y).(x - y) dy.

0

For y < x/2, A(x) - A(x - y) < yA(x - y) y\(y)• The rightmost bound shows that the integrand in the first integral is bounded by ey"(v)- A(y)a(y ) = ev'(y) b(y), an integrable function by assumption. The middle bound shows that it converges to b(y) for any fixed y since \ (x - y) -* 0. Thus by dominated convergence , the first integral has limit 1 . Since ) (x - y) < A (y) for y < x/2, we can use the same domination for the second integral but now the integrand has limit 0 . Thus B*2(x )/ B(x) - 1 has limit 1 + 0, proving B E S. Example 1.14 Consider the DFR Weibull case B(x) = e-x0 with 0 1, f ' L(y) dy ,,, L(x) y° (a - 1)xcl-1

258

CHAPTER IX. HEAVY TAILS

From this we get Proposition 1.17 If B has a density of the form b(x) = aL(x)/x°+1 with L(x) slowly varying and a > 1, then B(x) - L(x)/x" and )t(x) - a/x. Thus exa(x)b(x) - ea b(x) is integrable. However, the monotonicity condition in Proposition 1.13 may present a problem in some cases so that the direct proof in Proposition 1.4 is necessary in full generality. We conclude with a property of subexponential distributions which is often extremely important: under some mild smoothness assumptions, the overshoot properly normalized has a limit which is Pareto if B is regularly varying and exponential for distributions like the lognormal or Weibull. More precisely, let X W = X - xjX > x, 'y(x) = EXix>. Then: Proposition 1.18 (a) If B has a density of the form b(x) = aL(x)/xa with L(x) slowly varying and a > 1, then 7(x) x/(a - 1) and

P(X (,)/-Y(x) > y) (1 + y/(a - 1))^ ' (b) Assume that for any yo )t(x + y/A(x)) 1 A(x) uniformly for y E (0, yo] . Then 7(x) - 1/A(x) and P(X ixil'Y (x) > y) -* e-'; (c) Under the assumptions of either ( a) or (b), f O B(y) dy - y(x)B(x). Proof ( a): Using Karamata's theorem, we get

EX(x) - E(X - x)+ _ 1 °° P(X > x) P(X>x )J L

PX >y)dy

1 x L(y)/y-dy L(x)/((a1)x'-1) x ( )l ° J °° ()l a x

a-1 Further P ((a - 1)X(x)/x > y) = P(X > x[1 + y/(a - 1)] I X > x) L(x[1 + y/(a - 1)]) xa L(x) (x[1 + y/(a - 1)])a 1 1 . (1 + y/(a - 1))a .

(1.6)

259

2. THE COMPOUND POISSON MODEL

We omit the proof of (c) and that EX (x) - 1/.(x). The remaining statement (1.8) in (b) then follows from

P (A(x)X (x) > y)

= F(X > x + y/.A(x) I X > x) = exp {A(x) - A(x + y/A(x))}

fY

ex p = = + x) dx ex P - f yl a(x) a(x 0 0

A( x + u /A( x))

}

a(x) du

= exp {-y (1 + 0(1))} 0

The property (1.7) is referred to as 1/A(x) being self-neglecting. It is trivially verified to hold for the Weibull- and lognormal distributions , cf. Examples 1.14, 1.15. Notes and references A good general reference for subexponential distribution is Embrechts, Kliippelberg & Mikosch [134].

2 The compound Poisson model Consider the compound Poisson model with arrival intensity /3 and claim size distribution B. Let St = Ei ` Ui - t be the claim surplus at time t and M = sups>0 St, r(u) = inf it > 0; St > u}. We assume p = /3µB < 1 and are interested in the ruin probability V)(u) = P(M > u) = P(r(u) < oo). Recall that B0 denotes the stationary excess distribution, Bo(x) = f0 B(y) dy / µB. Theorem 2 .1 If Bo E S, then Vi(u) P Bo(u). P The proof is based upon the following lemma (stated slightly more generally than needed at present). Lemma 2.2 Let Y1, Y2, ... be i. i. d. with common distribution G E S and let K be an independent integer-valued r.v. with EzK < oo for some z > 1. Then P(Y1 + • • • + YK > u) - EK G(u). nG(u), u -a oo, and that for each Proof Recall from Section 1 that G*n (u) z > 1 there is a D < oo such that G*n(u) < G(u)Dzn for all u. We get p(yl+...+YK> = n)G* n(u ) -- n-0 = ^•P(K 1 •P(K= n)•n = EK, 0 G(u) L G(u) -u)nn-.


260

❑

using dominated convergence with >2 P(K = n) Dz" as majorant.

Proof of Theorem 2.1. The Pollaczeck-Khinchine formula states that (in the set-up of Lemma 2.2) M = Yl + • • • +YK where the Yt have distribution Bo and K is geometric with parameter p, P(K = k) = (1- p)p'. Since EK = p/(1- p) and EzK < oo whenever pz < 1, the result follows immediately from Lemma 2.2. ❑ The condition Bo E S is for all practical purposes equivalent to B E S. However, mathematically one must note that there exist (quite intricate) examples where B E S, Bo ¢ S, as well as examples where B ¢ S, Bo E S. The tail of Bo is easily expressed in terms of the tail of B and the function y(x) in Proposition 1.18, _

B(x^sx Bo(x) µ8 I aoB(y

)dy =

(x).

(2.1)

(^) - ?(xµ 8

In particular , in our three main examples (regular variation , lognormal , Weibull) one has

(

( ) B(x) - x^

lox - µ J

B(x) _ f or

B(x) = e-x'

Bo(x

) - µB(01 - 1)xa-1' vxe-(109x-11)2/202 2 +° /2 µB = eµ Bo(x) eµ+O2/2(log x)2 27r' = µB = F(1/0 )

Bo(x

1

xl-Qe-xp

) ,., r(1/Q)

From this , Bo E S is immediate in the regularly varying case, and for the lognormal and Weibull cases it can be verified using Pitman 's criterion (Proposition 1.13). Note that in these examples , Bo is more heavy-tailed than B . In general: Proposition 2.3 If B E S, then Bo(x)/B(x) -+ 00, x -4 00. Proof Since B(x + y)/B(x) -* 1 uniformly in y E [0, a], we have x+a

fx B(y)dy = a B0 (x) > lim inf lim inf x-+oo B(x) - x-400 PBB(x) PB Leta-+oo.

❑

Notes and references Theorem 2.1 is essentially due to von Bahr [56], Borovkov [73] and Pakes [280]. See also Embrechts & Veraverbeeke [136].

The approximation in Theorem 2.1 is notoriously not very accurate. The problem is a very slow rate of convergence as u -^ oo. For some numerical studies, see Abate,

261

3. THE RENEWAL MODEL

Choudhury & Whitt [1]. Kalashnikov [219] and Asmussen & Binswanger [27]. E.g., in [219] p. 195 there are numerical examples where tp(u) is of order 10-5 but Theorem 2.1 gives 10-10. This shows that even the approximation is asymptotically correct in the tail, one may have to go out to values of 1/'(u) which are unrealistically small before the fit is reasonable. In [1], also a second order term is introduced but unfortunately it does not present a great improvement. Somewhat related work is in Omey & Willekens [278], [279]. Based upon ideas of Hogan [200], Asmussen & Binswanger [27] suggested an approximation which is substantially better than Theorem 2.1 when u is small or moderately large.

3 The renewal model We consider the renewal model with claim size distribution B and interarrival distribution A as in Chapter V. Let U= be the ith claim , T1 the ith interarrival time and Xi = U; - Ti,

Snd) = Xl +... + Xn, M =

sup s$ , t9(u) = inf {n : Snd> > u} . {n= 0,1,...}

Then ik(u) = F ( M > u) = P(i9 (u) < oo). We assume positive safety loading, i.e. p = iB /µA < 1. The main result is: Theorem 3 . 1 Assume that (a) the stationary excess distribution Bo of B is subexponential and that (b) B itself satisfies B(x - y)/B (x) -> 1 uniformly on compact y -internals. Then l/i(u) 1 P

Bo(u)

u -+ 00. (3.1)

P [Note that (b) in particular holds if B E S.] The proof is based upon the observation that also in the renewal setting, there is a representation of M similar to the Pollaczeck-Khinchine formula. To

this end , let t9+ = i9(0) be the first ascending ladder epoch of

{Snd>

},

G+ (A) = P(Sq+ E A,,9+ < oo) = P(S,+ E A, T+ < oo) where r+ = T1 + • • • + T,y + as usual denotes the first ascending ladder epoch of the continuous time claim surplus process {St}. Thus G+ is the ascending ladder height distribution (which is defective because of PB < PA). Define further 0 = IIG+II = P(r9+ < oo). Then K

M=EY, i=1


262

where K is geometric with parameter 9, P(K = k) = (1 - 9)9'' and Y1,Y2,... are independent of K and i.i.d. with distribution G+/9 (the distribution of S,y+ given r+ < oo). As for the compound Poisson model, this representation will be our basic vehicle to derive tail asymptotics of M but we face the added difficulties that neither the constant 9 nor the distribution of the Yi are explicit.

Let F denote the distribution of the Xi and F1 the integrated tail, FI (x) _ fz ° F(y) dy, x > 0. Lemma 3 .2 F(x) - B(x), x -* oo, and hence FI(x) - PBBo(x). Proof By dominated convergence and (b),

B(x) _ J

O°

B(B(x)y) A(dy) f

1 . A(dy) = 1.

0 The lemma implies that (3.1) is equivalent to P(M > u) " -- FI(u),

u -a 00, (3.3)

and we will prove it in this form (in the next Section, we will use the fact that the proof of (3.1) holds for a general random walk satisfying the analogues of (a), (b) and does not rely on the structure Xi = Ui - Ti). Write G+( x) = G+ ( x, oo) = F(S,g+ > x, d+ < oo). Let further 19_ _ inf {n > 0: S^d^
N (this is possible by (b) and Lemma 3.2), and that U_(-n - 1, -n] < (1 + e)/1µc_ I for n > N. We then get lim sup G+(x) x-ro0 Fj(x) o F(x - y) U- (dy)

< lim sup

fN FI ( x)

X---)00

+ lim sup Z-Y00

N F(x - y) U_ (dy) 00

FI (x)

< lim sup F(x) U-(-N, 0] x-+00 FI(x)

00 + lim up 1 x) E F(x + n) U_ (-n - 1, -n] F1 ( n=N _1 1+e E F(x+n) 0 + limsup x-r00 FI(x) FAG- I n=N

E)2 r00 F(x + y) dy + e) lim sup - 1 I ,UG_ I x-,oo Fj(x) N

(1

J

(1 +6)2 I {IC_

I

lim sup X-400

FI(x + N) _ (1 + e)z (x) I Pi µ G_ I

Here in the third step we used that (b) implies B(x)/Bo(x) -+ 0 and hence F(x)/FI(x) -4 0, and in the last that FI is asymptotically proportional to Bo E S. Similarly, > (1 - e) z lim inf G+(x) -

FI (x)

Ip G_ I

❑

Letting a 10, the proof is complete.

Proof of Theorem 3.1. By Lemma 3.3, F(Y= > x) FI(x)/(OIp _ 1). Hence using dominated convergence precisely as for the compound Poisson model, (3.2) yields 00 F F I (u) P(M > u) _ E(1 - 0)0k k I(u) A;=1

BIp G_ I (1- 9)IpG_ I

Differentiating the Wiener-Hopf factorization identity (A.9) 1 - F[s] = (1 - O-[s])(1 - G+[s])

264


and letting s = 0 yields

-µF = -(1 - 1)6+[0] - (1 - IIG+II)µc_ = -(1 - 0)ua_ . Therefore by Lemma 3.2, FJ(u) UBBO(U) PBo(u) N

(1-0)Ipc_I

=

JUA - AB i-P

We conclude by a lemma needed in the next section: Lemma 3 .4 For any a < oo, P(M > u, S+9(u) - Se(u)_1 < a) = o(Fj(u)). Proof Let w(u) = inf {n : Sid) E (u - a, u), Mn < u}. Then P(M E (u - a, u)) > P(w(u) < oo)(i -lp (0))• On the other hand, on the set {M > u, Sty(u) - Sty(u)_I < a} we have w(u) < oo, and {Su,(u)+n - SS(u)}n=o,l,... must attain a maximum > 0 so that P(M > u, S+q(u) - So( u)_1 < a) < P (w(u) < oo)j/i(0) < 0(0) P(M E (u - a, u)). 1-0(0) But since P(M > u - a) N P(M > u), we have P(M E (u - a,u)) = o(P (M > u)) = o(FI(u)).

Notes and references Theorem 3.1 is due to Embrechts & Veraverbeke [136], with roots in von Bahr [56] and Pakes [280].

Note that substantially sharper statements than Lemma 3.4 on the joint distribution of (S,yiui_1,So(u)) are available, see Asmussen & Kliippelberg [36].

4 Models with dependent input We now generalize one step further and consider risk processes with dependent interclaim times, allowing also for possible dependence between the arrival process and the claim sizes. In view of the òne large claim' heuristics it seems reasonable to expect that similar results as for the compound Poisson and renewal models should hold in great generality even when allowing for such dependence.

4. MODELS WITH DEPENDENT INPUT

265

Various criteria for this to be true were recently given by Asmussen, Schmidli & Schmidt [47]. We give here one of them, Theorem 4.1 based upon a regenerative assumption, and apply it to the Markov-modulated model of Chapter VI. For further approaches, examples and counterexamples, see [47]. Assume that the claim surplus process {St}t>o has a regenerative structure in the sense that there exists a renewal process Xo = 0 < Xl x) ,,, Fo(Si > X), (4.3) where

Mnx) = sup

Sxn +t - Sxn =

o 1; (iii) For some o -field Y, X and N. are F-measurable and NX

Po

J:U=>x i=1

(iv) Po

o(B(x))

sup Zt > x / (0:5t<x

Then (4.6) holds with p = ,3PB. Proof It is easily seen that the r.v.'s

FNX U; and ENX Ui - X both have tails of

order EoNx • B(x), cf. the proof of Lemma 4.6 below. The same is true for Sl, since the tail of Zx is lighter than B(x) by (iv), and also for Mix) since Nx

Mix) < > UE + i=1

sup Zt.

o x) dx

IPF- I u

1 EoNxB(x) dx EoX(1 - p) Ju P Bo(u) 1-p

0

4. MODELS WITH DEPENDENT INPUT

269

Example 4 .3 As a first quick application, consider the periodic model of VI.6 with arrival rate /3(t) at time t (periodic with period 1) and claims with distribution B (independent of the time at which they arrive). Assume that B E S, Bo E S, i.e. (i) holds. The regenerative assumption is satisfied if we take Xo = Xi = 0, X2 = 1, X3 = 2,.. ., Zt - 0 (thus (iv) is trivial). The number N. of claims arriving in [0, 1) is Poisson with rate /3 = fo /3(s) ds so that (ii) holds, and taking F = o,(NX), (iii) is obvious. Thus we conclude that (4.6) ❑ holds. Example 4 .4 Assume that St = Zt - t + EN'I Ui where {>N`1 Ui - t} is standard compound Poisson and {Zt} an independent Brownian motion with mean zero and variance constant a2. Again , we assume that B E S, Bo E S; then (iv) holds since the distribution of supo2 7riaiBi

i=1

and we assume p = 01-4 B =

Ep

ri/3ipB;

i=1

< 1.

Theorem 4.5 Consider the Markov-modulated risk model with claim size distributions satisfying (4.9). Then (4.6) holds.

The key step of the proof is the following lemma.


