Advanced Series on Statistical Science & I Applied Probability
^^^A£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.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
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.
This page is intentionally left blank
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
We also refer to t/) ( u) and 0(u, T) as ruin probabilities with infinite horizon and finite horizon , respectively. They are the main topics of study of the present book. For mathematical purposes, it is frequently more convenient to work with the claim surplus process {St}t>0 defined by St = u - Rt. Letting T(u) = inf {t > 0 : Rt < 0} = inf It > 0 : St > u}, M =
(1.3)
sup St, MT = sup St, (1.4) O
1
2 CHAPTER I. INTRODUCTION be the time to ruin and the maxima with infinite and finite horizon, respectively, the ruin probabilities can then alternatively be written as ,b(u) =
P (r(u) < oo) = P(M > u),
(1.5)
i,i(u,T) =
F (MT > u) = P(r(u) < T).
(1.6)
Sofar we have not imposed any assumptions on the risk reserve process. However, the following set-up will cover the vast majority of the book: • There are only finitely many claims in finite time intervals. That is, the number Nt of arrivals in [0, t] is finite. We denote the interarrival times of claims by T2, T3, ... and T1 is the time of the first claim. Thus, the time of arrival of the nth claim is an = T1 + • • • + Tn, and Nt = min {n > 0 : 0rn+1 > 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<8.
(2.3)
2. CLAIM SIZE DISTRIBUTIONS 7 The mean EX is p/b and the variance Var X is p/b2. In particular, the squared coefficient of variation (s.c.v.)
VarX1 (EX )2 p is < 1 for p > 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)e`ix ,(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
CHAPTER I. INTRODUCTION
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.
CHAPTER I. INTRODUCTION
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 N2`i. 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
1
Since everything is non -negative, both sides are increasing in T, and letting T -* oo, (2.2) follows by monotone convergence. ❑ From Theorem 2.3 we obtain a likelihood ratio representation of the ruin probability V) (u) parallel to the martingale representation (1.1) in Proposition 1.1: Corollary 2.4 Under condition (a) of Proposition 1.1, ,O(u) = e-ryuE[e-'YC(u); T(u) < oo]. (2.4) Proof Letting G = {r(u) < oo}, we have F(G) = V )(u). Now just rewrite the r.h.s. of (2.2) by noting that 1 = e--rsr(„) = e-1'ue-7Ou). Lr(u) 11 The advantage of (2.4) compared to (1.1) is that it seems in general easier to deal with the (unconditional) expectation E[e-ryVu); r(u) < oo] occuring there than with the (conditional) expectation E[e'r{(u ) Jr(u) < oo] in (1.1). The crucial step is to obtain information on the process evolving according to F, and this problem will now be studied, first in the Markov case and next (Sections 4, 5) for processes with some random-walk-like structure. Consider a (time-homogeneous) Markov process {Xt} with state space E, say, in continuous time (the discrete time case is parallel but slightly simpler). In the context of ruin probabilities, one would typically have Xt = Rt, Xt = St, Xt = (Jt, Rt) or Xt = (Jt, St), where {Rt} is the risk reserve process, {St} = {u - Rt} the claim surplus process and {Jt} a process of supplementary variables possibly needed to make the process Markovian. A change of measure is performed by finding a process {Lt} which is a martingale w.r.t. each F,,, is non-negative and has Ey Lt = 1 for all x, t. The problem is thus to investigate which characteristics of {Xt} and {Lt} ensure a given set of properties of the changed probability measure. First we ask when the Markov property is preserved. To this end, we need the concept of a multiplicative functional. For the definition, we assume for simplicity that {Xt} has D-paths, is Markov w.r.t. the natural filtration {.Ft}
2. LIKELIHOOD RATIOS AND CHANGE OF MEASURE
29
on DE and define {Lt} to be a multiplicative functional if {Lt} is adapted to
{.Ft }, non-negative and (2.5)
Lt+8 = Lt•(Lso9t)
Px-a.s. for all x, s, t, where Ot is the shift operator. The precise meaning of this is the following: being .Ft-measurable, Lt has the form Lt = 'Pt ({x }0
Proof Since both sides of (2.5) are Tt+e measurable, (2.5) is equivalent to Ex[Lt+8Vt+8] = E8[Lt • (L8 o 91)Vt+8]
(2.6)
for any .Pt+8-measurable r.v. Vt+e, which in turn is the same as Ex[Lt+8Zt • (V8 o Bt)] = Ex[Lt • (L8 o 91)Z1 • (Y8 o et)] (2.7) for any Ft-measurable Zt and any .T9-measurable Y8. Indeed, since Zt • (Y8 o Ot ) is .Ft+8-measurable, (2.6) implies (2.7). The converse follows since the class of r.v.'s of the form Zt • (Y8 o 0t) comprises all r.v.'s of the form fl' f;(Xtitl) with all t(i) < t + s. Similarly, the Markov property can be written E.[Y,, o 9tI.Ft] = Ex,Y8f
t < s,
for any .F8-measurable r.v. Y8, which is the same as Ex[Zt(Y8 o Bt)] = E8[ZtEx,YB] for any Ft-measurable r.v. Zt. By definition of Px, this in turn means Ex[Lt+8Zt(V8 oet)] = Ex[LtZtExt[L8Y8]], or, since Ext [L8Y8] = E[(Y8 o et)(L8 o 8t)I.Ft], Ex[Lt+8Zt(Y8 o et)] = Ex[LtZt(Y8 o 0t)(L8 o Bt)],
(2.8)
which is the same as (2.7). 0
30
CHAPTER H. SOME GENERAL TOOLS AND RESULTS
Remark 2.6 For {u , }
to define a time-homogeneous Markov process, it xEE suffices to assume that {Lt} is a multiplicative functional with Ex Lt = 1 for all x, t. Indeed, then
E[Lt+B I.Ft] = LtE[L8 o 9t I. t] = LtExt L8 = Lt, (using the Markov property in the second step) so that the martingale property is automatic. ❑
Notes and references The results of the present section are essentially known in a very general Markov process formulation, see Dynkin [128] and Kunita [239]. A more elementary version along the lines of Theorem 2.5 can be found in Kuchler & Sorensen [240], with a proof somewhat different from the present one.
3 Duality with other applied probability models In this section, we shall establish a general connection between ruin probabilities and certain stochastic processes which occurs for example as models for queueing and storage. The formulation has applications to virtually all the risk models studied in this book. The result is a sample path relation, and thus for the moment no parametric assumptions (on say the structure of the arrival process) are needed. We work on a finite time interval [0, T] in the following set-up: The risk process {Rt}o
R
= Ro +
f
p(R8) ds - At where At =
U.
(3.1)
k: vk
The initial condition is arbitrary, Ro = u (say), and the time to ruin is 7-(u) = inf {t > 0: Rt < 0}. The storage process {Vt }o
3. DUALITY WITH OTHER APPLIED PROBABILITY MODELS
31
decreases at rate p(r) when Vt = r (i.e., V = -p(V)). That is, instead of (3.1) we have
Vt = At -
f
P(Vs)ds
where A= U= AT - AT_t, (3.2) k: ok
and we use the convention p(O) = 0 to make zero a reflecting barrier (when hitting 0, {Vt} remains at 0 until the next arrival). Note that these definitions make {Rt} right-continuous (as standard) and {Vt} left-continuous. The sample path relation between these two processes is illustrated in Fig. 3.1.
:.x......11 --4.__._...__.____•_..___ ._: 1} 0 011 =T-01N ^N-3 T-o
0
011
014
01N
Figure 3.1 Define r(u) = inf It > 0: Rt < 0} (r(u) = oo if Rt > 0 for all t < T) and let ii(u,T) =
inf Rt < 0 P (O
P(r(u) < T)
be the ruin probability. Theorem 3.1 The events {T(u) < T} and {VT > u} coincide. In particular, V)(u,T) = P(VT > u).
(3.3)
Proof Let rt' denote the solution of R = p(R) subject to r0(u) = u. Then rt°) > 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
<
RQ2,
and so on. Hence RQ„ > 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)<00 n=1
is known to be necessary and sufficient ((APQ] p. 176) but appears to be rather intractable. Remark 4 .3 The i.i.d. assumption on the Z1,. .. , ZN or, equivalently, the Y1, ... , YN in Theorem 4.1 is actually not necessary - the result is a sample path relation as is Theorem 3.1. Similarly, there is a more general version of Corollary 4.2. One then assumes Yn to be a stationary sequence, w.l.o.g. doubly ❑ infinite (n'= 0, ±1, ±2, ...) and defines Zn = -Y-n. Next consider change of measure via likelihood ratios. For a random walk, a Markovian change of measure as in Theorem 2.5 does not necessarily lead to a random walk: if, e.g., F has a strictly positive density and the Px corresponds to a Markov chain such that the density of X1 given Xo = x is also strictly positive, then the restrictions of Fx, Px to Fn are equivalent (have the same null sets) so that the likelihood ratio Ln exists. The following result gives the necessary and sufficient condition for {Ln} to define a new random walk:
35
4. RANDOM WALKS INDISCRETE OR CONTINUOUS TIME
Proposition 4.4 Let {Ln} be a multiplicative functional of a random walk with E_-L,, = 1 for all n and x. Then the change of measure in Theorem 2.5 corresponds to a new random walk if and only if (4.3)
Ln = h(Y1) ... h(Yn)
1Px-a.s. for some function h with Eh(Y) = 1. In that case, the changed increment distribution is F(x ) = E[h(Y); Y < x].
Proof If (4.3) holds, then n
n
Ex [f f
= Ex H fi a( YY) i=1 i_1
( Y=) h(YY)
H Ef=(Y=)h(Y=) =
II J fi(Y )P( d)
from which the random walk property is immediate with the asserted form of F. Conversely, the random walk property implies Ex f (Y1) = Eo f (Y1 ). Since L1 has the form g (Xo, Y1), this means E(g(x, Y ) f (Y)] = E[g(O, Y) f (Y)] for all f and x, implying g (x, Y) = h(Y ) a.s. where h (y) = g(0, y ). In particular, (4.3) holds for n = 1. For n = 2, we get
L2 = L1 (L1 o91 ) = h(Y1)g(X1,Y2) = h(1'i)h(I'a), ❑
and so onforn =3,4,....
A particular important example is exponential change of measure (h(y) = e°y-'(") where r. (a) = log F(a] is the c.g.f. of F). The corresponding likelihood ratio is Ln = exp {a (Y1 + • • • + Yn) - nrc(a )} (4.4) ({Ln} is the familiar Wald martingale , cf. e.g. Breiman [78] p. 100 ). We get: Corollary 4.5 Consider a random walk and an a such that c(a) = log F[a] = log Ee° ' is finite, and define Ln by (4.4). Then the change of measure in Theorem 2.5 corresponds to a new random walk with changed increment distribution
F(x) = e-'(a)
Jr
e"'F(dy) .
36
CHAPTER II. SOME GENERAL TOOLS AND RESULTS
Discrete time random walks have classical applications in queueing theory via the Lindley process representation of the waiting time , see Chapter V. In risk theory, they arise as models for the reserve or claim surplus at a discrete sequence of instants, say the beginning of each month or year , or imbedded into continuous time processes , say by recording the reserve or claim surplus just before or just after claims (see Chapter V for some fundamental examples). However, the tradition in the area is to use continuous time models. The appropriate generalization of random walks to continuous time is processes with stationary independent increments (Levy processes). The traditional formal definition is that {Xt} is R-valued with the increments Xt(1)_t(o), Xt(2)_t(l), ... , Xt (n)-t(n-1) being independent whenever t(O) < t(1 ) < ... < t(n) and with Xt( i)_t(i_l) having distribution depending only on t(i) - t(i - 1). An equivalent characterisation is {Xt} being Markov with state space R and
(4.5)
E [f (Xt+e - Xt)I.Ft] = Eof (X.).
Note that the structure of such a process admits a complete description. In discrete time, {Xt} is a random walk, i.e. given by a the increment distribution F(x) = P(Xn+l - Xn < x). In continuous time, {Xt} can be written as the independent sum of a pure drift {pt}, a Brownian component {Bt} (scaled by a variance constant) and a pure jump process {Mt}, Xt =Xo+pt+oBt+Mt. (4.6) More precisely, the pure jump process is given by its Levy measure v(dx), a positive measure on R with the properties
e
J
x2v(dx) < oo, e
v(dx) <
oo
(4.7)
f x:IxJ>e}
for all e > 0. Roughly, the interpretation is that the rate of a jump of size x is v(dx) (if f of Ixlv (dx) = oo, this description needs some amendments, but we omit the details ). The simplest case is 3 = JhvMM < oo, which corresponds to the compound Poisson case: here jumps of {Mt} occur at rate 0 and have distribution B = v/0 (in particular , the claim surplus process for the compound Poisson risk model , with premium rate p, corresponds to a process with stationary independent increments and u = -p, v2 = 0 and v = 3B). A general jump process can be thought of as limit of compound Poisson processes with drift by considering a sequence v(n) of bounded measures with v(n) T v; however, we are
4. RANDOM WALKS IN DISCRETE OR CONTINUOUS TIME 37 almost solely concerned with the compound Poisson case and shall therefore not treat the intricacies of unbounded Levy measures in detail. Now consider reflected versions of processes with stationary independent increments. First assume in the setting of Section 3 that {Rt} is the risk reserve process for the compound Poisson risk model with constant premium rate p(r) = 1. Then the storage process {Vt} has constant release rate 1, i.e. has upwards jumps governed by B at the epochs of a Poisson process with rate ,3 and decreases linearly at rate 1 in between jumps. A different interpretation is as the workload or virtual waiting time process in an M/G/1 queue, defined as a system with a single server working at a unit rate, having Poisson arrivals with rate ,Q and distribution B of the service times of the arriving customers. Here workload refers to the fact that we can interpret Vt as the amount of time the server will have to work until the system is empty provided no new customers arrive; virtual waiting time refers to Vt being the amount of time a customer would have to wait before starting service if he arrived at time t (this interpretation requires FIFO = First In First Out queueing discipline: the customers are served in the order of arrival). Corollary 4.6 In the compound Poisson risk model with constant premium rate p(r) - 1, O(u, T) = P(VT > u), where VT is the virtual waiting time at time T in an initially empty M/G/1 queue with the same arrival rate /3 and the service times having the same distribution B as the claims in the risk process. Furthermore, VT + V for some r.v. V E [0, oo], and b(u) = P(V > u). [The condition for V < oo a.s. is easily seen to be f3pB < 1, cf. Chapter III.] Processes with a more complicated path structure like Brownian motion or jump processes with unbounded Levy measure are not covered by Section 3, and the reflected version is then defined by means of the abstract reflection operator as in (4.2), WT = XT - min Xt (4.8) O
(assuming Wo = Xo = 0 for simplicity). Proposition 4.7 If {Xt} has stationary independent increments as in (4.6), then Ee'(xt-xo) = Eoeaxt = etx(a), where c(a) = ap + a2a2/2 + f
(ex - 1)v(dx)
(4.10)
00 provided the Levy measure of the jump part {Mt} satisfies f", jxJ v(dx) < oo.
38
CHAPTER IL SOME GENERAL TOOLS AND RESULTS
Proof By standard formulas for the normal distribution, Eea(µt + QBt) = et{aµ +a2OZ/2}.
By explicit calculation , we show in the compound Poisson case ( IlvIl < oo) in Proposition III.1.1 that (eax - 1 ) v(dx) .
E eaMt = exp fmoo
In the general case , use the representation as limit of compound Poisson processes. ❑ Note that (4.10) is the Levy-Khinchine representation of the c.g.f. of an infinitely divisible distribution (see, e.g., Chung [86]). This is of course no coincidence since the distribution of Xl - Xo is necessarily infinitely divisible when {Xt} has stationary independent increments. Theorem 4 . 8 Assume that {Xt} has stationary independent increments and that {Lt} is a non-negative multiplicative functional of the form Lt = g(t, Xt Xo) with E2Lt = 1 for all x, t. Then the Markov process given by Theorem 2. 5 has stationary independent increments as well. In particular, if Lt = e9(xt - xo)-tk ( e), then the changed parameters in the representation (4.6) are µ = µ + Oo2 ,
Q2 = v2,
v(dx) = e9xv (dx).
(4.11)
Proof For the first statement , we use the characterization (4.5) and get
E [f(Xt+B - Xt)I-'Ftl = E [f(Xt+B - Xt)L8 o 0tIFt]
= E [f (Xt+s - Xt)g(s, Xt +B - Xt)I Ftl = Eof (X8)g(s, X8) = Eof (X8)L8 = Eof (X8)• For the second, let e" (a ) = Eoeaxl. Then l e" (a)
=
Eo [ Li ea
"] =
e-K (9)Eo {e ( a+9)x1
I = er(a+o )-K(B)
J
R(a) = K(a + 0) - 4 o) aµ + ((a + 0 )
2 -
0 2 )o 2 /2+
a(µ + O 2) + a2a2 / 2 + J
r
J 00
w
(e (a + 9)x
-
a 9x )v(dx)
00
(eax - 1)eexv(dx).
39
5. MARKOV ADDITIVE PROCESSES
Remark 4.9 If X0 = 0, then the martingale {eex(t)-tk(e)} is the continuous ❑ time analogue of the Wald martingale (4.4). Example 4 .10 Let Xt be the claim surplus process of a compound Poisson risk process with Poisson rate ,3 and claim size distribution B, corresponding to p = -1, a = 0, v(dx) _ ,(3B(dx). Then we can write v(dx) _ /3eOxB(dx) = / (dx),
where ,(3 = ,3B[B], B(dx) = B[9] B(dx).