270

Lemma 4 . 6 Let (N1, ... , NP ) be a random vector in {0, 1, 2 , ...}P, X > 0 a r.v. and F a a-algebra such that (N1, ... , NP ) and X are .F-measurable. Let {Fi}t=1 P be a family of distributions on [0, oo) and define p Ni

Yx = EEX'i - X i=1 j=1

where conditionally upon F the Xi, are independent with distribution Fi for Xij. Assume EzN-1+"'+Np < oo for some z > 1 and all i, and that for some + cp distribution G on [0, oo) such that G E S and some c1, ... , cp with cl + > 0 it holds that Fi(x) - ciG(x). Then P

P(Yx > x) - c'(x)

where c = ciENi . i=1

Proof Consider first the case X = 0. It follows by a slight extension of results from Section 1 that P

P(Yo > x I Y) G( x) ci Ni,

P(Yo > x

I

+Np

^ ) < CG(x)zN1+

i =1

for some C = C(z) < oo. Thus dominated convergence yields

( P(Yo>x P(Yo>x .^•) G(x)

= E\

G(x)

P -^ E ciNi = C. i-1

In the general case, as x -a oo, P

P(YX

> x I.F) = P(Yo > X+x I •^)

G (x +x)>2ciNi i=1

P

- G( x )

> ciNi ,

i=1

and

P(Yx > x ^) < P(Y0 > x I.F) < CG(x)zn'1+,"+Np . The same dominated convergence argument completes the proof.

❑

Proof of Theorem 4.5. If Jo = i, we can define the regenerations points as the times of returns to i, and the rest of the argument is then just as the proof of Corollary 4.2. An easy conditioning argument then yields the result when Jo is ❑ random. For light-tailed distributions, Markov-modulation typically decreases the adjustment coefficient -y and thereby changes the order of magnitude of the ruin

5. FINITE-HORIZON RUIN PROBABILITIES

271

probabilities for large u, cf. VI.4. It follows from Theorem 4.5 that the effect of Markov-modulation is in some sense less dramatical for heavy-tailed distributions: the order of magnitude of the ruin probabilities remains ft°° B(x) dx. Within the class of risk processes in a Markovian environment, Theorem 4.5 shows that basically only the tail dominant claim size distributions (those with c, > 0) matter for determining the order of magnitude of the ruin probabilities in the heavy-tailed case. In contrast, for light-tailed distributions the value of the adjustment coefficient -y is given by a delicate interaction between all B. Notes and references Theorem 4. 5 was first proved by Asmussen, Floe Henriksen & Kliippelberg [31] by a lengthy argument which did not provide the constant in front of Bo(u) in final form. An improvement was given in Asmussen & Hojgaard [33], and the final reduction by Jelenkovic & Lazar [213]. The present approach via Theorem 4.1 is from Asmussen, Schmidli & Schmidt [47]. That paper also contains further criteria for regenerative input (in particular also a treatment of the delayed case which we have omitted here), as well as a condition for (4.6) to hold in a situation where the inter-claim times (T1,T2.... ) form a general stationary sequence and the U; i.i.d. and independent of (T1,T2.... ); this is applied for example to risk processes with Poisson cluster arrivals. For further studies of perturbations like in Corollary 4.2 and Example 4.4, see Schlegel [316].

5 Finite-horizon ruin probabilities We consider the compound Poisson model with p = /3pB < 1 and the stationary excess distribution Bo subexponential. Then O(u) - pl(1 - p)Bo(u), cf. Theorem 2.1. As usual, r(u) is the time of ruin and as in IV.7, we let PN"N = P(. I T(u) < oo). The main result of this section, Theorem 5.4, states that under mild additional conditions, there exist constants -Y(u) such that the F(u)distribution of r(u)/y(u) has a limit which is either Pareto (when B is regularly varying) or exponential (for B's such as the lognormal or DFR Weibull); this should be compared with the normal limit for the light-tailed case, cf. IV.4. Combined with the approximation for O(u), this then easily yields approximations for the finite horizon ruin probabilities (Corollary 5.7). We start by reviewing some general facts which are fundamental for the analysis. Essentially, the discussion provides an alternative point of view to some results in Chapter IV, in particular Proposition 2.3.

5a Excursion theory for Markov processes Let until further notice {St} be an arbitrary Markov process with state space E (we write Px when So = x) and m a stationary measure, i.e. m is a (or-finite)

272 CHAPTER IX. HEAVY TAILS measure on E such that

L for all measurable A C E and all t > 0. Then there is a Markov process {Rt} on E such that

fE m(dx)h(x)Exk(Rt) = Lm(dy)k(y)Eyh(St)

(5.2)

for all bounded measurable functions h, k on E; in the terminology of general Markov process theory, {St} and {Rt} are in classical duality w. r. t. m. The simplest example is a discrete time discrete state space chain, where we can take h, k as indicator functions, for states i, j, say, and (5.2) with t = 1 means m;rij = mjsji where r13,s=j are the transition probabilities for {St}, resp. {Rt}. Thus, a familiar case is time reversion (here m is the stationary distribution); but the example of relevance for us is the following: Proposition 5.1 A compound Poisson risk process {Rt} and its associated claim surplus process {St} are in classical duality w .r.t. Lebesgue measure. Proof Starting from Ro = x, Rt is distributed as x + t - >N` Ui, and starting from So = y, St is distributed as y - t + EI U; (note that we allow x, y to vary in, the whole of R and not as usual impose the restrictions x > 0, y = 0). Let G denote the distribution of ENt U, - t. Then (5.2) means

ffh(a,)k(x - z) dx G(dz) = ffh(y + z) k(y)dy G(dz). The equality of the l.h.s. to the r.h.s. follows by the substitution y = x - z.

❑

For F C E, an excursion in F starting from x E F is the (typically finite) piece of sample path' {St}o 0: St 0 F} .

We let QS be the corresponding distribution and Qx,y = Qx (. Sw(F.)- = y, w(Fc) < oo ) 'In general Markov process theory, a main difficulty is to make sense to such excursions also when Px(w(F°) = 0) = 1. Say {St} is reflected Brownian motion on [0,00), x = 0+ and F = (0, oo). For the present purposes it suffices , however , to consider only the case Px(w(F`) = 0) 0.

5. FINITE-HORIZON RUIN PROBABILITIES 273 y E F (in discrete time, Sw(Fo)_ should be interpreted as Sw(F^)_1). Thus, Qx y is the distribution of an excursion of {St} conditioned to start in x E F and terminate in y E F. QR and QRy are defined similarly, and we let Qy y refer to the time reversed excursion . That is,

Qx,y(-) = P ({SW(F`)-t-} 0 0, the one in (b) is the time reversed path. The theorem states that the path in (b) has the same distribution as an excursion of {Rt} conditioned to start in y < 0 and to end in x = 0. But in the risk theory example (corresponding to which the sample paths are drawn), this simply means the distribution of the path of {Rt} starting from y and stopped when 0 is hit. In particular: Corollary 5.3 The distribution of r(0) given r(0) < oo, S,(0)_ = y < 0 is the same as the distribution of w(-y) where w(z) = inf It > 0 : Rt = z}, z > 0. [note that w(z) < oo a.s. when p = ,13AB < 1] Proof of Theorem 5.2. We consider the discrete time discrete state space case only (well-behaved cases such as the risk process example can then easily be handled by discrete approximations). We can then view Qy,y as a measure on all strings of the form i0i1 ... in with i0, i1, ... , in E F, io = x, in = y, /^s x (S1 = Z1, ... , Sn = in = y; Sn+1 E Fc) nx,y(2p21 ...itt) = P Px(w(Fc) < 00, Sw(F)-1 = y)


274 note that Fx(w(Fc) < 00, S, (Fc)-1 = y) 00

E E Px (Si = 21i ... , S. = in = y; Sn+1 E Fc) n=1 i1,...,i„_iEF

Similarly, t' y and Qy x are measures on all strings of the form ipi l ... in with 20,ii ,...,in E F, i0 =

QyR x(2p21 ...

y,

in)

in =

-

x,

Pt' (R1 = ii, ... , Rn = in = x; Rn+1 E

F (w(Fc) < 00,

F`)

R ,(F y. Let Y = Yl = Sr+( 1) be the value of the claim surplus process just after the first ladder epoch , Z = Zl = ST+( 1)_ the value just before the first ladder epoch (these r.v.'s are defined w.p. 1 w .r.t. P(o) ), see Fig. 5.2.

U T(O) = T (u)

Y

Figure 5.2 The distribution of (Y, Z) is described in Theorem 111.2.2. The formulation relevant for the present purposes states that Y has distribution Bo and that conditionally upon Y = y, Z follows the excess distribution B(Y) given by B(Y) (x) _ B(y + x)/B(y). We are interested in the conditional distribution of T(u) = T(0) given {T(0) < oo, S,(o) > y} = {T(0) < oo, Y > y} , that is, the distribution w.r.t. P(") = P(. 7-(0) < oo, Y > u). Now the P(u,')distribution of Y-u is Bo"). That is, the P(u,')-density of Y is B(y)/[,UBBo(u)], y > u. Bo") is also the P(u,')-distribution of Z since

P(Z>aIY>u) =

1

°° B(y) B(y + a) dy FLBBo(u) B (y)

J°° (z) dy - B(a) +a PBBo(u)

276


Let {w(z)}Z^,o be defined by w(z) = inf It > 0: Rt = z} where {Rt} is is independent of {St}, in particular of Z. Then Corollary 5.3 implies that the P("'1)-distribution of T(u) = r(0) is that of w(Z). Now Bo E S implies that the Bo ")(a) -+ 0 for any fixed a, i.e. P(Z < a I Y > u) -3 0. Since w(z)/z a$. 1/(1 - p), z -^ oo, it therefore follows that T(u)/Z converges in Pi"'')probability to 1/(1 - p). Since the conditional distribution of Z is known (viz. Bo") ), this in principle determines the asymptotic behaviour of r(u). However, a slight rewriting may be more appealing. Recall the definition of the auxiliary function y(x) in Section 1. It is straightforward that under the conditions of Proposition 1.18(c)

Bo")(yY (u)) -+ P(W > y)

( 5.3)

where the distribution of W is Pareto with mean one in case ( a) and exponential with mean one in case (b). That is , Z/'y(u) -* W in Pi "' ')-distribution . r(u)/Z -4 1/(1 - p) then yields the final result T(u)/y(u) -+ W/(1 - p) in Pi"'')distribution. We now turn to the general case and will see that this conclusion also is true in P(")-distribution:

Theorem 5 . 4 Assume that Bo E S and that (5.3) holds. Then 7-(u)/-y(u) --^ W/(1 - p) in F(u) -distribution.

In the proof, let r+(1) = T(0),T+(2),... denote the ladder epochs and let Yk, Zk be defined similarly as Y = Y1, Z = ZI but relative to the kth ladder segment, cf. Fig. 5.3. Then, conditionally upon r+ (n) < oo, the random vectors (YI, Z1),. .. , (Y,,, Zn), are i.i.d. and distributed as (Y, Z). We let K(u) = inf In = 1, 2, ...: r+ (n) < oo, Y1 + • • • + Yn > u} denote the number of ladder steps leading to ruin and P("'n) = P(• I r(u) < oo, K(u) = n). The idea is now to observe that if K(u) = n, then by the subexponential property Yn must be large, i.e. > u with high probability, and YI,... , Yn_1 'typical'. Hence Z,, must be large and Z1,.. . , Zn_1 'typical' which implies that the first n-1 ladder segment must be short and the last long; more precisely, the duration T+ (n) - r+ (n - 1) of the last ladder segment can be estimated by the same approach as we used above when n = 1, and since its dominates the first n - 1, we get the same asymptotics as when n = 1.

5. FINITE-HORIZON RUIN PROBABILITIES 277

16

Z3

Z1

r+(1) T+(1) T+(1)

Figure 5.3 In the following, II ' II denotes the total variation norm between probability measures and ® product measure. Lemma 5.5 Ilp(u,n) (y1, ... , Y„-1, Yn - u) E •) - Bo (ri-1) ®B(,,u) II 0. Proof We shall use the easily proved fact that if A'(u), A"(u) are events such that P(A'(u) AA"(u)) = o(F (A'(u)) (A = symmetrical difference of events), then

IIP( I A'(u)) -

P(. I A"(u ))II -+ 0.

Taking A'(u) = {Y,, > u}, A"(u) _ {K(u)=n} = {Y1+

+Yn-1u},

the condition on A'(u) A A"(u) follows from Bo being subexponential (Proposition 1.2, suitably adapted). Further, P(. I A'(u)) = P(u,n),

P (Yj,...,Yn-1iYn - u) E • I A'(u)) = Bo (n-1) ®Bou) .


278 Lemma 5 .6

IIPIu'n )

((Z1'..., Zn) E •) - Bo (n-1) ®Bo'

0.

Proof Let (Y11, Z11),..., (Y,,, Zn), be independent random vectors such that the conditional distribution of Zk given Y.' = y is BM, k = 1, ... , n, and that Yk has marginal distribution B0 for k = 1, . . . , n - 1 and Y„ - u has distribution Bout That is, the density of Yn is B(y)/[IBBO(u)], y > u. The same calculation as given above when n = 1 shows then that the marginal distribution of Zn is Bou). Similarly (replace u by 0), the marginal distribution of Zk is Bo for k < n, and clearly Zi, ... , Zn are independent. Now use that if the conditional distribution of Z' given Y' is the same as the conditional distribution of Z given Y and JIF(Y E •) - P(Y' E •)II -* 0, then 11P(Z E •) - P(Z' E •)II -> 0 (here Y, Y', Z, Z' are arbitrary random vectors, in our example Y = (Y1, ... , Y") ❑ etc.). Proof of Theorem 5.4. The first step is to observe that K(u) has a proper limit distribution w.r.t. P(u) since by Theorem 2.1, n_1 < u, Y1 +... + Y" > u) Flul (K (u ) = n) _ Cu) P"F(1'i +...+y 1 p"F(Yn > u) P)Pn-1 P/(1 - P) Bo(u) for n = 1, 2, .... It therefore suffices to show that the P(u'")-distribution of T(u) has the asserted limit. Let {wl(z)},..., {wn(z)} be i.i.d. copies of {w(z)}. Then according to Section 5a, the F'-distribution of r(u) is the same as the P'-distribution of w1(Zl) + • • • + wn(Zn). By Lemma 5.6, wk(Zk) has a proper limit distribution as u -+ oo for k < n, whereas wn(Zn) has the same limit behaviour as when n = 1 (cf. the discussion just before the statement of Theorem 5.4). Thus F(u'n)(T(u) /7(u) >

y)

=

F(u'n)((wl (Z1)

+ ...

+wn(Z n))l7( u )

>

1y)

^' P(u'n)(wn (Zn)/7(u) > y) -4 NW/(1 - P) > y)

Corollary 5.7 O (u,,y(u)T) - 1 P PBo(u) • P(W/(1 - p) < y).

Notes and references Excursion theory for general Markov processes is a fairly abstract and advanced topic. For Theorem 5.2, see Fitzsimmons [144]), in particular his Proposition (2.1).

6. RESERVE-DEPENDENT PREMIUMS 279 The results of Section 5b are from Asmussen & Kluppelberg [36] who also treated the renewal model and gave a sharp total variation limit result . Extensions to the Markov-modulated model of Chapter VI are in Asmussen & Hojgaard [33]. Asmussen & Teugels [53] studied approximations of i (u, T) when T -+ oo with u fixed; the results only cover the regularly varying case.

6 Reserve-dependent premiums We consider the model of Chapter VII with Poisson arrivals at rate /3, claim size distribution B, and premium rate p(x) at level x of the reserve. Theorem 6 .1 Assume that B is subexponential and that p(x) -> 00, x -> oo. Then (6.1)

0 (u) Qf "O ^) dy. u

The key step in the proof is the following lemma on the cycle maximum of the associated storage process {Vt}, cf. Corollary II. 3.2. Assume for simplicity that {Vt} regenerates in state 0 , i.e. that fo p(x)-1 dx < oo, and define the cycle as a = inf{t>0: Vt=0, max VB>0I Vo=0^ o<s u is geometric with parameter q(u) = P(Mo > u I Vo = u). Hence f (u)r(u) = D(u) = Do(u) - P(MT > u) $B(u) Ft µ(1 - q ( u)) 1 - q(u) Now just use that p(x) -* oo implies q (x) -+ 0.

❑

Notes and references The results are from Asmussen [22], where also the (easier) case of p(x) having a finite limit is treated . It is also shown in that paper that typically, there exist constants c(u) -4 0 such that the limiting distribution of r(u)/c(u) given r(u) < oo is exponential.