Thus (since µ = p = -1, b = a = 0) the changed process is the claim surplus process of another compound Poisson risk process with Poisson rate ,l3 and claim ❑ size distribution B. Example 4 .11 For an example of a likelihood ratio not covered by Theorem 4.8, let the given Markov process (specified by the Px) be the claim surplus process of a compound Poisson risk process with Poisson rate 0 and claim size distribution B, and let the Px refer to the claim surplus process of another compound Poisson risk process with Poisson rate,3 =,3 and claim size distribution B # B. Recalling that U1, 0.2, ... are the arrival times and U1, U2.... the corresponding claim sizes , it is then easily seen that
Lt = H dB(Ui) i:o;
whenever the Radon-Nikodym derivative dB/dB exists (e.g. dB/dB = b/b when B, B have densities b, b with b(x) > 0 for all x such that b(x) > 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"sth^a) = 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 )h^a)1 = east-tK(a)h^a). ❑ 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.
5. MARKOV ADDITIVE PROCESSES
43
Proof By differentiation in Proposition 5.4, Ei [Steast h(a) + east k^a)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 k^a) 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
5. MARKOV ADDITIVE PROCESSES
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, , ^i = 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
^i = /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 <1, 0
CHAPTER II. SOME GENERAL TOOLS AND RESULTS
46
Proof of Theorem 5.11. First note that the ijth element of Ft[a] is h(.e)
e-tK(e)Ej [e(a+B)st E:[east Jt = j] = Ej[Lteas' ; Jt = A. ; Jt = j] = hie) ; In matrix notation , this means that Ft[a] = e-tw ( e)Ohc) Ft[a + 9]oh (e)
(5.8)
Consider first the discrete time case . Here the stated formula for P follows immediately by letting t = 1, a = 0 in (5.8). Further
Fib (dx )
=
P=(YI
E dx, Jl = j) = Ei[Lt; Yi E dx, Jl = j]
h(e) h(8) eey-K(B)p
:(Yi E dx, Ji = j)
h(e) h=e) eex-K ( e)Fi, (dx). This shows that F, is absolutely continuous w.r.t. F:j with a density proportional to eei . Hence the same is true for H=j and H;,; since Hij, H1, are probability measures , it follows that indeed the normalizing constant is H1 [0]. Similarly, in continuous time ( 5.8) yields et'[a]
= Ohie )et
(K[a +e]-K(e)I)Oh(°)
By a general formula (A.13) for matrix-exponentials , this implies
k[a] =
A -1
) (K[a + 0] - tc(0)' )Ah() = Oh(o) K[a + 0]Oh() - tc(0)I.
Letting a = 0 yields the stated expression for A. Now we can write K[a]
=A+A
) ( K[a + 0] - K [O])Oh(e) (0)
A + (tc(') (a + 0) - tc(') (8)/ld)ag h + ( 7 Aiiii (Bii[a + 0] - Bay [0]) That k(') (a + 0) - tc (') (0) corresponds to the stated parameters µ;, v= , v; (dx) of a process with stationary independent increments follows from Theorem 4.8.
6. THE LADDER HEIGHT DISTRIBUTION 47 Finally note that by (5.7),
h,)Ajjgij(Bij[a+0] - Bij[0]) = hjel)ijgijBij[0](Bij[a] - 1) = Aij4ij(Bij[a] - 1).
Notes and references The earliest paper on treatment of MAP's in the present spirit we know of is Nagaev [265]. Much of the pioneering was done in the sixties in papers like Keilson & Wishart [224], [225], [226] and Miller [260], [261], [262] in discrete time; the literature on the continuous time case tends more to deal with special cases. Though the literature on MAP's is extensive, there is, however, hardly a single comprehensive treatment; an extensive bibliography on aspects of the theory can be found in Asmussen [16]. Conditions for analogues of Corollary 5.3 for an infinite E are given by Ney & Nummelin [266]. For the Wald identity in Corollary 5.6, see also Fuh & Lai [149] and Moustakides [264]. The closest reference on exponential families of random walks on a Markov chain we know of within the more statistical oriented literature is Hoglund [203], which, however, is slightly less general than the present setting.
6 The ladder height distribution We consider the claim surplus process {St } of a general risk process and the time 7- (u) = inf {t > 0 : St > u} to ruin in the particular case u = 0 . Write r+ = T(0) and define the associated ladder height ST+ and ladder height distribution by
G+(x) = 11 (S,-, < x) = 11 (S,-+ < x, 7-+ < oo). Note that G+ is concentrated on (0, oo), i.e., has no mass on (-oo, 0], and is typically defective,
IIG+II = G+(oo) = P(T+ < oo) = 0(0) < 1 when 77 > 0 (there is positive probability that {St} will never come above level 0).
48 CHAPTER K. SOME GENERAL TOOLS AND RESULTS M ST+(2) Sr, = ST+(1)
Figure 6.1 The term ladder height is motivated from the shape of the process {Mt} of relative maxima, see Fig. 6.1. The first ladder step is precisely ST+, and the maximum M is the total height of the ladder, i.e. the sum of all the ladder steps (if rl > 0, there are only finitely many). On Fig. 6.1, the second ladder point is ST+(2) where r+(2) is the time of the next relative maximum after r+(1) = r+, the second ladder height (step) is ST+(2) - ST+(1) and so on. In simple cases like the compound Poisson model, the ladder heights are i.i.d., a fact which turns out to be extremely useful. In other cases like the Markovian environment model, they have a semi-Markov structure (but in complete generality, the dependence structure seems too complicated to be useful). In any case, at present we concentrate on the first ladder height. The main result of this section is Theorem 6.5 below, which gives an explicit expression for G+ in a very general setting, where basically only stationarity is assumed. To illustrate the ideas, we shall first consider the compound Poisson model in the notation of Example 1.2. Recall that B(x) = 1 - B(x) denotes the tail of B. Theorem 6 . 1 For the compound Poisson model with p = 01-LB < 1, G+ is given by the defective density g + (x) =,(3B(x ) = pbo(x) on (0,00 ). Here bo(x) _ B(x)/µB. For the proof of Theorem 6.1, define the pre-r+-occupation measure R+ by
R+(A) = E
o "o I(St E A,T+ > t)dt = E f 0T+I(St E A) dt.
(6.1)
f
The interpretation of R+(A ) is as the expected time {St} spends in the set A before T+. Thus, R+ is concentrated on (-oo, 0], i.e., has no mass on ( 0, oo). Also, by approximation with step functions , it follows that for g > 0 measurable, 0 f
T+
g(y)R+(dy) = E f g(St)dt. o 00
(6.2)
49
6. THE LADDER HEIGHT DISTRIBUTION Lemma 6 .2 R+ is the restriction of Lebesgue measure to (-00, 0].
Proof Let T be fixed and define St = ST - ST_t, 0 < t < T. That is, {St }o
St
S* t
a Figure 6.2(a): T+ > t
Figure 6.2(b): r+ < t Thus,
P(STEA,T+>T) = P(STEA,St<0,0
50
CHAPTER II. SOME GENERAL TOOLS AND RESULTS
Integrating w.r.t dT, it follows that R+ (A) is the expected time when ST is in A and at a minimum at the same time . But since St -4 -oo a. s., this is just the Lebesgue measure of A, cf. Fig. 6.3 where the bold lines correspond to minimal values.
Figure 6.3
Lemma 6 .3 G+ is the restriction of /3R+*B to (0, oo). That is, for A C (0, oo),
G+(A) = Q f 0 B(A - y)R+(dy) 00 Proof A jump of {St} at time t and of size U contributes to the event IS,, E A} precisely when r+ > t, U + St_ E A. The probability of this given { Su}u
t), and since the jump rate is /3, we get G+ (A) = f 00 /3 dt E[B(A - St _); T+ > t] 0 _ /3 f E[B( A - St);T+ > t] dt 0 T+
_ /3E f g( St) dt = 0 f g(y) R+(dy) 0 00
where g(y) = B(A - y) (here we used the fact that the probability of a jump at ❑ t is zero in the second step, and (6.2) in the last).
6. THE LADDER HEIGHT DISTRIBUTION
51
Proof of Theorem 6.1 With r+(y) = I(y < 0) the density of R+, Lemma 6.3 yields
g+ (x) = ,Q
f
r+(x - z)B(dz) _
f
I(x < z)B(dz) _ f (x).