Chapter X

Simulation methodology 1 Generalities This section gives a summary of some basic issues in simulation and Monte Carlo methods . We shall be brief concerning general aspects and refer to standard textbooks like Bratley, Fox & Schrage [77], Ripley [304], Rubinstein [310] or Rubinstein & Melamed [311] for more detail ; topics of direct relevance for the study of ruin probabilities are treated in more depth.

la The crude Monte Carlo method Let Z be some random variable and assume that we want to evaluate z = EZ in a situation where z is not available analytically but Z can be simulated. The crude Monte Carlo ( CMC) method then amounts to simulating i.i.d. replicates Zl,... , ZN, estimating z by the empirical mean (Z1 + • • + ZN)/N and the variance of Z by the empirical variance N s2

=

N

z) 2 = Zit NE i-i i-i

E(Z{

-

2.

According to standard central limit theory , vrN-(z - z) 4 N(0, 4Z), where a2 = Var(Z ). Hence 1.96s z f (1.2) is an asymptotic 95% confidence interval , and this is the form in which the result of the simulation experiment is commonly reported.

281

282

CHAPTER X. SIMULATION METHODOLOGY

In the setting of ruin probabilities, it is straightforward to use the CMC method to simulate the finite horizon ruin probability z = i,b(u, T): just simulate the risk process {Rt} up to time T (or T n 7-(u)) and let Z be the indicator that ruin has occurred,

Z = I inf Rt < 0 (0 0 and L(x) slowly varying. Then (cf. Theorem IX.2.1) V)(u) - p/(l - p)Bo(x), and the problem is to produce an estimator Z(u) with a variance going to zero not slower (in the logarithmic sense ) than Bo(u)2. A first obvious idea is to use conditional Monte Carlo: write i,b(u)

=

P (Xl

+•••+XK>u)

= EF[Xl + ...+XK > uIXl,...,XK-1] = EBo(u-X1 - ...-XK_1).

Thus, we generate only X1, ... , XK-1, compute Y = u - X1 - - XK_1 and let Z( 1)(u) = Bo (Y) (if K = 0, Z(1)(u) is defined as 0). As a conditional Monte Carlo estimator , Z(1) (u) has a smaller variance than Zl (x). However, asymptotically it presents no improvement : the variance is of the same order of magnitude F(x). To see this, just note that EZ(1)(u ) 2

> E[Bo (x - Xl - ... - SK-1)2; Xl > x, K > 2] = P2p(Xl > x) = P2Bo(x)

(here we used that by positivity of the X;, X1 + + XK_ 1 > x when X1 > x, and that Bo(y) = 1, y < 0). This calculation shows that the reason that this algorithm does not work well is that the probability of one single Xi to become large is too big. The idea of [27] is to avoid this problem by discarding the largest X; and considering only the remaining ones. For the simulation, we thus generate K and X1i ... , XK, form the order statistics X(1) < X(2) < ... < X(K)

throw away the largest one X(K), and let Z(2)(u)

=

P (SK

> u I X(l),X(2),...,X(K-1))

_

B0((u

- S( K_1)) V X(K-1))

/ Bo(X(K -1))

where S(K_l) = X(1) + X(2) + • • • + X(K_1). conditional probability, note first that

P(X(n) > x I X(1),X(2),...,X(n_1))

To check the formula for the

Bo(X(„_l) V X) Bo(X(n-1))

3. IMPORTANCE SAMPLING VIA LUNDBERG CONJUGATION 287 We then get

P(S"

> x I

X( 1), X(2), ...

, X(n-1))

X X(1), X(2),

P(X(TZ) + S(,_1) > P(X(n) > _

Bo((x

X

... ,

X (. -l))

- S(n_1) I X(1), X(2), ... , X(n-1))

- S (n-1)) V

X (. -l))

BO(X(n-1)) Theorem 2 . 1 Assume that Bo (x) = L(x)/x° with L(x) slowly varying. Then the algorithm given by { Z (2) (u) } is logarithmically efficient. Notes and references The proof of Theorem 2.1 is elementary but lengty, and we refer to [27]. The algorithm is sofar the only efficient one which has been developed for the heavy-tailed case. Asmussen , Binswanger and HOjgaard of [28] give a general survey of rare events simulation for heavy -tailed distributions , and that paper contains one more logarithmically efficient algorithm for the compound Poisson model using the Pollaczeck- Khinchine formula and importance sampling . However , it must be noted that a main restriction of both algorithms is that they are so intimately tied up with the compound Poisson model because the explicit form of the Pollaczeck-Khinchine formula is crucial (say, in the renewal or Markov- modulated model P(r+ < oo) and G+ are not explicit ). Also in other respects the findings of [28] are quite negative: the large deviations ideas which are the main approach to rare events simulation in the light-tailed case do not seem to work for heavy tails.

3 Importance sampling via Lundberg conjugation We consider again the compound Poisson model and assume the conditions of Ce-7", use the the Cramer-Lundberg approximation so that z(u) = '(u) - u is the = ST(") where ^(u) = e-7"ELe-7E(") representation 0(u) = e-7sr(u) instead of 13L, BL overshoot (cf. 111.5), and simulate from FL, that is, using = e-rysr(u). 0, B, for the purpose of recording Z(u) For practical purposes, the continuous-time process {St} is simulated by considering it at the discrete epochs {Qk} corresponding to claim arrivals. Thus, the algorithm for generating Z = Z(u) is: 1. Compute -y > 0 as solution of the Lundberg equation 0 = K(y) = )3(B[y] - 1) - y, and define )3L, BL by I3L = /3B[-y], BL(dx) = e7sB(dx)/B[y].

288


2. Let Sf-0

3. Generate T as being exponential with parameter ,l3 and U from B. Let S - S+U - T. 4. If S > u, let Z F e_'s. Otherwise, return to 3. There are various intuitive reasons that this should be a good algorithm. It resolves the infinite horizon problem since FL(,r(u) < oo) = 1. We may expect a small variance since we have used our knowledge of the form of 0(u) to isolate what is really unknown, namely ELe-ry£("), and avoid simulating the known part e-7". More precisely, the results of IV.7 tell that P(. r(u) < oo) and FL (both measures restricted to.F,(u)) are asymptotically coincide on {r(u)} < oo, so that changing the measure to FL is close to the optimal scheme for importance sampling , cf. the discussion at the end of Section 1b. In fact: Theorem 3.1 The estimator Z(u) = e-'rs* "u) (simulated from FL) has bounded relative error. Proof Just note that EZ(u)2 < e - 2ryu _ z (u)2/C2.

❑

It is tempting to ask whether choosing importance sampling parameters ,Q, b different from ,QL, BL could improve the variance of the estimator . The answer is no. In detail , to deal with the infinite horizon problem , one must restrict attention to the case 4µB > 1. The estimator is then

M(u) /3e-QT' dB Z(u) (Ui) j=1 )3 e $Ti dB where M(u) is the number of claims leading to ruin, and we have: Theorem 3.2 The estimator (3.1) (simulated with parameters ^3, B) is not logarithmically efficient when (/3, b) # (/3L, BL).

The proof is given below as a corollary to Theorem 3.3. The algorithm generalizes easily to the renewal model . We formulate this in a slightly more general random walk setting '. Let X1, X2, ... be i.i.d. with distribution F, let S,, = X1 + ... + X,,, M(u) = inf {n : S„ > u}, and assume that µF < 0 and that F[y] = 1, P'[-y] < oo for some ry > 0. Let FL (dx) = 'For the renewal model, Xi = U; -Ti, and the change of measure F -r FL corresponds to B -> BL, A -> AL as in Chapter V.

3. IMPORTANCE SAMPLING VIA LUNDBERG CONJUGATION

289

e7yF(dx). The importance sampling estimator is then Z( u) = e-'rSM( ). More generally, let F be an importance sampling distribution equivalent to F and

M(u) dF Z(u) _ I -(Xi) . (3.2) dF Theorem 3.3 The estimator (3.2) (simulated with distribution F of the X3 has bounded relative error when .P = FL. When F # FL, is not logarithmically efficient. Proof The first statement is proved exactly as Theorem 3 . 1. For the second, write

W(F IF) _ -F(XI)... -F(XM(u)). By the chain rule for Radon-Nikodym derivatives,

EFZ(u)2

= EeW2(FIF) = Ep [W2(FIFL)W2(FLIF)] = EL [W2 ( FIFL)w(FLIF)] = ELexp {Kl+...+KM(u)},

where

Kl og

(X) (j) 2

)

= -log dFL (Xi) - 2'X1 .

Here ELK; = c'- 2ryELXi, where

e' = -EL Iog dFL (Xi) > 0 by the information inequality. Since K1, K2, ... are i.i.d., Jensen's inequality and Wald's identity yield EpZ(u)2

> exp {EL(K1 + ... + KM(u))} = exp {ELM(u)(E - 2ryELXi)} .

Since ELM(u)/u -+ 1/ELXi, it thus follows that for 0 < e < e'/ELXi, EFZ(u)2 EFZ(u)2 lim sup z(u)2eeU = lim cop C2e-2,yu+elu u -+oo e-try' 1 > lim up C2e-2,yu = G,2 > 0,

290

CHAPTER X. SIMULATION METHODOLOGY ❑

which completes the proof.

Proof of Theorem 3.2. Consider compound Poisson risk process with intensities /3', /3", generic interarrival times T' , T", claim size distributions B', B" and generic claim sizes U', U". Then according to Theorem 3.3, all that needs to be shown is that if U' - T' = U" - T", then /3' B' = B". First by the memoryless distribution of the exponential distribution , U' - T' has a left exponential tail with rate /3' and U" - T" has a left exponential tail with rate /3'. This immediately yields /3' = 3". Next, from

P(U'-T'>x) ^ =

^

/3'e-Q'YR'( x + y) dy = ,3'eO'x f f

e-Q'zB (z) dz,

P (U" - T" > x)

J

/3"e-0 yB (x + y) dy = ,3"eQ x

0

J

e-Q zB (z) dz

x

(x > 0) and /3' = /3", U' - T' D U" - T", we conclude by differentiation that Bo(x)=B' (x)forallx > 0,i.e.B'=B".

❑

Notes and references The importance sampling method was suggested by Siegmund [343] for discrete time random walks and further studied by Asmussen [ 13] in the setting of compound Poisson risk models . The optimality result Theorem 3.1 is from Lehtonen & Nyrhinen [244], with the present (shorter and more elementary) proof taken from Asmussen & Rubinstein [45]. In [13], optimality is discussed in a heavy traffic limit y 10 rather than when u -+ oo. The extension to the Markovian environment model is straightforward and was suggested in Asmussen [ 16]. Further discussion is in Lehtonen & Nyrhinen [245]. The queueing literature on related algorithms is extensive , see e.g. the references in Asmussen & Rubinstein [45] and Heidelberger [190].

4 Importance sampling for the finite horizon case The problem is to produce efficient simulation estimators for '0 (u, T) with T < oo. As in IV.4, we write T = yu. The results of IV.4 indicate that we can expect a major difference according to whether y < 1/r,'(-y) or y > 1/r.'(-y). The easy case is y > 1/k'(-y) where O(u, yu) is close to zk(u), so that one would expect the change of measure F -4 FL to produce close to optimal results. In fact: Proposition 4.1 If y > 1/ic'('y), then the estimator Z(u) = e-7Sr(°)I(r(u) < yu) (simulated with parameters /3L, BL) has bounded relative error.

4. IMPORTANCE SAMPLING FOR THE FINITE HORIZON CASE

291

Proof The assumption y > 1/n'(-y) ensures that 1fi(u, yu)/z,(u) -* 1 (Theorem IV.4.1) so that z(u) = zP(u, yu) is of order of magnitude a-71. Bounding ❑ ELZ(u)2 above by a-7u, the result follows as in the proof of Theorem 3.1. We next consider the case y < 1/r.'(7). We recall that ay is defined as the solution of a'(a) = 1/y, that ryy = ay - yk(ay) determines the order of magnitude of z'(u, yu) in the sense that - log 4')u) -4 7y

u (Theorem IV.4.8), and that ryy > ry. Further

,O(u, yu) = e-ayu Eay Le-ay^(u)+r(u)K(ay); T(u) < yu] . (4.2) Since the definition of ay is equivalent to Eay r(u) - yu, one would expect that the change of measure F Pay is in some sense optimal. The corresponding estimator is Z(u) = e-avS' ( u)+T(u)K (ay)I(T( u) < yu),

(4.3)

and we have: Theorem 4.2 The estimator (4.3) (simulated with parameters /gay, Bay) is logarithmically efficient. Proof Since ryy > -y, we have ic(ay ) > 0 and get

Eay Z(u)2 =

Eay

[e

-

2aySr( u)+2r(u )r.(ay);

e-2ryyuEay

e

le-

T( u) < yu]

2ay^(u); T(u)
2 - to g x ( u ) u

so that (1.5) follows.

❑

Remark 4 . 3 Theorem IV.4.8 has a stronger conclusion than (4.1), and in fact, (4.1) which is all that is needed here can be showed much easier . Let Qy2 =


292

Vara„ (-r(u))/u so that (T(u) - yu)/(uyu1/2) . N(0,1) (see Proposition IV.4.2). Then z(u) =

Eay Z(u)

>

Eay

avS'(u)+T( u)k(av

Le-

1 ); yu - o ,u1/2 < r(u) < yu

l

= e- a yu +l/ur' (av)Eiàv re-av^(u)+(T(++)(U)

yu - o.yu1/2 e-7vu +avul/ 2r.(av)Eav l e- a vt(u); yu - Qyu1/2 < T(u) C yu e- ryyu

+oy u1/2K'(av)Eo

] l

l

1/2)

v

where the last step follows by Stam's lemma (Proposition IV.4.4). Hence lira inf log

-ryyu + vyu 1/2 tc(ay) - -7y x(u) > hm inf

u-+Oo U - u-aoo U

That lim sup < follows similarly but easier as when estimating En,, Z (u)2 above. 0 Notes and references The results of the present section are new. In Asmussen [13], related discussion is given in a heavy traffic limit q J. 0 rather than when u -3 oo.

5 Regenerative simulation Our starting point is the duality representations in 11.3: for many risk processes {Rt }, there exists a dual process { V t} such that i,b(u,T)

=

P

inf Rt < 0 = P(VT > u),

O) where Zi'i = w, is the cycle length,

Z2'> the time during the cycle where { Vt} exceeds u and zj = EZJ'), j = 1, 2. Then Z(1), ... , Z(N) are i . i.d. and

EZ1'i = z1 = Ew, EZ2'i = z2 = E

J0 'o I (Vt > u) dt .

Thus, letting

Z1 = (Zl1i +... + Z1N>) ,

Zl the LLN yields Z1

a$'

z 1, Z2

a4*

Z2 =

Z(1)

N (X21' + ... + Z2N)) ,

+...+Z(N)

z2,

(u) ?2 = E fo I(Vt > u) dt = 0( u ) zl Ew as N -> oo. Thus, the regenerative estimator z%(u) is consistent.

To derive confidence intervals , let E denote the 2 x 2 covariance matrix of Z('). Then (Z1-z1i Z2-z2 ) 4 N2(0,E).

Therefore , a standard transformation technique (sometimes called the delta method) yields 1 V 2 (h (Zi, Z2 - h (zl, z2)) -> N(O, oh) for h : R2 -^ R and Ch = VhEVh, Vh = (8h/8z1 8h/ 8z2). Taking h(zl, z2) z2/z1 yields Vh = (-z2/z2i 1/zl),

(^(u) - t(u)) 4

N(0, 02)

(5.3)

294


where

2

01 2

=

Z2

Z2

Eli

z1

+ 2 E22 - 2 E1 2 z1 z1

The natural estimator for E is the empirical covariance matrix N

S = N 1 12 (ZW - Z) ^Z(=) - z^ i=1

so a2 can be estimated by 2 2 = 72 S11+ 12 S22 - 2- g S12 (5.5)

Z1 Z1 Z1 and the 95% confidence interval is z1 (u) ± 1.96s/v"N-. The regenerative method is not likely to be efficient for large u but rather a brute force one. However , in some situations it may be the only one resolving the infinite horizon problem , say risk processes with a complicated structure of the point process of claim arrivals and heavy -tailed claims . There is potential also for combining with some variance reduction method. Notes and references The literature on regenerative simulation is extensive, see e.g. Rubinstein [310] and Rubinstein & Melamed [311].