0 Generalizing the set-up, we consider the claim surplus process {St }t>o of a risk reserve process in a very general set-up, assuming basically stationarity in time and space, {St+8 - S8 )t> o = {St }t>o for all s > 0. The sample path structure is assumed to be as for the compound Poisson case: {St*} is generated from interclaim times Tk and claim sizes Uk according to premium 1 per unit time, i.e. Nt
St
=>Uk
-t
where Nt = max{k = O,1,...:T1 +•••+Tk
k=1
The first ladder epoch r is defined as inf It > 0 : St > 0} and the corresponding ladder height distribution is
G+* (A) = P(S** E A) = P(ST+ E A,T+ < oo). The traditional representation of the input sequence {(TT, U k)} k=1 a is as a marked point process M *, i.e. as a point process on [0, oo) x (0, oo). The points in the plane (marked by x on Fig. 6 .4) are (ak, Uk) (k = 1, 2, ...) where ak = Ti + • • • + Tk , the first component representing time (the arrival time o,* ) and the second the mark (the claim size Uk ). The marked point process .M o 08 shifted by s is defined the obvious way, cf. Fig. 6 . 4 (the points in the plane are (ak - s, Uk) for those k for which ak - s > 0). We call M * stationary if M* o B8 has the same distribution as M* for all s > 0; obviously, this is equivalent to the risk process {St*} being stationary in the sense of (6.4). In the stationary case, we define the arrival rate as E# { k : ak E [0 , h]} /h (by stationarity, this does not depend on h).
CHAPTER II. SOME GENERAL TOOLS AND RESULTS
52
M* U. i
1
U2 Us
-1_ 0 or Q2 $
U3*1
L 0
7
X
11
1
I
Figure 6.4
Given a stationary marked point process M*, we define its Palm version M as a marked point process having the conditional distribution of M* given an arrival at time 0 , i.e. o, = 0 . We represent M by the sequence (Tk, Uk) k=1,2,... where TI = 0, and let T = T2 denote the first proper interarrival time . The two fundamental formulas connecting M* and M are
Eco(M) = aE E, V(M* o eak ), k: vk E [0, h]
Eco(M*) = 1 E f co(M o Bt)dt, where T is the first arrival time > 0 of M and h > 0 an arbitrary constant (in the literature, most often one takes h = 1). As above , the r. h.s. of (6.5) does not depend on h; letting h J. 0, Oh becomes the approximate probability F(ri < h) of an arrival in [0, h] and the sum approximately ^o(M*)I(ul < h). This more or less gives a proof that indeed (6.5) represents the conditional distribution of M* given vi = 0. Note also that (again by stationarity) the Palm distribution also represents the conditional distribution of M* o Ot given an arrival at time t. See, e.g., Sigman [348] for these and further aspects of Palm theory. Example 6 .4 Consider a finite Markov additive process (cf. Section 5) which has pure jump structure corresponding to pi = a = 0, vi(dx) = ,QiBi(dx). Assume {Jt} irreducible so that a stationary distribution 7r = (1i)iGE exists.
6. THE LADDER HEIGHT DISTRIBUTION
53
Interpreting jump times as arrival times and jump sizes as marks, we get a marked point process generated by Poisson arrivals at rate /3i and mark distribution Bi when Jt = i, and by some additional arrivals which occur w.p. qij when {Jt} jumps from i to j and have mark distribution Bij. A stationary marked point process M* is obtained by assigning Jo distribution Tr. If Jt_ = i, an arrival for M* occurs before time t + dt w.p.
dt
A + E Aijgij j#i
Thus the arrival rate for M* is
1] it A + E Aijgij iEE i#i
Given that an arrival occurs at time t , the probability aij of Jt - = i, Jt = j is iri(3i /,O for i = j and iriAijgij/,O for i # j. It follows that we can describe the Palm version M as follows . First choose (Jo_, Jo) w.p. aij for (i, j) and let the initial mark Ul have distribution Bi when i = j and Bij otherwise. After that, let the arrivals and their marks be generated by {Jt} starting from Jo = j. Note in particular that the Palm distribution of the mark size (i.e., the distribution of Ul) is the mixture
B = E
aii Bi + aij Bij J =
iEE
j#i
!i
iEE 0
J,6iBi + Aijgij Bij j#i
Theorem 6 . 5 Consider a general stationary claim surplus process {St }t>o, let U0 be a r. v. having the Palm distribution of the claim size and F (x) = F(Uo < x) its distribution . Assume that St -* - oo a.s. and that p = 0EU0 < 1. Then the ladder height distribution G+ is given by the (defective) density g+(x) = ,OF(x). Before giving the proof, we note: Corollary 6.6 Under the assumptions of Theorem 6.5, the ruin probability ,/,*(0) with initial reserve u = 0 is p = /3EU0. This follows by noting that
iP*(0) = IIG+JI =
J0 "o g+(x)dx = ,O fo "o F(x)dx = ,OEU0.
54
CHAPTER H. SOME GENERAL TOOLS AND RESULTS
By (6.5), V` (0) = E E Uk k: ak E [0,1]
here the r . h.s. has a very simple interpretation as the average amount of claims received per unit time . The result is notable by giving an explicit expression for a ruin in great generality and by only depending on the parameters of the model through the arrival rate 0 and the average ( in the Palm sense) claim size EU0. The last property is referred to as insensitivity in the applied probability literature. Proof of Theorem 6.5. A standard argument for stationary processes ([78] p. 105) shows that one can assume w.l.o.g. that M* and M have doubly infinite time (i.e., are point processes on (-oo , oo) x (0 , oo)). We then represent M by the mark (claim size ) Uo of the arrival at time 0, the arrival times 0 < 0'1 < Q2 < ... in (0, oo ) and the arrival times 0 > 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
p(t) = P(St EA,Su< 0,0
6. THE LADDER HEIGHT DISTRIBUTION
{
55
Su}0
U0
A
U0
tt
\t
N u>0
U_1
Figure 6.5
where
it = { St < Su, 0 < u < t } is the event that { Su } has a relative minimum at t . In Fig. 6.5, time instants corresponding to such minimal values have been marked with bold lines in the path of { St}, and we let L(dy) be the random measure L(A) = fo°° I(St E A; NIt)dt . Since So = U0, the support of L has right endpoint U0, and since by assumption St -* -oo a.s., t -a oo, the left endpoint of the support is -oo. A sample path inspection just as in the proof of Lemma 6 . 2 therefore immediately shows that L(dy) is Lebesgue measure on (-oo, Uo], cf. Fig. 6.5 where the boxes on the time axis correspond to time intervals where {St } is at a minimum belonging to A and split A into pieces corresponding to segments where {Su} is at a relative minimum. Thus,
G' (A) = 3 f P(St E A; Mt)dt = i3EL(A) o"o
56
CHAPTER II. SOME GENERAL TOOLS AND RESULTS
= OE f 0 I(Uo>y)I (yEA)dy = Q f IP (Uo>y)dy A 0o a fA P(y) dy•
0 Notes and references Theorem 6.5 is due to Schmidt & co-workers [48], [147], [263] (a special case of the result appear in Proposition VI.2.1). A further relevant reference related to Corollary 6.6 is Bjork & Grandell [67].
Chapter III
The compound Poisson model We consider throughout this chapter a risk reserve process {Rt } t>o in the terminology and notation of Chapter I, and assume that • { Nt}t>o is a Poisson process with rate j3. • the claim sizes U1, U2, ... are i. i.d. with common distribution B, say, and independent of {Nt}. • the premium rate is p = 1.
Thus , {Rt} and the associated claims surplus process {St} are given by Nt
Rt = u+t -EUi,
Nt
St = u-Rt = EUi -t.
i=1
i=1
An important omission of the discussion in this chapter is the numerical evaluation of the ruin probability. Some possibilities are numerical Laplace transform inversion via Corollary 3.4 below , exact matrix-exponential solutions under the assumption that B is phase-type (see further VIII. 3), Panjer's recursion ( Corollary XI. 4.6) and simulation methods ( Chapter X). For finite horizon ruin probabilities , see Chapter IV. It is worth mentioning that much of the analysis of this chapter can be carried over in a straightforward way to more general Levy processes . A common view of the literature is to consider such processes as perturbed compound Poisson risk processes , i.e. being of the form Rt = Rt+Bt + Jt where {Rt } is a compound
57
CHAPTER III. THE COMPOUND POISSON MODEL
58
Poisson risk process, {Bt} a Brownian motion and {Rt} a pure jump process, say stable Levy motion. We do not spell out in detail such generalizations. See e.g. Dufresne & Gerber [126], Schmidli [319], [324], Furrer [150], and Schlegel [316].
1 Introduction For later reference, we shall start by giving the basic formulas for moments, cumulants , m.g.f.'s etc. of the claim surplus St - u - Rt. Write pB^) = EUn' YB = Pali = EU, P = PAB = 1/(1 + rl) Proposition 1.1 (a) ESt = t(13µ$ - 1) = t(p - 1);
(b) Var St
= t,6pBa);
(c) Ee8St = et" (8) where c(s) = f3(B[s] - 1) - s; (d) The kth cumulant of St is tf3p(k) for k > 2. Proof It was noted in Chapter I that p - 1 is the expected claim surplus per unit time, and this immediately yields (a). A more formal proof goes as follows: Nt
r
Nt
ESt = E > U k - t = E E [ U k k=1
Nt - t
k=1
= E[Ntµs] - t = fltpB - t = t(p - 1). The same method yields also the variance as Nt
Var St
Nt
Ne
= Var E Uk = Var E ^ Uk Nt +EVar [
Uk Nt
1 k=1
k=1 k=1
Var [Ntµs] + E[NtVar U] = 113µs + t13Var U = tf3pB2). For (c), we get Ee8st =
00 e-8t c` Ee8 (U1+...+Uk)P(Nt = k) k=O B[s]k , e - )3t (fit' k
e-8t
t} = etk(8) exp {-st -'3t + B[s]f
k=O
Finally, for (d) just note that the kth cumulant of St is tic(k) (0), where K(k) (0) is the kth derivative of is at 0, and that B(k)[0] = Pak). 0
59
1. INTRODUCTION
The linear way the index t enters in the formulas in Proposition 1.1 is the same as if {St} was a random walk indexed by t = 0, 1, 2,.... The connections to random walks are in fact fundamental, and there are at least two ways to exploit this: Recalling that ok is the time of the kth claim, we have Sok - Sok_l = Uk - Tk, where Tk is the time between the kth and the (k - 1)th claim. Obviously, the Uk - Tk are i.i.d. so that {Sok } is a random walk with mean EU-ET = EU- 1 = ,3EU0-1 = -1µs where rt is the safety loading. In this way, we get a discrete time random walk imbedded in the claim surplus process {St}, which is often used in the literature for obtaining information about {St} and the ruin probabilities. For example, obviously 0(u) = F(maxk Sok > u). We return to this approach in Chapter V. The point of view in the present chapter is, however, rather to view {St} directly as a random walk in continuous time, meaning that the increments are stationary and independent, cf. II.4. Here is one immediate application: Proposition 1.2 (DRIFT AND OSCILLATION)
(a) No matter the value of 77,
St/ta3'p-1 ast ->oo;
(b) If 77 < 0, then St 4 co; (c) If 77 > 0, then St -00; (d) If 17 = 0, then lien inft, St = -oo, lim supt. St = oo.
For the proof, we need the following lemma: Lemma 1.3 If nh < t < (n + 1)h, then
Snh - h < St < S(n+1)h + h. Proof We first note that for u, v > 0,
S„+V > S„ - V. Indeed, Sn+0 - S„ attains its minimal value when there are no arrivals in (u, u + v], and the value is then precisely v. In particular, if t = nh + v with 0 < v < h, then St> Snh-V>Snh-h. The right hand inequality in (1.3) is proved similarly.
CHAPTER III. THE COMPOUND POISSON MODEL
60
Proof of Proposition 1.2. For any fixed h, {Snh}n=o,1,... is a discrete time random walk, and hence by the strong law of large numbers, Snh/n a4' ESh = h(p - 1). Thus using Lemma 1.3, we get
lim inf St t->oo
= lim inf inf
St
t n-roo nh
h l++m of Sn 7t h =h -ESh = p - 1. A similar argument for lim sup proves (a), and (b), (c) are immediate consequences of (a). Part (d) follows by a (slightly more intricate) general random walk result ([APQ], p. 169) stating that lim infra, 0 Snh = -00, lim supn_,,,. Snh ❑ = 00 (the lemma is not needed for (d)). Corollary 1.4 The ruin probability 0(u) is 1 for all u when 77 < 0, and < 1 for all u when 77 > 0. Proof The case of 17 < 0 is immediate since then M = oo by Proposition 1.2. If rl > 0, it suffices to prove 4'(0) = F(M > 0) < 1. However, if P(M > 0) = 1, then {St} upcrosses level 0 a.s. at least once. Considering the next downcrossing (which occurs w.p. 1 since St -4 -oo) and repeating the argument, it is seen that upcrossing occurs at least twice, hence by induction i.o. This contradicts ❑ St-4-00. There is also a central limit version of Proposition 1.2: Proposition 1.5 The limiting distribution of St - t - 1) as t -4 oo is normal vtwith mean zero and variance )3µsz) Proof Since {St}t>o is a Levy process (a random walk in continuous time), {Snh}n=o,1,._. is a discrete-time random walk for any h > 0, and hence it folz lows from standard central limit theory and the expression Var(St) = tf3pB (Proposition 1.1(b)) that the assertion holds as t -4 oo through values of the form t = 0, h, 2h.... The general case now follows either by another easy application of Lemma 1.3, or by a general result on discrete skeletons ([APQ] p. ❑ 307). Remark 1 .6 Often it is of interest to consider size fluctuations, where the size of the portfolio at time t is M(t). Assuming that each risk generates claims at Poisson intensity /3 and pays premium 1 per unit time, this case can be reduced to the compound Poisson model by an easy operational time transformation ❑ T-1(t) where T(s) = )3 fo M(t)dt. Notes and references
All material of the present section is standard.
61
2. THE POLLACZECK-KHINCHINE FORMULA
2 The Pollaczeck-Khinchine formula The time to ruin r(u) is defined as in Chapter I as inf It > 0: St > u}, and we shall here exploit the decomposition of the maximum M as sum of ladder heights, cf. Fig. 11.6.1. We assume throughout rl > 0 or, equivalently, p < 1. It is crucial to note that for the compound Poisson model, the ladder heights are i. i. d.. This follows simply by noting that the process repeats itself after reaching a relative maximum. The decomposition of M as a sum of ladder heights now yields: 00
Theorem 2 . 1 The distribution of M is (1- IIG +II)EG+ , where G+ is given n=0
by the defective density g+ (x) = 3B (x) = pbo(x) on (0, oo ). B(x)/aB.
Here bo(x) _
Proof The probability that M is attained in precisely n ladder steps and does not exceed x is G+ (x)(1 - IIG+II) (the parenthesis gives the probability that there are no further ladder steps after the nth ). Summing over n, the formula for the distribution of M follows . The expression for g+ was proved in Theorem 11.6.1. 0 Alternatively, we may view the ladder heights as a terminating renewal process and M becomes then the lifetime. Combined with i/i(u) = P ( M > u), Theorem 2.1 provides a representation formula for 0(u), which we henceforth refer to as the Pollaczeck-Khinchine formula. Note that the distribution B0 with density bo is familiar from renewal theory as the limiting stationary distribution of the overshoot (forwards recurrence time ), [APQ] Ch. IV.3-4 or A. 1e. Thus , we can rewrite the PollaczeckKhinchine formula as 00 (u) = P (M > u) = (1 - P) E PnBon(u) ,
(2.1)
n=0
representing the distribution of M as a geometric compound. As a vehicle for computing tIi(u), (2.1) is not entirely satisfying because of the infinite sum of convolution powers, but we shall be able to extract substantial information from the formula, nevertheless. The following results generalizes the fact that the conditional distribution of the deficit ST(o) just after ruin given that ruin occurs (i.e., that r(0) < oo) is Bo: taking y = 0 shows that the conditional distribution of (minus) the surplus -ST(o)- just before ruin is again B0, and we further get information about the joint conditional distribution of the surplus and the deficit. Note that this
62
CHAPTER III. THE COMPOUND POISSON MODEL
distribution is the same as the limiting joint distribution of the age and excess life in a renewal process governed by B, cf. Theorem A1.5. Theorem 2 . 2 The joint distribution of (-ST(o )_, ST(o)) is given by the following four equivalent statements: (a) 11 (-ST(o)_ > x, ST(o) > y; 7r(0 ) < oo) = Q
B(z) dz; f +b (b) the joint distribution of (-ST( o)-, ST(o )) given r (0) < oo is the same as the distribution of (VW, (1 - V)W) where V, W are independent, V is uniform on (0, 1) and W has distribution Fw given by dFyy/ dB(x) = x/µB; (c) the marginal distribution of -ST(o)_ is Bo , and the conditional distribution of ST(o) given -ST(o)_ = y is the overshoot distribution B(Y) given by Bov)(z) _ Bo (y + z )/Bo(y); (d) the marginal distribution of ST(o)_ is B0, and the conditional distribution of -ST(o)_ given ST(o)_ = z is Bo z) The proof is given in IV.2 and it gives an alternative derivation of the distribution of the deficit ST(o) Notes and references The Pollaczeck-Khinchine formula is standard in queueing theory, see for example [APQ], Feller [143] or Wolff [384]. The proof of Theorem 11.6.1 is traditionally carried out for the imbedded discrete time random walk, where it requires slightly more calculation. As shown in Theorem 11.6.5, the form of G+ is surprisingly insensitive to the form of {St} and holds in a certain general marked point process set-up. However, in this setting there is no decomposition of M as a sum of i.i.d. ladder heights so that the results do not appear not too useful for estimating 0(u) for u>0. Theorem 2.2(a) is from Dufresne & Gerber [125]. Again, there is a general marked point process version, cf. Asmussen & Schmidt [49]. For the study of the joint distribution of the surplus ST(u)_ just before ruin and the deficit ST(„)- just after ruin, see Schmidli [323] and references there. In the risk theory literature, the Pollaczeck-Khinchine formula is often referred to as Beekman 's convolution formula, cf. Beekman [61], [62].