6 Sensitivity analysis We return to the problem of 111 . 9, to evaluate the sensitivity z/i( (u ) = (d/d() 0(u) where ( is some parameter governing the risk process . In 111.9, asymptotic estimates were derived using the renewal equation for z /i(u). We here consider simulation algorithms which have the potential of applying to substantially more complex situations. Before going into the complications of ruin probabilities , consider an extremely simple example , the expectation z = EZ of a single r.v. Z of the form Z = ^p(X) where X is a r . v. with distribution depending on a parameter (. Here are the ideas of the two main appfoaches in today 's simulation literature: The score function ( SF) method . Let X have a density f (x, () depending on C. Then z(() = f cp(x) f (x, () dx so that differentiation yields zS d( fco(x)f(x,C)dx = f w(x) d( f ( x, () dx f Ax) (dl d()f (x' () f ( z, () dx = E[SZ] f(X,0

295

6. SENSITIVITY ANALYSIS where

S = (d/d()f (X, () = d log f (X, C) f(X,() d( is the score function familiar from statistics . Thus, SZ is an unbiased Monte Carlo estimator of z(. Infinitesimal perturbation analysis (IPA) uses sample path derivatives. So assume that a r.v. with density f (x, () can be generated as h(U, () where U is uniform(0,1). Then z(() = Ecp(h(U, ()),

zc = E [d( co(h(U, C)), = E [`d (h(U, ()) d( hc(U, C), where h( (u, () = (8/8()h (u, () Thus, cp' (h(U, ()) h((U, () is an unbiased Monte Carlo estimator of zS. For example , if f (x, () _ (e-Sx, one can take h (U, () = - log U/(, giving h( (U, () = log U/(2. The derivations of these two estimators is heuristic in that both use an interchange of expectation and differentiation that needs to be justified. For the SF method, this is usually unproblematic and involves some application of dominated convergence . For IPA there are, however , non-pathological examples where sample path derivatives fail to produce estimators with the correct expectation. To see this, just take cp as an indicator function , say W(x) = I(x > xo) and assume that h(U, () is increasing in C. Then , for some Co = (o(U), cp(h(U, ()) is 0 for C < Co and 1 for C > Co so that the sample path derivative cp'(h(U, ()) is 0 w . p. one. Thus , IPA will estimate zS by 0 which is obviously not correct. In the setting of ruin probabilities , this phenomenon is particularly unpleasant since indicators occur widely in the CMC estimators . A related difficulty occurs in situations involving the Poisson number Nt of claims: also here the sample path derivative w.r.t. /3 is 0. The following example demonstrates how the SF method handles this situation. Example 6 .1 Consider the sensitivity tka(u) w.r.t. the Poisson rate /3 in the compound Poisson model. Let M(u) be the number of claims up to the time r(u) of ruin (thus, r(u) = Tl + • • • +TM(u)). The likelihood ratio up to r(u) for two Poisson processes with rates /3, /3o is M(u)

Oe

-(3T:

11 /3oe-OoT; I(r(u)

< oo) .

296


Taking expectation, differentiating w.r.t. j3 and letting flo = 0, we get

1 M(u) 00(u)

= E

(_Ti)I(T(U) u. That is, Y'a(U) = P(T

(u)

= r+(a)) =

1 - P(•r(u, a) = r(u)),

wherel T(u) = inf {t > 0 : Rt < 0} ,

T+(a) = inf It > 0 : Rt > al,

T(u, a) = r(u) A T+(a).

Besides its intrinsic interest , Oa(U ) can also be a useful vehicle for computing t/i(u) by letting a -* oo. Consider first a Bernoulli random walk, defined as Ro = u (with u E {0,1,.. .... }), R„ = u+X,+• • •+X,, where X1, X2, ... are i.i.d. and {-1,1}-valued with P(Xk = 1) = 9.

'Note that in the definition of r(u ) differs from the rest of the book where we use r(u) = inf {t > 0 : Rt < 0} ( two sharp inequalities ); in most cases , either this makes no difference (P(R.,(u) = 0 ) = 0) or it is trivial to translate from one set-up to the other, as e.g. in the Bernoulli random walk example below.

297

CHAPTER XI. MISCELLANEOUS TOPICS

298

Proposition 1.1 For a Bernoulli random walk with 0 0 1/2, C1_0\a- (1-B)u

I\ e

oJ 0- (u)

=

1 oa ' ()i

a

=

u,

u + 1,.... (1.1)

o

If 0 = 1/ 2, then 'Oa(u) _

au a

We give two proofs , one elementary but difficult to generalize to other models, and the other more advanced but applicable also in some other settings. Proof 1. Conditioning upon X1 yields immediately the recursion 'a(1) = 1-9+00a(2), _ (1 - o)T/la (1) + 8z/'u(3),

tba(2)

7/la(a - 2)

= (1-9)4/'0(a-3)+9ba(a-1),

Oa(a - 1)

= (1 - o)'t/1a(a - 2),

❑

and insertion shows that ( 1.1) is solution.

Proof 2. In a general random walk setting , Wald's exponential martingale is defined as in 11.(4.4) by ea(u+Xl+...+Xn) F[ a]n n=0,1,...

where a is any number such that Ee°X = F[a] 1/2, a el u

1h (u) =

\1

If 9 < 1/ 2, then Vi(u) = 1. ❑

Proof Let a-+ oo in (1.1).

Proposition 1.3 Let {Rt} be Brownian motion starting from u and with drift p and unit variance . Then for p 0 0, ,•

a-2µa

-

e-2µu

,ba(u) = e-2µa - 1

If p = 0, then

'Oa (U)

--

a-u a

Proof Since

Eea(R°- u) =

et(a2

/2 +aµ)

the Lundberg equation is rye/2-'yp = 0 with solution y = 2p. Applying optional stopping to the exponential martingale {e-7R, } yields e-7u

= Ee-7R° = e°Wa(u) + e-7a(1 - 0a(u)),

and solving for 9/la(u) yields Z/)a(u) = (e -76 - e-7u)/(e-7° - 1) for p # 0. If p = 0, {Rt} is itself a martingale and just the same calculation as in the ❑ proof of Proposition 1.1 yields 't/la(u) = (a - u)/u.

Corollary 1.4 For a Brownian motion with drift u > 0, i1(u) = e-211 . If p u) where MT = maxou,STu,ST>U), P(MT > u) = P(ST > u) + P(ST < u, MT > u) = P (ST > u) + P (ST > u, MT > U)

= P(ST > u) + P(ST > u) (1.9) = 2P(ST > u).

Corollary 1.8 Let {Rt} be Brownian motion with drift - µ so that {St} is Brownian motion with drift µ . Then the density and c.d.f. of -r(u) are ( U2 Pµ (T(u ) E dT) =

Pµ (T(u) < T)

2^T -3/2 exp µu - 2

( 1.10)

, + µ2T) } ,

)

!....µ%T (1.11) = 1 - 4) I = - µ ✓T I + e2µ"4) ( - VIT

Proof For p = 0, (1.11 ) is the same as (1.8 ), and (1 . 10) follows then by straightforward differentiation. For µ # 0, the density dPµ / dP0 of St is eµst-tµ2/2, and hence Pµ('r(u) E dT)

= Eo [e µsr(,.)_ _( u)µ2 /2; T(u) E dT, = eµu-Tµ2/2Po (T(

u) E dT) 2

eµu-Tµ2/2

u T-3/2 ex p u 27r p 1-2 T


302

which is the same as (1.10). (1.11) then follows by checking that the derivative of the r.h.s. is (1.10) and that the value at 0 is 0. ❑ Small modifications also apply to Bernoulli random walks: Proposition 1.9 For Bernoulli random walk with 9 = 1/2, Vi(u,T) = P(ST = u) + 2P (ST > u),

(1.12)

whenever u, T are integer-valued and non-negative. Here {2-T( (v-}TT)/2) v=-T,-T+2,...,T-2,T

P(ST = v) = 0 otherwise. Proof The argument leading to ( 1.9) goes through unchanged, and (1.12) is the same as ( 1.9). The expression for F ( ST = v) is just a standard formula for the ❑ binomial distribution. The same argument as used for Corollary 1.8 also applies to the case 9 54 1/2, but we omit the details. We finally consider a general diffusion {Rt} on [0, oo) with drift µ(x) and variance a2 (x) at x. We assume that u(x) and a2 (x) are continuous with a2 (x) > 0 for x > 0. Thus, close to x {Rt} behaves as Brownian motion with drift µ = u(x) and variance a2 = a2(x), and in a similar spirit as in VII.3 we can define the local adjustment coefficient y(x) as the one -2µ(x)/a2(x) for the locally approximating Brownian motion. Let

s(y) = ef0 ry(.T)dx, S(x) = f x s(y)dy, S(oo) = f c s(y)dy. 0 0

(1.13)

The following results gives a complete solution of the ruin problem for the diffusion subject to the assumption that S(x), as defined in (1.13) with 0 as lower limit of integration, is finite for all x > 0. If this assumption fails, the behaviour at the boundary 0 is more complicated and it may happen, e.g, that 0(u), as defined above as the probability of actually hitting 0, is zero for all u > 0 but that nevertheless Rt ^4 0 (the problem leads into the complicated area of boundary classification of diffusions, see e.g. Breiman [78] or Karlin & Taylor [222] p. 226). Theorem 1.10 Consider a diffusion process {Rt} on [0, oo), such that the drift µ(x) and the variance a2(x) are continuous functions of x and that a2(x) > 0

1. RANDOM WALK; BROWNIAN MOTION; TWO BARRIERS

303

for x > 0. Assume further that S (x) as defined in (1.13) is finite for all x > 0. If (1.14) S(oo) < 00, then 0 < 2l.(u) < 1 for all u > 0 and (1 . 15)

^ S^ Conversely, if (1.14) fails, then,0(u) = 1 for all u > 0.

Lemma 1.11 Let 0 < b < u < a and let t&0,b(u) be the probability that {Rt} hits b before a starting from u. Then YIa,b(u) = S(a) - S(u) (1.16) S(a) - S(b) Proof Recall that under mild conditions on q, E„ q(Rdt) = q(u)+Lq(u)dt, where Lq(u) = 0'22u) q "(u) + p(u)q(u) is the differential operator associated with the diffusion. If b < u < a, we can ignore the possibility of ruin or hitting the upper barrier a before dt, so that Y)n,b('u) = Eu &0,b(Rdt), and we get Wo,b('u) = Eu , O,b (Rdt) = Oa,b(u) + L,ba,b(u)dt,

i.e LVa, b = 0. Using s'/ s = -2p/a2, elementary calculus shows that we can rewrite L as Lq(u)

d

1a2 (u)s(u)d

?

[ s (u)

] . (1.17)

Hence L,ba,b = 0 implies that VQ b/s is constant, i.e. Wa,b = a+/3S. The obvious boundary conditions '0a,b(b) = 1, 1', ,b(a) = 0 then yield the result. 0 Proof of Theorem 1.10. Letting b J. 0 in (1.16) yields 4b (u) = 1 - S(u)/S(a). Letting a T oo and considering the cases S(oo) = oo, S(oo) < oo separately ❑ completes the proof. Notes and references All material of the present section is standard. A classical reference for further aspects of Bernoulli random walks is Feller [142]. For generalizations of Proposition 1.6 to Markov-modulated models , see Asmussen & Perry [42]. Further references on two-barrier ruin problems include Dickson & Gray [116], [117]. A good introduction to diffusions is in Karlin & Taylor [222]; see in particular pp. 191-195 for material related to Theorem 1.10. In view of (1.16), the function S(x) is

304


referred to as the natural scale in the general theory of diffusions (in case of integrability problems at 0, one works instead with a lower limit 5 > 0 of integration in (1.13)). Another basic quantity is the speed measure M , defined by the density 1/va(u)s(u) showing up in (1.17). Markov-modulated Brownian models , with the drift and the variance depending on an underlying Markov process , is currently an extremely active area of research; much of the literature dels with the pure drift case, correponding to piecewise linear paths or , equivalently, variance 0, which is motivated from the study of modern ATM (asynchronous transfer mode ) technology in telecommunications. The emphasis is often on stationary distributions , but by duality, information on ruin probabilities can be obtained . See Asmussen [20] and Rogers [305] for some recent treatments and references to the vast literature.

2 Further applications of martingales Consider the compound Poisson model with adjustment coefficient ry and the following versions of Lundberg 's inequality (see Theorems 111.5.2, 111 .6.3, IV.4.5): _ z/'(u) < e 7u,

(2.1)

C_e-7u < t(u) < C+e _7u,

(2.2)

where C_ =

B(x) _ B(x) sup 2no fy° e7(Y )B(dy)' f2e7(Y-2)B(dy)' C+ i/i(u, yu) '+/1(u) - t&(u, yu)

- (7) , (2.4)

where W (ay) = y, 7y = ay - ytc (ay). (2.5) A martingale proof of (2.1 ) was given already in II.1, and here are alternative martingale proofs of the rest . They all use the fact that ( tx(a) l ( e-aRt -

I.

= e-au + aSt-tx(a)

Lo

I.

Lo

is a martingale (cf. Remark 11.4.9 ) and optional stopping applied to the stopping time r(u) A T, yielding e-au = Ee- aRo - o•K(a) = Ee - aR,(,.)AT - (T(u)AT) r.(a)

(2.6)

2. FURTHER APPLICATIONS OF MARTINGALES

305

(we cannot use the stopping time r(u) directly because P(-r(u) = oo) > 0 and also because the conditions of the optional stopping time theorem present a problem). Proof of ( 2.2): As noted in Proposition II.1.1 , it follows easily from (2.6) with = 'y that e--yu

- E [e- 7R,( u ) I

T(U)

r, when Rt_ = r. Equivalently, -Rt has distribution B(r + dy)/B(r). Hence ^00 H( dt, dr

E [e-7Rr (u) Jr(u) < ool

JO Zoo

) f e7'B(r + dy) B(r) Jo

^00 ^00 H(dt, dr) e 7( y-r)B(dy) B(r) fr oo o 0 >

dr) 1 = 1 I0 /oH(dt, C+ C+

From this the upper inequality follows, and the proof of the lower inequality is similar. ❑ Proof of (2.3), (2.4): We take a = ay in (2.6). For (2.3), we have tc(ay) > 0 and we can bound (2.6) below by 1 E Le-7Rr(,.)-r(u)r.(ay)I T(u) < yu] P(r(u) < yu)

> e-yu"(ay ) iYj(u,yu)

(using RT(u) < 0), so that i/1(uL yu) < e-ayu , eyuk (ay) = e-7yu e

Similarly for (2.4), we have ic(ay ) < 0 and use the lower bound E [e-7Rr („)- T(u)K(ay) I yu < r(u) < T] F(yu < r(u) < T) > e- yuk (ay)(u&(u,T) -

V,(u,yu))•

Letting T -+ oo yield e_ayu

Notes and references

> e-yur4ay)(0(u) -

See II.1.

b(u, yu))•


306

3 Large deviations The area of large deviations is a set of asymptotic results on rare event probabilities and a set of methods to derive such results. The last decades have seen a boom in the area and a considerable body of applications in queueing theory, not quite so much in insurance risk. The classical result in the area is Cramer's theorem. Cramer considered a random walk Sn = X1 + ... + X. such that the cumulant generating function r.(B) = log EeOX 1 is defined for sufficiently many 0, and gave sharp asymptotics for probabilities of the form P (S,,/n E I) for intervals I C R. For example, if x > EX1, then

C

S.

P

> ^ x e -nn 1 n 0o 2xn

(3.1)

where we return to the values of 0, ri, v2 later. The limit result (3.1) is an example of sharp asymptotics : - means (as at other places in the book) that the ratio is one in the limit (here n -* oo). However , large deviations results have usually a weaker form, logarithmic asymptotics , which in the setting of (3.1) amounts to the weaker statement lim 1 log P I Sn > x I = -17. n--roo n n ///

Note in particular that (3.1) does not capture the \ in (3. 1) but only the dominant exponential term - the correct sharp asymptotics might as well have +,3na with a < 1. Thus , large deviations results been, e.g., cle - nn or C2e-,?n typically only give the dominant term in an asymptotic expression . Accordingly, logarithmic asymptotics is usually much easier to derive than sharp asymptotics but also less informative . The advantage of the large deviations approach is, however , its generality, in being capable of treating many models beyond simple random walks which are not easily treated by other models , and that a considerable body of theory has been developed. og gn if For sequences fn, gn with fn -+ 0 , gn -4 0, we will write fn 1-

lim 109 fn = 1

n-ioo

log gn

(later in this section, the parameter will be u rather than n). Thus, (3.2) can be rewritten as F (Sn/n > x) 1-g a-'fin. Example 3.1 We will go into some more detail concerning (3.1), (3.2).