3 Special cases of the Pollaczeck-Khinchine formula The model and notation is the same as in the preceding sections. We assume rt > 0 throughout.
3. SPECIAL CASES OF POLLACZECK-KHINCHINE
63
3a The ruin probability when the initial reserve is zero The case u = 0 is remarkable by giving a formula for V)(u) which depends only on the claim size distribution through its mean: Corollary 3.1 0(0) = p = Nl2B = 1 1 +71 Proof Just note that (recall that T+ = r(0)) 00
z/^(0) = I' (-r+ < oo) =
IIG+II
= )3 f(x)dx =l3LB•
Notes and references The fact that tp(u) only depends on B through µB is often referred to as an insensitivity property. As shown in 11.6, the formula for P(O) holds in a more general setting; a further relevant reference is Bjork & Grandell [67].
3b Exponential claims Corollary 3.2 If B is exponential with rate S, then,0(u) = pe-(a-A)" Proof The distribution Bo of the ascending ladder height ( given that it is defined ) is the distribution of the overshoot of {St} at time r+ over level 0. But claims are exponential , hence without memory, and hence this overshoot has the same distribution as the claims themselves . I.e., B0 is exponential with rate S and the result can now be proved from the Pollaczeck -Khinchine formula by elementary calculations . Thus , Bon is the Erlang distribution with n phases and thus the density of M at x > 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,S,+
zu P(M > u - y)G+(dy ) =
f U V(u
- y)G+(dy)
For the last identity in (3.3), just insert the explicit form of G+. The case of (3.4) is similar (equivalently, (3.4) can be derived by elementary algebra from (3.3)). ❑
65
3. SPECIAL CASES OF POLLACZECK-KHINCHINE Corollary 3.4 The Laplace transform of the ruin probability is
fo
00
e-8uiP(u)du - 3 - /3B[-s] - Ps s(,(3 - s - )3B[-s])
(3.5)
Proof We first find the m. g.f. Bo of B0 as m e8u B(du) = B[s] - 1 Bo[s] = f oc, eau B(u) du = f PB 3PB SPB 0 o
(3.6)
Hence Ee8M
00 = (I - p) E p"Bo[s]" = 1 - p = (1 - p)s , (3.7) s +,3 - ,(3B[s] 1 - pBo[s] n-o
f ao e-8' ( u)du = a-8uP (M > u)du = 0 o
1
( 1+
(1 - Ee-8M)
(1 - p)s
s /3 - s - /3B[-s] which is the same as (3.5). Corollary 3.5 The first two moments of M are 2 EM - PPB2) EM2 = PPB) + QZPBl 2(1 - p)2 3(1 - P)PB 2(1 - P)pB'
(3.8)
Proof This can be shown, for example, by analytical manipulations (L'Hospital's rule) from (3.7). We omit the details (see, e.g., [APQ] pp. 206-207). 0 Notes and references Corollary 3.3 is standard , see e.g. [APQ] pp. 111-112 or Feller [143]. The approach there is to condition upon the first claim occuring at time t and having size x , which yields the survival probability as 00
f
u }t
Z(u) = f f3e-Rtdt 0 from which (3.4) can be derived by elementary but tedious manipulations. Of course, it is not surprising that such arguments are more cumbersome since the ladder height representation is not used. Also (3.7) and Corollary 3.5 can be found in virtually any queueing book. In fact, either of these sets of formulas are what many authors call the Pollaczeck-Khinchine formula. In view of (3.5), numerical inversion of the Laplace transform is one of the classical approaches for computing ruin probabilities. Some relevant references are Abate & Whitt [2], Embrechts, Griibel & Pitts [132], Griibel [179] and Thorin & Wikstad [370] (see also the Bibliographical Notes in [307] p. 191).
66
CHAPTER III. THE COMPOUND POISSON MODEL
3d Deterministic claims Corollary 3.6 If B is degenerate at p, then p) 1: e-p(k -u/,u) [p(k - u/p)]k k-o
k!
Proof By replacing {St} by {Stu/p} if necessary, we may assume p = 1 so that the stated formula in terms of the survival probability Z(u) = 1 - z/'(u) takes the form
L^J L. e-O('-u) [)3(k - u)]k
Z(u)
k! k-0
The renewal equation (3.4) for Z( u) means
f lhu Z(u) = 1-,3+ 1-8+
J0
Z(u-y),3I( 0
uu ✓ u-lhu
1-a+/3
Z(y)/3I(0
J0 uZ(y)dy
0
U Z(y) dy
1-13+0
1 < u < oo
For 0 < u < 1, differentiation yields Z'(u) _ /3Z(u) which together with the boundary condition Z(0) = 1-/3 yields Z(u) _ (1-/3) eAu so that (3.9) follows for 0 < u < 1. Assume (3.9) shown for n - 1 < u < n and let Z(u ) denote the r.h.s. of (3.9). For n < u < n + 1, differentiation yields Z(u) _ /3Z(u) - /3Z(u - 1), u) [N(k - u)]k Z^ d 1 u )=e-R(k_ a) n ( du ( k! (1 -
Q (k - u) a)Qea" + (1 - Q) 3e1
[O(k - u)]k k!
k= n
(1
- u)]k-1
-
L3) 1: e_O(k-u) NIN (k (k - 1)! k=1
u-1
k-u+1) [/3( k - u + 1 )]k = QZ(u) - )3(1 - Q) k=0 E e-0( k! = /32(u) - /32(u - 1).
4. CHANGE OF MEASURE VIA EXPONENTIAL FAMILIES 67 Since Z(n) = 2(n) by the induction hypothesis, it follows that Z(u) = 2(u) for n
4 Change of measure via exponential families If X is a random variable with c.d.f. F and c.g.f. K(a) = logEe'X =
109f
00 eaxF(dx) = logF[a], 00
the standard definition of the exponential family {F9} generated by F is FB(dx) = e°x-K(e)F(dx), (4.1) or equivalently, in terms of the c.g.f. of F9, co(a) = rc(a + 9) - r.(9). (4.2) (Here 9 is any such number such that r.(9) is well-defined.) The adaptation of this construction to stochastic processes with stationary independent increment as {St} has been carried out in 11.4, but will now be repeated for the sake of self-containedness. We could first tentatively consider the claim surplus X = St for a single t, say t = 1: recall from Proposition 1.1 that c(a) = /3(B[a] - 1) - a, and define rce by (4.2). The question then naturally arises whether ie is the c.g.f. corresponding to a compound Poisson risk process in the sense that for a suitable arrival intensity 00 and a suitable claim size distribution BB we have no(a) = rc(a + 9) - rc(9) = ,Qe(Bo[a] - 1) - a. The answer is yes: inserting in (4.2) shows that the solution is Ox
[O]0]. (4.4) ,3B = ,3B[9], B9(dx) = B[9] B(dx), or equivalently BB[a] = B[^+ Repeating for t 54 1, we just have to multiply (4.3) by t, and thus (4.4) works as well. Formalizing this for the purpose of studying the whole process {St}, we set up
68
CHAPTER III. THE COMPOUND POISSON MODEL
Definition 4.1 Let P be the probability measure on D[0, oo) governing a given compound Poisson risk process with arrival intensity,3 and claim size distribution B, and define 09, BB by (4.4). Then FB denotes the probability measure governing the compound Poisson risk process with arrival intensity,0e and claim size distribution Be; the corresponding expectation operator is E9. The following result (Proposition 4.2, with T taking the role of n) is the analogue of the expression exp{8(x1 + • • • + xn) - nr.(9)} (4.5) for the density of n i.i.d. replications from Fe (replace x by xi in (4.1) and multiply from 1 to n). Let FT = o(St : t < T) denote the o•-algebra spanned by the St, t < T, and PBT) the restriction of PB to FT.
Proposition 4.2 For any fixed T, the PBT) are mutually equivalent on.FT, and dP(T)
dP^T)
= exp {BST - Tic (0)}
. (4.6)
That is, for G E FT,
F(G) = Po (G) = EB [exp {-BST + Ttc(0)} ; G]. (4.7) Proof We must prove that if Z is FT-measurable, then EBZ = E [Ze9ST _T"(9)I .
(4.8)
By standard measure theory, it suffices to consider the case where Z is measurable w.r.t. .FTn) = Q(SkTIn : k = 0,1, . .. , n) for a given n. But let Xk = SkT/n - S(k_1)Tln. Then the Xk are i.i.d. with common c.g.f. Ti(a)/n, Z is measurable w.r.t. v(Xi, ... , Xn), and thus (4.8) follows by discrete exponential family theory, in particular the expression (4.5) for the density. The identity (4.7) now follows by taking Z = e-BST+TK(e)I(G) ❑ Theorem 4 .3 Let T be any stopping time and let G E FT, G C {T < oo}. Then P(G) = Fo(G) = EB [exp {-BST + TK(O)} ; G]. (4.9) Proof We first note that for any fixed t, Eee-BSt + tk(B) = 1.
(4.10)
69
5. LUNDBERG CONJUGATION
Now assume first that G C Jr < T} for some deterministic T. Then G E FT, and hence (4.7) holds. Given FT, t = T -,r is deterministic. Thus by (4.10),
Ee [exp { -BST +Trc(9)} I(G) FT)] = 1, so that PG = EeE0 [exp { -9ST+Trc(9)}I(G)I FT)]
= Ee [exp { -BST + rrrc(O)} I(G)EB [ exp {-9 (ST - ST) + (T - r)rc(9)}I .FT]] = EB [exp { -BST + Trc(9)} I(G)] . Now consider a general G. Then GT = G n Jr < T} satisfies GT E FT, GT C_ Jr < T}. Thus, according to what has just been proved, (4.9) holds with G replaced by GT. Letting T t oo and using monotone convergence then shows ❑ that (4.9) holds for G as well.
5 Lundberg conjugation Being a c.g.f., c(a) is a convex function of a. The behaviour at zero is given by the first order Taylor expansion c(a)
r. (0) + rc'(0)a = 0 + ES1 a = a (p - 1) _ -1 + a. 77
Thus, subject to the basic assumption ij > 0 of a positive safety loading, the typical shape of rc is as in Fig. 5.1(a). (b) KL(a)
(a) rc (a)
'Y
-'Y
Figure 5.1 It is seen that typically) a ry > 0 satisfying 0 = r.(-Y) = 13(B['Y] - 1) - 7 1 Some discussion further supporting this statement is given in the next section.
70 CHAPTER III. THE COMPOUND POISSON MODEL exists . Equation (5.1) is known as the Lundberg equation and plays a fundamental role in risk theory ; an equivalent version illustrated in Fig. 5.2 is B(7) = 1 + ^. (5.2)
7
s
Figure 5.2 As support for memory, we write FL instead of F7, ,QL instead of /37 and so on in the following . Note that KL (a) = /L (BL [a] - 1) - a = i(a + 7),
(5.3)
cf. Fig. 5.1(b). An established terminology is to call -y the adjustment coefficient but there are various alternatives around, e.g. the Lundberg exponent. Example 5 .1 Consider the case of exponential claims, b[s] = 5/(b - s). It is then readily seen that the non-zero solution of (5.1) (or (5.2)) is 7 = 5-/3. Thus B[7] = 6/,3, and (4.4) yields /3L = b and that BL is again exponential with rate bL =,3. Thus, Lundberg conjugation corresponds to interchanging the rates of the interarrival times and the claim sizes. ❑ It is a crucial fact that when governed by FL, the claim surplus process has positive drift > 0, (5.4) ELS1 = #L(0) cf. Fig. 5.1(b). Taking T = r(u), G = {T(u) < oo} in Theorem 4.3, we further note that ( 5.1) is precisely what is needed for one of the terms in the exponent
5. LUNDBERG CONJUGATION 71 to vanish so that Theorem 4.3 takes a particular simple form, V) (u) = P(T(u) < oo) = EL [exp {-ryS,(u)} ; T(u) < oo] . Letting e(u) = ST(u) - u be the overshoot and noting that PL(T(u) < oo) = 1 by (5.4), we can rewrite this as 0(u) = e-"ELe-7^(u). (5.5) Theorem 5 .2 (LUNDBERG'S INEQUALITY) For all u > 0, V)(u) < e-7u. Proof Just note that e(u) > 0 in (5.5). 0 Theorem 5 .3 (THE CRAMER-LUNDBERG APPROXIMATION) i'(u) - Ce-7u as u -4 oo, where
C - 1-p - 1- P -Y j o' xeryxOB (x) dx /3k [-Y] - 1
(5.6)
Proof By renewal theory, see A .1e, e(u) has a limit i;(oo) (in the sense of weak convergence w.r.t. PL ) with density 1 - G+ L) (x) G+L) (x) IL(+) µ+L)
where G+L) is the FL- ascending ladder height distribution and µ+L) its mean. Since a-7' is continuous and bounded, we therefore have ELe-7t(u) -+ C where C ELe-7 (00) = µ+) f
ry^+L)
e-7-(1 - G+L)(x)) dx
00 f 0 (1 - e-7x)G+(dx), (5.7) 0
J
and all that is needed to check is that ( 5.7) is the same as (5.6 ). To this end, take first 0 = ry, T = T+, G = {S,+ E A} in Theorem 4.3. Then P(ST+ E A) = EL [exp { -7S?+} ; ST+ E A] ,
which shows that G(L) (dx) = e7xG +(dx) = e7x /3 (x) dx.
(5.8)
CHAPTER III. THE COMPOUND POISSON MODEL
72
In principle, this solves the problem of evaluating (5.7), but some tedious (though elementary) calculations remain to bring the expressions on a final form. Noting that SIG(L)II = 1 because of (5.4), we get
L
00 (1 - e- -")G + (dx ) = 1 -
J0
00
3B(x) dx = 1-p.
Using (5.8) yields +L)
J0
xel'B ( x) dx
(5.10)
where VW =
JI c* e° (x) dx = a (B[a] - 1)
(5.11)
so that I 7B ['Y]-(B[7]-1) BI [7]-Q VP (7) 72 7 (using (5.1)) and
7µ+L) = 'y/3 [7] 7 1/0 = /3B ['y] - 1 .
(5.12)
Example 5 .4 Consider first the exponential case b(x) = Se-ax. Then 0(u) = pe-(a_Q)u where p = /3/S. From this it follows, of course, that 7 = S -,3 (this was found already in Example 5.1 above) and that C = p. A direct proof of C = p is of course easy: B ['y]
C
d S S (S-7 )2 d7S --y
S 02'
1-p 1-p _ 1-p /3B' [7] 2 - 1 = - p. ^7- -1 P-1
The accuracy of Lundberg's inequality in the exponential case thus depends on how close p is to one, or equivalently of how close the safety loading 77 is to zero. ❑
5. LUNDBERG CONJUGATION
73
Remark 5.5 Noting that PL - 1 = ,3LIBL - 1 = #ci (0 ) = k (ry) _ ,QB' ['Y] - 1 , we can rewrite the Cramer-Lundberg constant C in the nice symmetrical form G, _'(0)1 - 1 - p K'(7) PL-1
(5.13)
In Chapter IV, we shall need the following result which follows by a variant of the calculations in the proof of Theorem 5.3: 1 - aB[ry - a] - 1 Lemma 5 . 6 For a # ry, ELe-a^ (°°) = 7 aK'(7) 7 - a Proof Replacing 7 by a in (5.7) and using ( 5.8), we obtain 1 (I 1 - ^ e('r-a) x,3 (x)dx) (L ) ELe-a^*) = a \\\ f
using integration by parts as in (3.6) in the last step . Inserting (5.12), the result follows. ❑ Notes and references The results of this section are classical, with Lundberg's inequality being given first in Lundberg [251] and the Cramer-Lundberg approximation in Cramer [91]. Therefore, extensions and generalizations are main topics in the area of ruin probabilities, and in particular numerous such results can be found later in this book; in particular, see Sections IV.4, V.3, VI.3, VI.6.
The mathematical approach we have taken is less standard in risk theory (some of the classical ones can be found in the next subsection). The techniques are basically standard ones from sequential analysis, see for example Wald [376] and Siegmund [346].
5a Alternative proofs For the sake of completeness, we shall here give some classical proofs, first one of Lundberg's inequality which is slightly longer but maybe also slightly more elementary:
74 CHAPTER III. THE COMPOUND POISSON MODEL Alternative proof of Lundberg 's inequality Let X the value of {St} just after the first claim , F(x) = P(X < x). Then , since X is the independent difference U - T between an interarrival time T and a claim U, ,3+ry F'[7} = Ee7 ( U-T) = Ee7U • Ee-7T = B['Y] a = 1' where the last equality follows from c(ry) = 1. Let 0( n) (u) denote the probability of ruin after at most n claims. Conditioning upon the value x of X and considering the cases x > u and x < u separately yields
,0(n +1) (u) = F(u) +
Ju
0 (n) (u - x) F(dx).