3. LARGE DEVIATIONS

307

Define rc* as the convex conjugate of rc, rc*(x) = sup(Ox - r.(0)) e (other names are the entropy, the Legendre-Fenchel transform or just the Legendre transform or the large deviations rate function). Most often, the sup in the definition of rc* can be evaluated by differentiation: rc*(x) = Ox - rc(0) where 0 = 0(x) is the solution of x = rc'(0), which is a saddlepoint equation - the mean rc'(0) of the distribution of X1 exponentially tilted with 0, i.e. of P(X1 E dx) = E[e9X1-K.(e)i XI E dx],

(3.3)

is put equal to x. In fact, exponential change of measure is a key tool in large deviations methods. Define ,q = rc* (x). Since

P

nn

>

x)

=

E {e_8

' ( 9).

S

rtn

> x

1,

replacing Sn in the exponent and ignoring the indicator yields the Chernoff bound P Sn > x 1 < e-°n (3.4) n Next, since Sn is asymptotically normal w.r.t. P with mean nx and variance no, 2 where o2 = o2(x) = rc"(0), we have

P(nx < Sn < nx + 1.960/) -* 0.425, and hence for large n P(Sn/n > x) >

>

E [e- 9S„+n' ( 9); nx < Sn < nx + 1.96o /]

0.4 e-nn +1.sseo f

which in conjunction with (3.4) immediately yields (3.2). More precisely, if we replace Sn by nx + o / V where V is N(0,1), we get P(Sn/n > x)

E [e-9nx +nK(9)-9" '; V > 0 e-9o^y

e- tin f o o')

= e-tin

1 Bo 27rn

1

1 e-y2/2 dy

21r


308

which is the same as (3.1), commonly denoted as is the saddlepoint approximation. The substitution by V needs, however, to be made rigorous; see Jensen ❑ [215] or [APQ] p. 260 for details. Further main results in large deviations theory are the Gartner-Ellis theorem, which is a version of Cramer's theorem where independence is weakened to the existence of c(O) = limn,o log Ee9Sn /n, Sanov's theorem which give rare events asymptotics for empirical distributions, Mogulskii's theorem which gives path asymptotics, that is, asymptotics for probabilities of the form P ({S[nti/n}o µ + r7) < -0(1i + r7) + i(p(0 +'Y))/p n-+oo n

and by Taylor expansion, it is easy to see that the r.h.s. can be chosen strictly negative by taking p close enough to 1 and 0 close enough to 0. The rest of the argument is as before. ❑ Proof of Theorem 3.2. We first show that lim inf„_,,,,, log zl'(u)/u > -'y. Let r7 > 0 be given and let m = m(77) = [u(1 + 77)/µ] + 1. Then

V, ( U)

P(S,n > u ) =

Em 1e- -YS- +r-- W;

km e-7Sm+n.m(7). S. >

[

1

[em

mµ

+17] m(7);

Em

Sm > u]

S,n - > - µ?7

m µ 1 + rl


310

>

]Em I e- YS +^c

IL exp

1-_

(7); I

`S- - I < µl1 I 1+77 M

1+277 S,n Yµ 1 + m + r ('Y) } U n \ 77

m µ

µ7 1

< 1+ 77 )

Here E,,,(•) goes to 1 by Lemma 3. 3, and since Ic,n(ry)/u -4 0andm/u-* (1 + r7)/µ, we get lum inf

z/i(u) +12r7 >_1-ry + 77

Letting r7 J. 0 yields liminfu __,,,. logO(u)/u > -ry. For lim supu,0 log i'(u )/u < -'y, we write 'i/I(u) _ E00P(T(u) = n) = Il + I2 + I3 + I4 n=1

where n(b) Lu(1-0/µJ Ii = 1: F(T(u) = n), I2 = F(T(u) = n), n=n(b)+1

n=1

Lu(1 +6) /µJ 13

00

P(T(u) n), 14 = = E Lu(1-6)/aJ+1 Lu(1+6)/µJ+l

F( T (u)

=

= n)

and n(S) is chosen such that icn('y )/n < 6 A (- log z) /2 and Sn Fn\

n

> la+ 8 I < zn >lb+S) n(E); this is possible by (iii), (iv) and Lemma 3.3. Obviously,

P(T(u) = n) < P(Sn > u) = En [e-7S,.+wn(7); Sn > u] < e-Yu+Kn(7)pn(Sn > u) (3.7) so that

n(b) I1 < e-'Yu E en.(-Y), n=1

311

3. LARGE DEVIATIONS Lu(1-6)/µJ

I2

e'n(Y)P(Sn > u)

< e-"u n=n(6)+1

Lu(1-6)/µJ ^, e-ryu e-n logz/2p n nt n, -µ n=n(6)+1 \

0. Assuming that the further regularity conditions can be verified, Theorem 3.2 then immediately yields the estimate log F( sup Skh > u) a-7u (3.12) k=0,1,...

for the ruin probability z/-'h(u) of any discrete skeleton {Skh}k=0,1,.... The problem is whether this is also the correct logarithmic asymptotics for the (larger) ruin probability O(u) of the whole process, i.e. whether P ( sup St > u ltg a ^" 0 0 such that ic('y) = 0 and that r. (9) < oo for 9 < 'y + C. If the nth claim arrives at time Qn = s, it contributes to St by the amount Un(t - s). Thus by (3.14),

Kt (0)

t (Ee9U"it-8i - 1) ds - 9t = /3 t (Ee8U° i8l - 1) ds - 9t, J J0 0

and since EeOUn(8) -+ Ee°U^ as s -* oo, we have rct (9)/t -4 ic (9). Since the remaining conditions of Theorem 3.2 are trivial to verify, we conclude that Cu) log e-7 u (cf. the above discussion of discrete skeletons). It is interesting and intuitively reasonable to note that the adjustment coefficient ry for the shot - noise model is the same as the one for the Cramer -Lundberg model where a claim is immediately settled by the amount Un. Of course, the Cramer- Lundberg model has the larger ruin probability. 0 Example 3 . 7 Given the safety loading 77, the Cramer-Lundberg model implicitly assumes that the Poisson intensity /3 and the claim size distribution B (or at least its mean µB) are known. Of course , this is not realistic . An apparent solution to this problem is to calculate the premium rate p = p(t) at time t based upon claims statistics . Most obviously, the best estimator of /3µB based upon Ft-, where Ft = a(A8 : 0 < s < t), At = ;'`1 U;, is At - It. Thus, one would take p(t) = (1 + rt)At-/ t, leading to

St = At-(1+77)

Jo t

S8 ds.

(3.15)

315

3. LARGE DEVIATIONS With the Qi the arrival times, we have Nt

St =

Ui

t

N.

Ui

- (1 +i) f > i= 1

i=1 o

Nt

/

t

1 - (1 + r7) log t (3.16)

ds = E Ui

01i

s

i=1

Let ict (a) = log Eeast . It then follows from (3.14) that

rt _ 13

(a [1_( i+77)log]) ds_flt = t (a)

Jo

K(a)

_

o 1 O (a[I + (1 + 77) log u]) du

(3.17)

-)3. (3.18)

f

Thus (iii) of Theorem 3.2 hold, and since the remaining conditions are trivial to verify, we conclude that t,b(u) IN a-'Yu (cf. again the above discussion of discrete skeletons) where y solves ic('y) = 0

It is interesting to compare the adjustment coefficient y with the one y* of the Cramer-Lundberg model, i.e. the solution of (3.19)

/3(Eelu - 1) - (1 + 17)0µB = 0. Indeed, one has y

>

y'

(3.20)

with equality if and only if U is degenerate. Thus, typically the adaptive premium rule leads to a ruin probability which is asymptotically smaller than for the Cramer-Lundberg model . To see this , rewrite first rc as

te(a) _ /3E

1

eau 1 - /3. (3.21)

1 +(1+77)aUJ

This follows from the probabilistic interpretation Si

EN '1 Yi where

Yi = Ui( 1+(1 +r7)log ©i) = Ui(1-(1 +17)Vi) where the Oi are i .i.d. uniform (0,1) or , equivalently, the Vi = - log Oi are i.i.d. standard exponential , which yields eau f 1 t(1+n )audtl = E r Ee°Y = E [O(1+n)aueaul = E [eau J L Jo J L1+(l+r))aUJ

316


Next, the function k(x) = e7*x - 1 - (1 + ri)y*x is convex with k(oo) = 00, k(0) = 0, k'(0) < 0, so there exists a unique zero xo = xo(r7) > 0 such that k(x) > 0, x > x0, and k(x) < 0, 0 < x < x0. Therefore

e7'U _ k(U) E [1+(1+77)y*U] - 1 E [1+(1+77)y*U] 0 k (+ *y B(+ 1 + (1(+71)y*y B(dy) L xa 1 +

f

+ (1 + rl) Y* xo jJxo k(y) B(dy

) + f' k(y) B(dy) } = 0,

using that Ek(U) = 0 because of (3.19). This implies n(y*) < 0, and since tc(s), a* (s) are convex with tc'(0) < 0 , rc*' (0 ) < 0, this in turn yields y > y*. Further, y = y* can only occur if U - xo. 11 Notes and references Some standard textbooks on large deviations are Bucklew [81], Dembo & Zeitouni [105] and Shwartz & Weiss [339]. In addition to Glynn & Whitt [163], see also Nyrhinen [275] for Theorem 3.2. For Example 3.7, see Nyrhinen [275] and Asmussen [25]; the proof of (3.20) is due to Tatyana Turova.

Further applications of large deviations idea in risk theory occur in Djehiche [122], Lehtonen & Nyrhinen [244], [245], Martin-L6f [256], [257] and Nyrhinen [275].

4 The distribution of the aggregate claims We study the distribution of the aggregate claims A = ^N' U; at time t, assuming that the U; are i.i.d. with common distribution B and independent of Nt. In particular, we are interested in estimating P(A > x) for large x. This is a topic of practical importance in the insurance business for assessing the probability of a great loss in a period of length t, say one year. Further, the study is motivated from the formulas in IV.2 expressing the finite horizon ruin probabilities in terms of the distribution of A.

The main example is Nt being Poisson with rate fit. For notational simplicity, we then take t = 1 so that p,, = P(N = n) = e-(3an

However, much of the analysis carries over to more general cases, though we do not always spell this out.

4. THE DISTRIBUTION OF THE AGGREGATE CLAIMS

317

4a The saddlepoint approximation We impose the Poisson assumption (4.1). Then Ee"A = e'(") where x(a) _ 0(B[a] - 1). The exponential family generated by A is given by Pe(A E dx) = E [eeA -K(9); A E dx] . In particular, no(a) = logE9e'A = rc(a + 9) - ic(9) = ,3e(bo[a] - 1) where )30 = ,3B[9] and Be is the distribution given by

eox B9(dx) = B [9] B(dx). This shows that the Pe-distribution of A has a similar compound Poisson form as the F-distribution, only with 0 replaced by a9 and B by B9. The analysis largely follows Example 3.1. For a given x, we define the saddlepoint 9 = 9(x) by EBA = x, i.e. K'(0) _ ic'(9) = x. Proposition 4.1 Assume that lim8T8. B"[s] = oo, B"' [s] lim (B",[s])3/2 = 0, 818' where s' = sup{s : B[s] < oo}. Then as x -* oo, e-9x+K(°) P(A > x)

B 2ir /3 B" [9] Proof Since EBA = x, Vare(A) = s;"(0) = ,3B"[9], (4.2) implies that the limiting Pe-distribution of (A - x)//3B"[9] is standard normal. Hence

P(A > x) = E e [e-9A+ ic(9); A > x)] = e-ex+K( e)E9 [e - 9(A-x); A > x)

x)]]

e-ex+K(e ) e-e AB°[ely 1 e-v2/2 dy 0 2^ 00 -9x+p(e) e e-ze-z2/(2BZpB „[9)) dz 9 27r/3B" [9] fo e-ex+w ( e)

J

oo z

e-ex+w(B)

dz

0 27r /3B" [9] o e 9 2 /3B" [9]

318


It should be noted that the heavy-tailed asymptotics is much more straightforward. In fact, just the same dominated convergence argument as in the proof of Theorem 2.1 yields: Proposition 4.2 If B is subexponential and EzN < oo for some z > 1, then P(A > x) - EN B(x). Notes and references Proposition 4.1 goes all the way back to Esscher [141], and (4.2) is often referred to as the Esscher approximation. The present proof is somewhat heuristical in the CLT steps. For a rigorous proof, some regularity of the density b(x) of B is required. In particular, either of the following is sufficient: A. b is gamma-like, i.e. bounded with b(x) - ycix °-ie-6x B. b is log-concave, or, more generally, b(x) = q(x)e-h(z), where q(x) is bounded away from 0 and oo and h (x) is convex on an interval of the form [xo,x') where x' = sup {x : b(x) > 0}. Furthermore 00 b(x)Sdx < oo for some ( E (1, 2). For example, A covers the exponential distribution and phase-type distributions, B covers distributions with finite support or with a density not too far from a-x° with a > 1. For details, see Embrechts et al. [138], Jensen [215] and references therein.

4b The NP approximation In many cases , the distribution of A is approximately normal . For example, under the Poisson assumption (4.1), it holds that EA = ,l3pB, Var(A) _ ^3p.2i and that (A - (3µB)/(0µB^))1/2 has a limiting standard normal distribution as Q -^ oo, leading to

P(A > x) :; 1 - (D X - Q{AB

(4.3)

The result to be surveyed below improve upon this and related approximations by taking into account second order terms from the Edgeworth expansion. Remark 4 . 3 A word of warning should be said right away : the CLT (and the Edgeworth expansion) can only be expected to provide a good fit in the center of the distribution . Thus , it is quite questionable to use (4.3) and related results ❑ for the case of main interest , large x. The (first order) Edgeworth expansion states that if the characteristic function g(u) = Ee"`}' of a r.v. Y satisfies 9(u) ti e-u2/2(1 + ibu3) (4.4)


319

where b is a small parameter, then P(Y < y) 4(y) - 6(1 - y2)^P(y)• Note as a further warning that the r.h.s. of (4.5) may be negative and is not necessarily an increasing function of y for jyj large. Heuristically, (4.5) is obtained by noting that by Fourier inversion, the density of Y is

9(y) =

°° _1 e-iuy f(u) du 2x _. f °o

1 e-'uye -u2/2(1 + iSu3) du 27r

_ cc(y) - 5 (y3 - 3& (y), and from this (4.5) follows by integration. In concrete examples , the CLT for Y = Y6 is usually derived via expanding the ch.f. as u2 u3 u4 9(u) = Ee'uY = exp {iuci - 2X2 - 2K3 + 4i 64 + .. . where Kl , ,c2i... are the cumulants ; in particular, s;l = EY,

K2 = Var (Y), K3 = E(Y - EY)3.

Thus if EY = 0, Var(Y) = 1 as above , one needs to show that 163, K4 .... are small. If this holds , one expects the u3 term to dominate the terms of order u4, u5, ... so that 1(u)

3 exp { - 2 2 - i 3 K3 } Pt^ exp - 2 ^ \1 - i 6 r 1 3

so that we should take b = -ic3/6 in (4.5). Rather than with the tail probabilities F(A > x), the NP (normal power) approximation deals with the quantile al_E, defined as the the solution of P(A < yl-e) = 1 - e. A particular case is a.99, which is often denoted VaR (the Value at Risk). Let Y = (A - EA)/ Var(A) and let yl_E, zl_e be the 1 - e-quantile in the distribution of Y, resp. the standard normal distribution. If the distribution of Y is close to N(0,1), yl-E should be close to zl_E (cf., however, Remark 4.3!), and so as a first approximation we obtain

a1_E = EA + yl-e Var(A) .: EA + zl_E Var(A) . (4.6)


320

A correction term may be computed from (4.5) by noting that the 4;(y) terms dominate the S(1 - y2)cp( y) term. This leads to

1-

-t( yl -E) - 6 (1 - yi- E)A1✓ l -E)

E

4)(yl -E)

- 5(1 - zl-

E)ô(zl -E)

^' ..

4(z1-E) + ( yl-E - zl -E)V(zl_E) - S(1 - zl-E)W(zl-E)

=

1 - E + (yl- E - zl-E )w(zl _E)

- S(1 - zi- E )Azl -E)

which combined with S = -EY3/6 leads to

1

q^

Y1 - E = z1-E + S(zi_E

Using Y = (A - EA ) /

- 1)EY3.

Var(A), this yields the NP approximation

6(Z1 _E - 1) E (A - EA)3 a1_E = EA + z1_E(Var (A))1/2 + 1

Var(A)

Under the Poisson assumption (4.1), the kth cumulant of A is /3PBk' and so s;k = /3µB^1 / (,6pBki) d/2. In particular , k3 is small for large /3 but dominates 1c4, K5 .... as required . We can rewrite (4.7) as 1 (3) a1-E = Qµa +z1 - E(/3PB^1 )1^2 + s(z1-E - 1)^ 2) µ'E

Notes and references We have followed largely Sundt [354]. Another main reference is Daykin et at. [101]. Note, however, that [101] distinguishes between the NP and Edgeworth approximations.

4c Panjer 's recursion Consider A =

EN 1 U%, let pn

= P(N = n), and assume that there exist

constants a, b such that

Pn =

(a+

n

) Pn_i , n = 1, 21 ....

For example, this holds with a = 0, b = /3 for the Poisson distribution with rate /3 since Pn

ê-Q

,3n-i /3

= -Pn-1 n! n (n - 1)! n


321

Proposition 4.4 Assume that B is concentrated on {0, 1, 2,.. .} and write gj = 2 , . . fj = P(A = j), j = 0,1..... Then fo = >20 9onpn and

fi = 1 E

(a + b!) 1-ag k_1 3

gkfj- k

, j = 1, 2, ... .

(4.10)

In particular, if go = 0, then j

f o = po, fj = E (a+ b

)9kfi_k

, j = 1, 2.....

(4.11)

k =1

Remark 4.5 The crux of Proposition 4.4 is that the algorithm is much faster than the naive method, which would consist in noting that (in the case go = 0)

fj = pn9jn

(4.12)

n=1

where g*n is the nth convolution power of g, and calculating the gj*n recursively by 9*1 = 9j, j-1

g; n =

9k(n-1 )9j -k •

(4.13)

k=n-1

Namely, the complexity (number of arithmetic operations required) is O(j3) for (4.12), (4.13) but only O(j2) for Proposition 4.4. ❑ Proof of Proposition 4.4. The expression for fo is obvious. By symmetry, E[a +bU=I >Ui =j l

(4.14)

i=1 J

is independent of i = 1, . . . , n. Since the sum over i is na + b, the value of (4.14) is therefore a + b/n. Hence by (4.9), (4.12) we get for j > 0 that

fj

a

-

b +

n

*n

n p

n-lgj

00 U I n *n = E a+b-1 Ui=j pn-19j n=1

j

i=1

CC)

EE n=1 Ia

n

+b Ul i=1

Ui

=j pn_1


322 00 J

EE (a + bk I gkg3 _ k lien-i n=ik=0 (a+bk l gkE g j'`kpn = E (a+b!)9kfi_k n=0 k=0 k=0 Î 1 E(a+b. agofj+ k Jgkfj-k, k=i /

and (4.9) follows . (4.11) is a trivial special case. and

❑

If the distribution B of the Ui is non-lattice , it is natural to use a discrete approximation . To this end, let U(;+, U(h) be U; rounded upwards, resp. downwards , to the nearest multiple of h and let A}h) = EN U. An obvious modification of Proposition 4.4 applies to evaluate the distribution F(h) of A(h) letting f( ) = P(A() = jh) and

g(h)

= P (U(h2 = kh) = B((k + 1)h) - B(kh ), k = 0, 1, 2, ... ,

gkh+

= P (U4;+ = kh) = B(kh) - B (( k - 1)h) = gk - l,-, k = 1, 2, ... .

Then the error on the tail probabilities (which can be taken arbitrarily small by choosing h small enough ) can be evaluated by 00

00

f! h)

< P(A > x ) f (h) j=Lx/hl j=Lx/hl Further examples ( and in fact the only ones , cf. Sundt & Jewell [355]) where (4.9) holds are the binomial distribution and the negative binomial (in particular, geometric ) distribution . The geometric case is of particular importance because of the following result which immediately follows from by combining Proposition 4.4 and the Pollaczeck-Khinchine representation: Corollary 4.6 Consider a compound Poisson risk process with Poisson rate 0 and claim size distribution B. Then for any h > 0, the ruin probability zb(u) satisfies 00 00 f^,h) Cu) < E ff,+, j=Lu/hJ j=Lu/hJ

(4.15)

5. PRINCIPLES FOR PREMIUM CALCULATION

323

where f^ +, f^ h) are given by the recursions (h)

3 (h)

(h)

fj,+ = P 9k fj-k,+ ' I = 17 2, .. . k=1

(h)

=

P

3 (h)

(h)

f9,- - (h) gk,-fA-k,- e 1 - ago,- k=1

j = 1+2,

starting from fo + = 1 - p, f(h) 07 = (1 - p)/(1 - pgoh-) and using g(kh)

1 (k+1)h

=

Bo((k + 1 ) h) - Bo(kh ) = - f AB

gkh+

Bo(kh ) - Bo((k - 1 ) h) = 9kh)1 ,

B(x) dx, k = 0, 1, 2, ... ,

kh

k = 1,2 .....

Notes and references The literature on recursive algorithms related to Panjer's recursion is extensive, see e.g. Dickson [115] and references therein.

5 Principles for premium calculation The standard setting for discussing premium calculation in the actuarial literature does not involve stochastic processes, but only a single risk X > 0. By this we mean that X is a r.v. representing the random payment to be made (possibly 0). A premium rule is then a [0, oo)-valued function H of the distribution of X, often written H(X), such that H(X) is the premium to be paid, i.e. the amount for which the company is willing to insure the given risk. The standard premium rules discussed in the literature (not necessarily the same which are used in practice!) are the following: The net premium principle H(X) = EX (also called the equivalence principle). As follows from the fluctuation theory of r.v.'s with mean, this principle will lead to ruin if many independent risks are insured. This motivates the next principle, The expected value principle H(X) = (1 + 77)EX where 77 is a specified safety loading. For 77 = 0, we are back to the net premium principle. A criticism of the expected value principle is that it does not take into account the variability of X which leads to The variance principle H(X) = EX+77Var(X). A modification (motivated from EX and Var(X) not having the same dimension) is

324


The standard deviation principle H(X) = EX +rl

Var(X).

The principle of zero utility. Here v(x) is a given utility function, assumed to be concave and increasing with (w.lo.g) v(O) = 0; v(x) represents the utility of a capital of size x . The zero utility principle then means v(0) = Ev (H(X) - X);

(5.1)

a generalization v(u) = Ev (u + H(X) - X ) takes into account the initial reserve u of the company. By Jensen 's inequality, v(H(X) - EX) > Ev(H(X) - X) = 0 so that H(X) > EX. For v(x) = x, we have equality and are back to the net premium principle. There is also an approximate argument leading to the variance principle as follows. Assuming that the Taylor approximation

v(H(X) - X) ^ 0 +v'(0)(H (X) - X) + v 0 (H(X) - X)2 ,/2 is reasonable , taking expectations leads to the quadratic v"H(X )2 + H(X) (2v' - 2v"EX) + v"EX2 - 2v'EX = 0 (with v', v" evaluated at 0) with solution

H(X)=EX-v^±V(- ^ )2-Var(X). Write ( vI ) 2

\

-Var(X) v^ - 2v^Var(X)/ I - (

, Var(X) )2

If v"/v' is small, we can ignore the last term. Taking +f then yields H(X) ,:: EX -

2v'(0) VarX;

since v"(0) < 0 by concavity, this is approximately the variance principle. The most important special case of the principle of zero utility is The exponential principle which corresponds to v(x) = (1 - e-6x)/a for some a > 0. Here (5.1) is equivalent to 0 = 1 - e-0H(X)EeaX, and we get

H(X) = 1 log Ee 0X . a

325

5. PRINCIPLES FOR PREMIUM CALCULATION

Since m.g.f.'s are log-concave, it follows that H,, (X) = H(X) is increasing as function of a. Further, limQyo Ha (X) = EX (the net premium princiHa (X) = b (the premium ple) and, provided b = ess supX < oo, lim,, but is clearly not maximal loss principle is called the H(X) = b principle very realistic). In view of this, a is called the risk aversion The percentile principle Here one chooses a (small ) number a, say 0.05 or 0.01, and determines H(X) by P(X < H(X)) = 1 - a (assuming a continuous distribution for simplicity). Some standard criteria for evaluating the merits of premium rules are 1. 77 > 0, i .e. H(X) > EX. 2. H(X) < b when b (the ess sup above ) is finite 3. H(X + c) = H(X) + c for any constant c

4. H(X + Y) = H(X) + H(Y) when X, Y are independent 5. H(X) = H(H(XIY)). For example , if X = EN U= is a random sum with the U; independent of N, this yields

H

C^

U; I = H(H(U)N)

(where, of course, H(U) is a constant). Note that H(cX) = cH(X) is not on the list! Considering the examples above, the net premium principle and the exponential principle can be seen to the only ones satisfying all five properties. The expected value principle fails to satisy, e.g., 3), whereas (at least) 4) is violated for the variance principle, the standard deviation principle, and the zero utility principle (unless it is the exponential or net premium principle). For more detail, see e.g. Gerber [157] or Sundt [354]. Proposition 5.1 Consider the compound Poisson case and assume that the premium p is calculated using the exponential principle with time horizon h > 0. That is, N,,

Ev I P - E U; i =1

= 0 where

v(x) = 1(1 - e-°x a

Then ry = a, i.e. the adjustment coefficient 'y coincides with the risk aversion a.

326


Proof The assumption means

0 a (1 - e-areo (B[a1-1)

l

i.e. /3(B[a] - 1) - ap = 0 which is the same as saying that a solves the Lundberg ❑ equation. Notes and references The theory exposed is standard and can be found in many texts on insurance mathematics, e.g. Gerber [157], Heilman [191] and Sundt [354]. For an extensive treatment, see Goovaerts et al. [165].

6 Reinsurance Reinsurance means that the company (the cedent) insures a part of the risk at another insurance company (the reinsurer). Again, we start by formulation the basic concepts within the framework of a single risk X _> 0. A reinsurance arrangement is then defined in terms of a function h(x) with the property h(x) < x. Here h(x) is the amount of the claim x to be paid by the reinsurer and x - h(x) by the the amount to be paid by the cedent. The function x - h(x) is referred to as the retention function. The most common examples are the following two: Proportional reinsurance h(x) = Ox for some 0 E (0, 1). Also called quota share reinsurance. Stop-loss reinsurance h(x) = (x - b)+ for some b E (0, oo), referred to as the retention limit. Note that the retention function is x A b. Concerning terminology, note that in the actuarial literature the stop-loss transform of F(x) = P(X < x) (or, equivalently, of X), is defined as the function

b -* E(X - b)+ =

f

(s - b)F(dx) _ f

(x) dx.

6 00

An arrangement closely related to stop-loss reinsurance is excess-of-loss reinsurance, see below. Stop-loss reinsurance and excess-of-loss reinsurance have a number of nice optimality properties. The first we prove is in terms of maximal utility: Proposition 6.1 Let X be a given risk, v a given concave non-decreasing utility function and h a given retention function. Let further b be determined by E(X b)+ = Eh(X). Then for any x,

Ev(x - {X - h(X)}) < Ev(x - X A b).

327

6. REINSURANCE

Remark 6 .2 Proposition 6.1 can be interpreted as follows. Assume that the cedent charges a premium P > EX for the risk X and is willing to pay P1 < P for reinsurance. If the reinsurer applies the expected value principle with safety loading q, this implies that the cedent is looking for retention functions with Eh(X) = P2 = P1/(1 + 77). The expected utility after settling the risk is thus

Ev(u + P - P1 - {X - h(X)}) where u is the initial reserve . Letting x = u + P - P1, Proposition 6.1 shows that the stop-loss rule h (X) = (X - b)+ with b chosen such that E(X - b)+ ❑ = P2 maximizes the expected utility. For the proof of Proposition 6.1, we shall need the following lemma: Lemma 6 .3 (OHLIN'S LEMMA) Let X1, X2 be two risks with the same mean, such that Fj(x) < F2 (x), x < b, Fi(x) ? F2(x), x > b for some b where Fi(x) = P(Xi < x). Then Eg(X1) < g(X2) for any convex function g. Proof Let Yi=XiAb, Zi=Xivb.

Then P(Yl < x) _ Fi(x) b so that Y1 is larger than Y2 in the sense of stochastical ordering . Similarly, P(Zl < x) _

0 = P(Z2 < x) x < b Fi(x) > F2(x) = P(Z2 < x) x > b

so that Z2 is larger than Zl in stochastical ordering. Since by convexity, v(x) = g(x) - g(b) - g'(b)(x - b) is non-increasing on [0, b] and non-decreasing on [b, oo), it follows that Ev(Y1) < Ev(Y2), Ev(Zi) < Ev(Z2). Using v(Yi) + v(Zi) = v(Xi), it follows that

0 < Ev(X2) - Ev(Xi) = Eg(X2) - Eg(X1), using EX1 = EX2 in the last step.

❑

Proof of Proposition 6.1. It is easily seen that the asssumptions of Ohlin' s lemma hold when X1 = X A b, X2 = X - h(X); in particular, the requirement EX1


328

= EX2 is then equivalent to E(X - b)+ = Eh(X). Now just note that -v is convex. ❑ We now turn to the case where the risk can be written as N

X = Ui i=1

with the Ui independent; N may be random but should then be independent of the Ui. Typically, N could be the number of claims in a given period, say a year, and the Ui the corresponding claim sizes. A reinsurance arrangement of the form h(X) as above is called global; if instead h is applied to the individual claims so that the reinsurer pays the amount EN h(Ui), the arrangement is called local (more generally, one could consider EN hi(Ui) but we shall not discuss this). The following discussion will focus on maximizing the adjustment coefficient. For a global rule with retention function h* (x) and a given premium P* charged for X - h* (X), the cedents adjustment coefficient -y* is determined by

(6.2)

1 = Eexp {ry*[X - h*(X) - P*]}, for a local rule corresponding to h(u) and premium P for X look instead for the ry solving

[ X_P_^

J _f

1 = Eexp

N 1 h (Ui), we

[ Ei [U - h(Ui)] -P

= Eexp{ry

h(Ui)]

J}

l (6.3) This definition of the adjustment coefficients is motivated by considering ruin at a sequence of equally spaced time points, say consecutive years, such that N is the generic number of claims in a year and P, P* the total premiums charged in a year, and referring to the results of V.3a. The following result shows that if we compare only arrangements with P = P*, a global rule if preferable to a local one. Proposition 6.4 To any local rule with retention function h(u) and any N

P > E X - N h(Ui) 4 =1

there is a global rule with retention function h* (x) such that N

Eh*(X) = Eh(U1) i=1

and 'y* > ry where ry* is evaluated with P* = P in (6.3).

(6.4)

6. REINSURANCE

329

Proof Define N

h* (x) = E > h(Ui) X = x ; then (6.5) holds trivially. Applying the inequality Ecp(Y ) > EW(E (YIX )) (with W convex ) to W(y ) = eryy, y = Ei [Ui - h(Ui)] - P, we get N

1 = Eexp

ry E[Ui i-i

- h(Ui)] - P

> EexP{7[X - h * (X) - P]}.

But since ry > 0, ry* > 0 because of (6.4), this implies 7* > 7.

❑

Remark 6.5 Because of the independence assumptions , expectations like those in (6.3), (6.4), (6.5) reduce quite a lot. Assuming for simplicity that the Ui are i.i.d., we get EX = EN • EU, N

E X - h( UU) = EN • E[U - h(U)],

Eexp

7 [E ' [Ui - h(Ui)] - P I = EC [7]N,

(6.6)

i-i where C[ry] = Ee'r(u-4(u)), and so on.

❑

The arrangement used in practice is, however, as often local as global. Local reinsurance with h(u) = (u - b)+ is referred to as excess-of-loss reinsurance and plays a particular role: Proposition 6.6 Assume the Ui are i. i.d. Then for any local retention function u - h(u) and any P satisfying (6.4), the excess -of-loss rule hl (u) = (u - b)+ with b determined by E(U - b)+ = Eh(U) (and the same P) satisfies 71 > ry. Proof As in the proof of Proposition 6.4, it suffices to show that

Eexp

'UiAb- P } < 1 = Eexp E[Ui- h(Ui)-P JJJ l:='l {ry

{ry i-i

]

or, appealing to (6.6), that 01[ry] < 0[-y] where 0[-y] = Ee'r(U^') . This follows by taking Xl = U A b, X2 = U - h(U) (as in the proof of Proposition 6.4) and ❑ g(x) = e7x in Ohlin's lemma.

330


Notes and references The theory exposed is standard and can be found in.many texts on insurance mathematics, e.g. Bowers et at. [76], Heilman [191] and Sundt [354]. See further Hesselager [194] and Dickson & Waters [120]. The original reference for Ohlin's lemma is Ohlin [277]. The present proof is from van Dawen [99]; see also Sundt [354].

Appendix Al Renewal theory la Renewal processes and the renewal theorem By a simple point process on the line we understand a random collection of time epochs without accumulation points and without multiple points. The mathematical representation is either the ordered set 0 < To < T1 < ... of epochs or the set Y1, Y2, ... of interarrival times and the time Yo = To of the first arrival (that is, Y,, = T„ - T„_1). The point process is called a renewal process if Yo, Y1, ... are independent and Y1, Y2, ... all have the same distribution, denoted by F in the following and referred to as the interarrival distribution; the distribution of Yo is called the delay distribution. If Yo = 0, the renewal process is called zero-delayed. The number max k : Tk_j < t of renewals in [0, t] is denoted by Nt. The associated renewal measure U is defined by U = u F*" where F*" is the nth convolution power of F. That is, U(A) is the expected number of renewals in A C R in a zero-delayed renewal process; note in particular that U({0}) = 1. The renewal theorem asserts that U(dt) is close to dt/µ, Lebesgue measure dt normalized by the mean to of F, when t is large. Technically, some condition is needed: that F is non-lattice, i.e. not concentrated on {h, 2h,.. .} for any h > 0. Then Blackwell 's renewal theorem holds, stating that U(t+a)-U (t) -^ a, t -00

(A.1)

(here U(t) = U([0, t]) so that U(t + a) - U(t) is the expected number of renewals in (t, t +a]). If F satisfies the stronger condition of being spread-out (F*' is nonsingular w .r.t. Lebesgue measure for some n > 1), then Stone 's decomposition holds : U = U, + U2 where U1 is a finite measure and U2(dt) = u(t)dt where 331

332

APPENDIX

u(t) has limit 1/µ as t -4 oo. Note in particular that F is spread-out if F has a density f. A weaker (and much easier to prove) statement than Blackwell's renewal theorem is the elementary renewal theorem, stating that U(t)/t --> 1/p. Both result are valid for delayed renewal processes, the statements being

EN(t + a) - EN(t) - a, resp.

ENt -4 1

lb Renewal equations and the key renewal theorem The renewal equation is the convolution equation U Z(u - x)F(dx),

Z(u) = z(u) +

(A.2)

f where Z(u) is an unknown function of u E [0 , oo), z(u) a known function, and F(dx) a known probability measure . Equivalently, in convolution notation Z = z + F * Z. Under weak regularity conditions (see [APQJ Ch. IV), (A.2) has the unique solution Z = U * z, i.e.

Z(u) =

J0 u z(x)U(dx).

(A.3)

Further, the asymptotic behavior of Z(u) is given by the key renewal theorem: Proposition A1.1 if F is non-lattice and z (u) is directly Riemann integrable (d.R.i.; see [APQ] Ch. IV), then

Z(u) -i

f0

z(x)dx .

(A.4)

µF

If F is spread- out, then it suffices for (A.4) that z is Lebesgue integrable with limZ.i". z(x) = 0. In 111.9, wee shall need the following less standard parallel to the key renewal theorem: Proposition A1.2 Assume that Z solves the renewal equation (A.2), that z(u) has a limit z(oo) (say) as u -4 oo, and that F has a bounded density2. Then

Z(u) -4 z(oo), u

-4 00.

u PF

2This condition can be weakened considerably , but suffices for the present purposes

(A.5)

APPENDIX

333

Proof The condition on F implies that U(dx) has a bounded density u(x) with limit 1/µF as x -* oo. Hence by dominated convergence, Z(u) U

=

1 u 1 u f z(u - x)u(x) dx = z(u( 1 - t))u(ut) dt 0 0

J

f z(oo) • 1 dt = z(OO). 0 PF µF 11

In risk theory, a basic reason that renewal theory is relevant is the renewal equation II.(3.3) satisfied by the ruin probability for the compound Poisson model. Here the relevant F does not have mass one (F is defective). However, asymptotic properties can easily be obtained from the key renewal equation by an exponential transformation also when F(dx) does not integrate to one. To this end, multiply (A.2) by e7x to obtain Z = z +P * Z where Z(x) = e'Y'Z(x), z(x) = e7xz(x), F(dx) = e7xF(dx). Assuming that y can be chosen such that f °° Ox F(dx) = 1, i.e. that F is a probability measure, results from the case fo F(dx) = 1 can then be used to study Z and thereby Z. This program has been carried out in III.5a. Note, however, that the existence of y may fail for heavy-tailed F.

1c Regenerative processes Let {T,,} be a renewal process. A stochastic process {Xt}t>0 with a general state space E is called regenerative w.r.t. {Tn} if for any k, the post-Tk process {XT,k+t }t>o is independent of To, T1,. .. , Tk (or, equivalently, of Yo, Y1 , . . • , Yk ), and its distribution does not depend on k. The distribution F of Y1, Y2.... is called the cycle length distribution and as before, we let µ denote its mean. We let FO, Eo etc. refer to the zero-delayed case. The simplest case is when {Xt} has i.i.d. cycles. The kth cycle is defined as {XTk+t}o t]. Hence

t t lt ) = f U(dy)z(t y) = f U(t dy Eoe(t )z(y) < c ^ l z(k) 0

0

k=o

where c = sup., U(x + 1) - U(x) (c < oo because it is easily seen that U(x + 1) - U(x) < U( 1)). Since z ( k) < E[Yi ; Y1 > t] -4 0, the sum is o(t) so that Eo£(t)/t -+ 0 . In the general case, use

t E^(t)/t = E[Yo - t; Yo > 0] + f Eo^ (t - y)P(Yo E dy) . 0

If Markov renewal theory By a Markov renewal process we understand a point process where the interarrival times Yo , Y1i Y2, ... are not i.i.d. but governed by a Markov chain {Jn} (we

336

APPENDIX

assume here that /the state space E is// finite) in the sense that

P(Y.

< yIJ)

=

Fij( y)

on {Jn= i,

Jn +1=j}

where J = a(JO, J1 i ...) and (Fij )i,jEE is a family of distributions on (0, oo). A stochastic process {Xt}t>o is called semi-regenerative w.r.t. the Markov renewal process if for any n, the conditional distribution of {XT„+t}t>o given Yo, Y1, . . . , Yn, Jo, . . . , Jn_1, Jn = i is the same as the P; distribution ofjXt}t>o itself where Pi refers to the case Jo = i. A Markov renewal process {Tn} contains an imbedded renewal process, namely {Twk } where {Wk } is the sequence of instants w where Jo., = io for some arbitrary but fixed reference state io E E. The semi-regenerative process is then regenerative w.r.t. IT,,,,}. These facts allow many definitions and results to be reduced to ordinary renewal- and regenerative processes. For example, the semi-regenerative process is called non-lattice if {T,,,,} is non-lattice (it is easily seen that this definition does not depend on i). Further: Proposition A1.7 Consider a non-lattice semi-regenerative process. Assume that uj = EjYo < oo for all j and that {J„} is irreducible with stationary distribution (v3)jEE. Then Xt 4 Xo,, where the distribution of X,,. is given by Eg(X00) = 1

YO vjEj f g(Xt) dt

µ jEE o

where p = ujEEViAj.

Notes and references Renewal theory and regenerative processes are treated, e.g., in [APQ], Alsmeyer [5] and Thorisson [372].

A2 Wiener-Hopf factorization Let F be a distribution which is not concentrated on (-oo, 0] or (0 , oo). Let X1, X2, ... be i.i .d. with common distribution F, Sn = X1 + • • • + Xn the associated random walk, and define r+=inf{n>0: Sn>0}, T_=inf{n>0: SnG+, U- =EGn, and the T+- and r_ pre-occupation measures T+-1

R+(A) = E E I(Sn E A),

r_-1

R_(A) = E I(Sn E A).

n -0

n=0

The basic identities are the following: Theorem A2.1 (a) F = G+ + G_ - G+ * G_: (b) G_ (A) = f °° F(A - x)R_ (dx), A C (-oo, 0); (c) G+(A) = f °. F(A - x)R+(dx), A C (0, oo); (d) R+ = U_; (e) R_ = U+. Proof Considering the restrictions of measures to (-oc, 0] and (0, oo), we may rewrite (a) as

G_ (A) = G+(A) =

F(A) + (G+ * G_)(A), A C (-oo, 0], F(A) + (G+ * G_)(A), A C (0, oo)

(A.7) (A.8)

(e.g. (A.7) follows since G+(A) = 0 when A C (-oo, 0]). In (A.7), F(A) is the contribution from the event {T_ = 1} = {X1 < 0}. On {T_ > 2}, define w as the time where the pre-T_ path S1, ... , Sr_ _1 is at its minimum . More rigorously, we consider the last such time (to make w unique) so that {w=m,T_=n} = {S,-S.. >0, 0<j<m, S,-S.>0, m<j0, 0<j<m, SmEdu) = P(T+=m, ST+Edu). Aso, clearly (Sj -Sm>0, m < j 2

F (7-- = n, S,_ E A) n-1

P(r_=nw=m

f

Sm

EduSrEA)

m=1 n-1

F(r+=mSr+Edu).F(r_n_mSrEA_u). m=1 f

S mming over n = 2,3.... and reversing the order of summation yields P(T_ > 2, ST_ E A) P(T+ = m, S,+ E du) E P(S,_ = n - m, ST_ E A - u) f0m m=1 00 n=m+1

J0 OO P(S,+ E du)P(S,_ E A - du) (G+ * G-)(A)• C llecting terms, (A.7) follows, and the proof of (A.8) is similar. (b) follows from 00

G+ (A) _ E F(Sn E A, -r+ = n) n=1

C-0 E fF(Sk< 0,0 0, letting P = I + Q/i and truncating the series in the identity

= e-17t 00 Pn(,]t)n (A.14) E n n=0

which is easily seen to be valid as a consequence of eqt = en(P-r)t = e-ntenpt The idea which lies behind is uniformization of a Markov process {Xt}, i.e. construction of {Xt} by realizing the jump times as a thinning of a Poisson process {Nt } with constant intensity 77. To this end, assume that Q is the intensity matrix for {Xt} and choose q with rt > max J%J = max -qii• 1,3

(A.15)

i

Then it is easily checked that P is a transition matrix , and we may consider a new Markov process {Xt} which has jumps governed by P and occuring at epochs of {Nt} only (note that since pii is typically non-zero , some jumps are dummy in the sense that no state transition occurs ). However , the intensity matrix Q is the same as the one Q for {Xt} since a jump from i to j 1-1 i occurs at rate qij = 77pij = q22. The probabilistic reason that (A.14) holds is therefore that the t-step transition matrix for {fft} is °O

eQt = E e-nt (,7t) n=0

n

Pn

n!

(to see this, condition upon the number n of Poisson events in [Olt]) -

❑

Example A3.3 (DIFFERENTIAL EQUATIONS) Letting Kt = eQt, we have k = QK (or KQ) which is a system of p2 linear differential equations which can be solved numerically by standard algorithms (say the Runge-Kutta method) subject to the boundary condition Ko = I.

In practice, what is needed is quite often only Zt = TreQt (or eQth) with it (h) a given row (column) vector. One then can reduce to p linear differential equations by noting that k = ZQ, Zo = a (Z = QZ, Zo = h). The approach is in particular convenient if one wants eQt for many different ❑ values of t. Here is a further method which appears quite appealing at a first sight: Example A3 .4 (DIAGONALIZATION) Assume that Q has diagonal form, i.e. p different eigenvalues Aj i ... , Ap. Let vi,... , vp be the corresponding left

342

APPENDIX

(row) eigenvectors and hl,..., hp the corresponding right (column) eigenvectors, v5Q = Aivi, Qhi = vihi. Then vihj = 0, i # j, and vihi ¢ 0, and we may adapt some normalization convention ensuring vihi = 1. Then P

Q =

P

eQt =

P

> Aihivi = E Aihi (9 vi, i= 1 i=1 P

E e\`thivi = E ea:thi ® vi. i=1

(A.16) (A.17)

i=1

Thus, we have an explicit formula for eQt once the A j, vi, hi have been computed; this last step is equivalent to finding a matrix H such that H-1QH is a diagonal matrix, say A = (Ai)diag, and writing eQt as eQt = He°tH-1 = H (e\it)di.g H-1. (A.18)

Namely, we can take H as the matrix with columns hl,..., hp.

❑

There are, however, two serious drawbacks of this approach: Numerical instability : If the A5 are too close, (A.18) contains terms which almost cancel and the loss of digits may be disasterous. The phenomenon occurs not least when the dimension p is large. In view of this phenomenon alone care should be taken when using diagonalization as a general tool for computing matrix-exponentials. Complex calculus : Typically, not all ai are real, and we need to have access to software permitting calculations with complex numbers or to perform the cumbersome translation into real and imaginary parts. Nevertheless, some cases remain where diagonalization may still be appealing.

Example A3.5 If Q= ( 411 ( q21

q12 q22

is 2 x 2, the eigenvalue, say Al, of largest real part is often real (say, under the conditions of the Perron-Frobenius theorem), and hence A2 is so because of A2 = tr(Q). Everything is nice and explicit here: 411+q2+-D' )12_g11+q2-^^ where (411-422 D = z ) + 4412421. 2 2

343

APPENDIX

Write 7r (= v1) for the left eigenvector corresponding to a1 and k (= hl) for the right eigenvector. Then 7r =

(ir1

7r2 ) = a (q21

Al

- q, 1) ,

k -

C k2

) =b

( A1

q 1 Q11 /

where a , b are any constants ensuring//Irk = 1, i.e. l ab (g12g21 +

(A1

-

411) 2)

= 1.

Of course, v2 and h2 can be computed in just the same way, replacing ai by A2. However, it is easier to note that 7rh2 = 0 and v2k = 1 implies v2 = (k2 - k1), h2 = Thus, eqt = eNlt ( ir1ki i2k1 \ ir1 k2 72 k2

+ e

azt

7r2k2 -i2k1 -7ri k2 7r1 k1

(A.19)

Example A3 .6 A particular important case arises when Q = -q1 qi ) q2 -q2 J is an intensity matrix. Then Al = 0 and the corresponding left and right eigenvectors are the stationary probability distribution 7r and e. The other eigenvalue is A = A2 = -q1 - Q2i and after some trivial calculus one gets eQt =

ir =

7r 1

112

+ eat

7r1 7r2 / (7fl 7r2) =

(

7r2

-1r2

-7r1 IF,

q2 ql qi +q 2 9l +q2

where

(A.20)

(A.21)

Here the first term is the stationary limit and the second term thus describes the rate of convergence to stationarity. ❑

Example A3.7 Let 3

9

2 14 7 11 2 2

344

APPENDIX

Then 7 T4 -2 =52,

D= 2+ 11)'

x1 -3/2 - 11/2 + 5 -1, A2 = -3/2 - 11/2 - 5 - - -6, 2 2 1 3 2 2)'

1=ab(142+(-1+2)2 ) = tab, ir =a(2 9

9

k=b

14

14 =b -1+ 2

ir1

k1

7r1 k2

ir2

k1

2

_

7r2 k2

9 2 10

9 70

5 7

1 '

10

e4" = e_,.

9 9 10 10 + 7 1 10 10

e_6u

10 1 10 7 10

9 70 9 10 0

A4 Some linear algebra 4a Generalized inverses A generalized inverse of a matrix A is defined as any matrix A- satisfying AA-A = A. (A.22) Note that in this generality it is not assumed that A is necessarily square, but only that dimensions match , and a generalized inverse may not unique. Generalized inverses play an important role in statistics. They are most often constructed by imposing some additional properties , for example

AA+A = A, A+AA+ = A+, (AA+)' = AA+, (A+A)' = A+A. (A.23)

APPENDIX

345

A matrix A+ satisfying (A.23) is called the Moore-Penrose inverse of A, and exists and is unique (see for example Rao [300]). E.g., if A is a possibly singular covariance matrix (non-negative definite), then there exists an orthogonal matrix C such that A = CDC' where

0 0 D =

AP Here we can assume that the A , are ordered such that Al > 0,. .. , Am > 0, Am+1 = ... _ A,, = 0 where m < p is the rank of A, and can define

/ail

A+ = C

0 0

0

A' 0 0 0

0

0 0 0

C' .

01

In applied probability, one is also faced with singular matrices , most often either an intensity matrix Q or a matrix of the form I-P where P is a transition matrix. Assume that a unique stationary distribution w exists . Rather than with generalized inverses , one then works with Q = (Q - eir )-1, (I - P)- = (I - P + e7r)-1 (here ( I - P + e7r )- 1 goes under the name fundamental matrix of the Markov chain). These matrices are not generalized inverses but act roughly as inverses except that 7r and e play a particular role - e.g. ( Q - eir )- 1Q = Q(Q - eir)-1 = I - ew. Here is a typical result on the role of such matrices in applied probability: Proposition A4.1 Let A be an irreducible intensity matrix with stationary row vector it, and define D = (A - e ® 7r)-1. Then for some b > 0,

lt o

eAx dx =

te7r + D(eAt - I)

(A.24)

= te7r - D + O(e-bt), (A.25)

346

APPENDIX

t

2

xe Ax dx

= eir + t(D + e-7r) + D(eAt - I) - DZ(ent - I) (A.26) 2

= 2 e7r + tD - 2e7r - D + D2 + O(e-bt).

(A.27)

Proof Let A(t), B(t) denote the l.h.s. of (A.24), resp. the r.h.s. Then A(O) _ B(O) = 0, B'(t) = e7r + DAeAt = eir + (I - eir)eAt = eAt = A'(t). (A.26) follows by integration by parts: t f t /' xeAx dx = [x {xe7r + D(eAx - I)}, - J {xe^r + D(e - I)} dx. o Finally, the formulas involving O(e-6t) follow by Perron-Frobenius theory, see below. ❑ 4b The Kronecker product ® and the Kronecker sum

We recall that if A(1) is a k1 x ml and A(2) a k2 x m2 matrix, then the Kronecker (tensor) product A(') ®A(2) is the (k1 x k2) x (ml x m2) matrix with (il i2) (jl j2)th entry a;91a(2) . Equivalently, in block notation i2h A®B= ( a11B a21 B

a12B a22 B

Example A4.2 Let it be a row vector with m components and h a column vector with k components. Interpreting 7r, h as 1 x m and k x 1 matrices, respectively, it follows that h ® it is the k x m matrix with ijth element hi7rj . I.e., h ® it reduces to hit in standard matrix notation. Note that h ® it has rank 1; the rows are proportional to it, and the columns to h, and in fact any rank 1 matrix can be written on this form. For example,

()®(6

f 6/ 7f 8^ 7 8 )=! ^)( 6 7 8 )=(6^ 7^ 8^) \ ❑

Example A4.3 Let

2 A= 4

3 Vf' N7 5 )' B= ( 8 ).

347

APPENDIX

Then

A®B =

2 f 20- 3v'6- 3vV/72f 20- 3V8- 3f 4v/6 4vf7 5v/-6 504f 4-,A9- 5v'-8 5vf9-

11 A fundamental formula is (A1B1C1) ®(A2B2C2) = (A1 (9 A2)(B1 (9 B2)(C1®C2).

(A.28)

In particular, if Al = vi, A2 = v2 are row vectors and C1 = h1, C2 = h2 are column vectors, then v1B1h1 and v2B2h2 are real numbers, and v1B1h1 • v2B2h2 = v1B1h1 ® v2B2h2 = ( v1(&v2 )( B1(&B2 )( h1(&h2 ) .(A.29) If A and B are both square (k1 = ml and k2 = m2), then the Kronecker sum is defined by

A(1) ®A(2) = A(1) ®Ik2 + k ®A(2). (A.30) A crucial property is the fact that the functional equation for the exponential

eA+B = eAeB function generalizes to Kronecker notation (note that in contrast typically only holds when A and B commute): Proposition A4.4 eA® B = eA ®eB. Proof We shall use the binomial formula

t / l (A ®B)t = I k Ak 0 B1-k

(A.31)

k=0

Indeed,

(AED B)1 = (A®I+I(9 B)l is the sum of all products of t factors, each of which is A ® I or I ® B; if A ® I occurs k times, such a factor is Ak (&B 1-k according to (A.29), and the number of such factors is precisely given by the relevant binomial coefficient.

Using (A.31), it follows that 0o

e® ® e B

_An

oo

J

oo Bn

®_

Ak ®Bl-k

r

7 I F n! = ` k! (I - k)! ( n-0 n=0 t=0 k=0

^. (A B)' = eA®B e! L 1=0

0

APPENDIX

348

Remark A4.5 Many of the concepts and results in Kronecker calculus have p(2) is the intuitive illustrations in probabilistic terms. Thus , p = P(1) ® {X }, X ) }, where transition matrix of the bivariate Markov chain {X n1),

n2

n1 )

{X(2) } are independent Markov chains with transition matrices P(1), P(2), and Q = Q(1) ® Q

(2)

= Q(1) ® I + I ® Q(2)

(A.32)

is the intensity matrix of the bivariate continuous Markov process {Yt(1), Yt(2) independent Markov processes with intensity matri{y(2) } are {Y(1) }, first term on the r . h.s. represents ces Q( 1), Q(2); in the definition (A.32), the {Yt(2) } transitions in the {Yt(1) } component and the second transitions in the component , and the form of the bivariate intensity matrix reflects the fact that Yt(2) } cannot change state in both components at due to independence ,

where

{Yt(1),

the same time. A special case of Proposition A4.4 can easily be obtained by probabilistic be the s-step transition reasoning along the same lines . Let P8f P(Sl), P(t) Yt(2) }. From what has been said about matrices of {Yt( 1), Yt(2 ) }, { 1't(1) }, resp . { On the other hand, independent Markov chains, we have P8 = Pal) ® p(2). P8 = exp {sQ} = exp {s (Q(1) ®Q(2)) } , Ps 1) = exp {sQ ( 1) } > p(2 ) = exp {sQ(2) } can therefore be rewritten as Taking s = 1 for simplicity , P8 = Pal ) ® P82) exp {Q ( 1) ® Q(2)1 = eXp {Q( 1) } ® exp {Q(2) }

Also the following formula is basic: B are both square such that a +,3 < 0 Lemma A4 .6 Suppose that A and of B. Let further it, v whenever a is an eigenvalue of A and 0 is an eigenvalue be any row vectors and h, k any column vectors. Then 2 0

ire At h • ve

Bt kdt = (^®v)(A®B)-1(e A®Ba - I)(h ® k).

(A.33)

APPENDIX

349

Proof According to (A.29), the integrand can be written as ( 7r (9 v)( eAt ® eBt )(h ®k ) =

( 7r

®v)(eA (DBt)(h (& k).

Now note that the eigenvalues of A ® B are of the form a +,3 whenever a is an eigenvalue of A and 3 is an eigenvalue of B, so that by asssumption A ® B is ❑ invertible, and appeal to (A.12).

4c The Perron-Frobenius theorem Let A be a p x p-matrix with non-negative elements. We call A irreducible if the pattern of zero and non-zero elements is the same as for an irreducible transition matrix. That is, f o r each i, j = 1, ... , p there should exist io, il,... , in such that io = i, i,, = j and atk_li,. > 0 for k = 1, . . . , n. Similarly, A is called aperiodic if the pattern of zero and non-zero elements is the same as for an aperiodic transition matrix. Here is the Perron-Frobenius theorem, which can be found in a great number of books, see e.g. [APQ] X.1 and references there (to which we add Berman & Plemmons [63]): Theorem A4.7 Let A be a p x p-matrix with non-negative elements. Then: (a) The spectral radius Ao = max{JAI : A E sp(A)} is itself a strictly positive and simple eigenvalue of A, and the corresponding left and right eigenvectors v, h can be chosen with strictly positive elements; (b) if in addition A is aperiodic, then IN < Ao for all A E sp(A), and if we normalize v, h such that vh = 1, then

An = Aohv+O(µ") = Aoh®v+O(µ")

(A.34)

for some u. E (0, ao). Note that for a transition matrix, we have AO = 1, h = e and v = 7r (the stationary row vector). .The Perron-Frobenius theorem has an analogue for matrices B with properties similar to intensity matrices: Corollary A4.8 Let B be an irreducible3 p x p-matrix with non-negative offdiagonal elements. Then the eigenvalue Ao with largest real part is simple and real, and the corresponding left and right eigenvectors v, h can be chosen with 3By this, we mean that the pattern of non-zero off-diagonal elements is the same as for an irreducible intensity matrix.

350

APPENDIX

strictly positive elements. Furthermore, if we normalize v, h such that vh = 1, then eBt = ea0thv + O(eµt) = eA0th ® v + O(et t) (A.35) for some p E (-oo, Ao). Note that for an intensity matrix, we have A0 = 0, h = e and v = 7r (the stationary row vector).

Corollary A4.8 is most often not stated explicitly in textbooks, but is an easy consequence of the Perron-Frobenius theorem. For example, one can consider A = 77I + B where rl > 0 is so large that all diagonal elements of A are strictly positive (then A is irreducible and aperiodic), relate the eigenvalues of B to those of B via (A. 10) and use the formula 00 Antn

-me at e Bt = e

= e - n t AL n=0

n!

(cf. the analogy of this procedure with unformization, Example A3.2).

A5 Complements on phase-type distributions 5a Asymptotic exponentiality In Proposition VIII.1.8, it was shown that under mild conditions the tail of a phase-type distribution B is asymptotical exponential. The next result gives a condition for asymptotical exponentiality, not only in the tail but in the whole distribution. The content is that B is approximately exponential if the exit rates ti are small compared to the feedback intensities tij (i # j). To this end, note that we can write the phase generator T as Q - (ti)diag where Q = T + (ti)diag is a proper intensity matrix (Qe = 0). I.e., the condition is that t is small compared to Q. Proposition A5.1 Let Q be a proper irreducible intensity matrix with stationary distribution a, let t = (ti)iEE # 0 have non-negative entries and define T(°) = aQ - (ti)ding.

Then for any (3, the phase-type distribution B(a)

with representation (,(3, T(°)) is asymptotically exponential with parameter t* _ r EiEE aiti as a -4 oo, Bi° (x) -+ a-t*x

Proof Let { 4 } be the phase process associated with B(a) and (°) its lifelength, let {Yti°i } be a Markov process with initial distribution a and intensity

351

APPENDIX

matrix aQ , and write Yt = Yt(1),

((1) etc. We can assume that Jta) = Yt(°), t < (a), and that Yt(a) = Yat for all t. Let further V be exponential with intensity V and independent of everything else. We can think of ( ( a) as the first event in an inhomogeneous Poisson process ( Cox process ) with intensity process

{t

Y( a) } v>0

((a) =

. Hence we can represent ( (a) as

inf { t > O : f tY( )dv=V

} ^l = inf { t > O : t adv = V }

jat inf{t > 0: tydv =aV} = JJJ a

where o (x) = inf {t >0: fo tY dv = x}.

J

l

J

By the law of large numbers for

Markov processes , fo tY dv/t a$' t*, and this easily yields a(x)/x a-' 1/t*. Hence O ((a) aa. v/ t-. We shall , in fact , prove a somewhat more general result which was used in the proof of Proposition VI.1.9. In addition to the asymptotic exponentiality, it states that the state, from which the phase process is terminated , has a limit distribution: Proposition A5.2 Pi (c(a) > x, J(()) _ = i) -+ a-t•x

t tt' .

Proof Assume first ti > 0 for all i and let I. = YQ(x). Then {Ix} is a Markov process with to = Yo. Conditioning upon whether { Yt} changes state in [0, dx/ti] or not, we get dx F (Idx = j) = (1 + qij t )Sij + qij dt,x (1 - bij) Hence the intensity matrix of { Ix} is (qij/ti)i,jEE, from which it is easily checked that the limiting stationary distribution is (aiti/t*)iEE• Now let a' -4 oo with a in such a way that a' < a, a'/a -+ 1, a - a' -+ oo (e.g. a' = a - aE where 0 < e < 1). Then a(a'V)/a (aV) a' 1. Since JJ(.)_ = Y(a) = 1'aS(a) = Ya(av)^ it follows that Pi ((,(a) > x , JÔ)_ = j)

Pi (v(aaV) > x,YQ(av) = j)

Pi ( ci(a'V) > x,Yj(av) = j f

352

APPENDIX

rr Ia(a'V) Ei I ( > x) P

L

at

(Yo (aV)

,., Et II I a(a^V) > x) at' .+ a-t*x • a't' L ` at t* t*

J

Reducing the state space of {Ix } to {i E E : t, > 0}, an easy modification of the argument yields finally the result for the case where t; = 0 for one or more ❑

i.

Notes and references Propositions A5.1 and A5.2 do not appear to be in the literature. However, these results are in the spirit of rare events theory for regenerative processes (e.g. Keilson [223], Gnedenko & Kovalenko [164] and Glasserman & Kou [162]). See also Korolyuk, Penev & Turbin [238].

5b Discrete phase-type distributions The theory of discrete phase-type distributions is a close parallel of the continuous case, so we shall be brief. A distribution B on {1, 2, ...} is said to be discrete phase-type with representation (E, P, a) if B is the lifelength of a terminating Markov chain (in discrete time) on E which has transition matrix P = (p,j) and initial distribution a. Then P is substochastic and the vector of exit probabilities is p = e - Pe. Example A5.3 As the exponential distribution is the simplest continuous phasetype distribution, so is the geometric distribution, with point probabilities bk = (1 - p)k-1 p, k = 1, 2, ..., the simplest discrete phase-type distribution: here E has only one element, and thus the parameter p of the geometric distribution ❑ can be identified with the exit probability vector p. Example A5.4 Any discrete distribution B with finite support, say bk = 0, K}, a = b = (bk)k=1,...,x k > K, is discrete phase-type. Indeed, let E and Pkj

j=k-1, 1 k=1 1 0k>1, otherwise, ' pk 0 k>1

11 Theorem A5.5 Let B be discrete phase-type with representation (P, a). Then: (a) The point probabilities are bk = aPk-lp; (b) the generating function b[z] _ E' , zkbk is za(I - zP)-'p; (c) the nth moment k 1 k"bkis 1)"n!aP-"p.

APPENDIX

353

5c Closure properties Example A5.6 (CONVOLUTIONS) Let B1, B2 be phase-type with representations (E(1),a(1),T(1)), resp. (E(2),a(2),T(2)). Then the convolution B = B1 * B2 is phase-type with representation (E, a, T) where E = E(1) + E(2) is the disjoint union of E(1) and E(2), and a=1), a' - { 0, _

i E

E(1) T(1)

i E E(2) ,

t(1)a(2)

T= ( 0 T(2) )

(A.36)

in block-partitioned notation (where we could also write a as (a (1) 0)). A reduced phase diagram (omitting transitions within the two blocks) is

am

E(1)

t(1) a(2)

(2)

t(2)

Figure A.2 The form of these results is easily recognized if one considers two independent phase processes { Jt 1) }, { Jt 2) } with lifetimes U1 , resp. U2, and piece the processes together by it =

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