We claim that this implies /,(n) (u) < e 7u, which completes the proof since Vi(u) = limniw 1/J(n) (u). Indeed , this is obvious for n = 0 since 00)(u) = 0. Assuming it proved for n, we get „/, (n+1)(u) <
F(u) +
e-7(u-=) F(dx)
Ju
00
00
<
e-7u
e7x F(dx)
f
u
+ fu
e - 7(u -z) F(dx)
o0
= e- 7uE[ 'Y] = e -7u.
Of further proofs of Lundberg's inequality, we mention in particular the martingale approach, see II.1. Next consider the Cramer-Lundberg approximation. Here the most standard proof is via the renewal equation in Corollary 3.3 (however, as will be seen, the calculations needed to identify the constant C are precisely the same as above): Alternative proof of the Cramer-Lundberg's approximation Recall from Corollary
3.3 that (u) = )3
J OO B(x) dx + J U Vi(u - x)/3 (x) dx. u
0
Multiplying by e7u and letting Z(u) = e7" -O(u),
z(u) = e7u/
J
B(x)dx, F(dx) = e7x,QB(x)dx,
u
we can rewrite this as u Z(u) =
z(u)
f +
J
- x) • e7'/B(x) dx, e7(u-x ),Y' • l•(u 1
0
= z(u) +
J0 u Z(u - x)F(dx),
6. MORE ON THE ADJUSTMENT COEFFICIENT 75 i.e. Z = z+F*Z. Note that by (5.11) and the Lundberg equation, ry is precisely the correct exponent which will ensure that F is a proper distribution (IIFII = 1). It is then a matter of routine to verify the conditions of the key renewal theorem (Proposition A1.1) to conclude that Z (u) has the limit C = f z(x)dx/µF, so that it only remains to check that C reduces to the expression given above. However, µF is immediately seen to be the same as a+ calculated in (5.10), whereas
L
00
z(u) du =
f J
/3e7udu "o
J "o B(x) dx = J "o B(x)dx J y,0eludu u
0
0
B(x)^ (e7x - 1) dx = ^' (B[7] - 1) - As] [0 -µs] = l y P^
using the Lundberg equation and the calculations in (5.11). Easy calculus now gives (5.6). ❑
6 Further topics related to the adjustment coefficient 6a On the existence of y In order that the adjustment coefficient y exists, it is of course necessary that B is light-tailed in the sense of I.2a, i.e. that b[a] < oo for some a > 0. This excludes heavy-tailed distributions like the log-normal or Pareto, but may in many other cases not appear all that restrictive, and the following possibilities then occur: 1. B[a] < oo for all a < oo. 2. There exists a* < oo such that b[a] < oo for all a < a* and b[a] = 00 for all a > a*. 3. There exists a* < oo such that fl[a] < oo for all a < a* and b[a] = 00 for all a > a*. In particular , monotone convergence yields b[a] T oo as a T oo in case 1, and B[a] T oo as a f a* in case 2 (in exponential family theory , this is often referred to as the steep case). Thus the existence of y is automatic in cases 1 , 2; standard examples are distributions with finite support or tail satisfying B(x) = o(e-ax)
76 CHAPTER III. THE COMPOUND POISSON MODEL for all a in case 1, and phase-type or Gamma distributions in case 2. Case 3 may be felt to be rather atypical, but some non-pathological examples exist, for example the inverse Gaussian distribution (see Example 9.7 below for details). In case 3, y exists provided B[a*] > 1+a*/,3 and not otherwise, that is, dependent on whether 0 is larger or smaller than the threshold value a*/(B[a*] - 1). Notes and references Ruin probabilities in case 3 with y non-existent are studied, e.g., by Borovkov [73] p. 132 and Embrechts & Veraverbeeke [136]. To the present authors mind, this is a somewhat special situation and therefore not treated in this book.
6b Bounds and approximations for 'y Proposition 6.1 ry <
2(1 - aps) 2µs OMB PB)
Proof From U > 0 it follows that B[a] = Eea' > 1 + µsa + pB2)a2/2. Hence 1 = a(B[7] - 1) > Q (YPB +72µs)/2) = 3µs + OYµa2) 2 (6.1) 7 'Y from which the results immediately follows.
❑
The upper bound in Proposition 6.1 is also an approximation for small safety loadings (heavy traffic, cf. Section 7c): Proposition 6.2 Let B be fixed but assume that 0 = ,3(77) varies with the safety loading such that 0 = 1 Then as 77 .0, µB(1 +rl) 2) -Y = -Y(77) 277 PB Further, the Cramer-Lundberg constant satisfies C = C(r1) - 1. Proof Since O(u) -+ 1 as r7 , 0, it follows from Lundberg's inequality that y -* 0. Hence by Taylor expansion, the inequality in (6.1) is also an approximation so that OA-Y] - 1) N Q (711s + 72µB2) /2) = p + 3,,,(2) B 'y 7 2 2(1 - p) _ 271µB QPB PB)
6. MORE ON THE ADJUSTMENT COEFFICIENT 77 That C -4 1 easily follows from -y -4 0 and C = ELe-7V°O) (in the limit, b(oo) is distributed as the overshoot corresponding to q = 0 ). For an alternative analytic proof, note that C - 1-P = rlµB 73B' [7] - 1 B' [ry) - 1/0 711µB
77
7PBIPB
µB +7µB2 ) - µB(1 +77 ) 'l = 1. 277-q
- 77
13 Obviously, the approximation (6.2) is easier to calculate than -y itself. However, it needs to be used with caution say in Lundberg's inequality or the Cramer-Lundberg approximation, in particular when u is large.
6c A refinement of Lundberg 's inequality The following result gives a sharpening of Lundberg 's inequality (because obviously C+ < 1) as well as a supplementary lower bound: Theorem 6 .3 C_e-ryu < ,)(u) < C+ e-ryu where
= B(x) = C_ x>o f °° e7( Y-x)B(dy )' C+
B(x) xuo f 0 e'r( v-x)B(dy)
Proof Let H(dt, dx ) be the PL-distribution of the time -r(u) of ruin and the reserve u - S7(„)_ just before ruin . Given r(u) = t, u - ST (u)- = x, a claim occurs at time t and has distribution BL(dy)/BL(x), y > x. Hence ELe-7£(u)
J
0
f
e--Y(Y- x)
H(dt, dx)
o
°o
f H(dt, dx)
fX
00 f°° B(dy) x
BL dy BL(x)
L ^ H(dt, dx) f e7B( x)B(dy) Jo oc, < C+
J0o" 0J o" H(dt, dx) = C.
The upper bound then follows from ik(u) = e-7uELe-Vu), and the proof of the ❑ lower bound is similar.
78 CHAPTER III. THE COMPOUND POISSON MODEL Example 6.4 If B(x) = e-ax, then an explicit calculation shows easily that B(x) _ e-6X fz ° e7(Y-x)B(dy) f x' e(6-,6)(Y-x)8e-sydy = 5 = P. Hence C_ = C+ = p so that the bounds in Theorem 6.3 collapse and yield the exact expression pe-y" for O(u). ❑ The following concluding example illustrates a variety of the topics discussed above (though from a general point of view the calculations are deceivingly simple: typically, 7 and other quantities will have to be calculated numerically). Example 6.5 Assume as for (3.1) that /3 = 3 and b(x) = 2 .3e-3x + 2 .7e-7x, and recall that the ruin probability is 245e-u + 3e 5-su *(u) = 3 Since the dominant term is 24/35 • e-", it follows immediately that 7 = 1 and C = 24/35 = 0.686 (also, bounding a-S" by a-" confirms Lundberg's inequality). For a direct verification, note that the Lundberg equation is
7 = /3(B['Y]-1)
= 3\
2.337
+2.777-1
which after some elementary algebra leads to the cubic equation 273 - 1472 + 127 = 0 with roots 0, 1, 6. Thus indeed 7 = 1 (6 is not in the domain of convergence of B[7] and therefore excluded). Further, 181B = 1-3 2.3+2.71 =
1-P = B
[7] -
1
3
1
7
I
7'
_ 17
2 (3-a )2 + 2 (7 - a)2 «=7=1
36 '
2 1-p _ 7 _ 24 3.17-1 35* 36 For Theorem 6.3, note that the function QB[Y]-1
f°°{L 3e_3x+ -u
f. 0c, ex
• 7e-7x 1 dx
J
I -2 . 3e-3x + 2 . 7e-7x l dx l
J
3 + 3e-4u
9/2 + 7/2e-4u
7. VARIOUS APPROXIMATIONS FOR THE RUIN PROBABILITY 79 attains its minimum C_ = 2/3 = 0.667 for u = oo and its maximum C+ = 3/4 = 0.750 for u = 0, so that 0.667 < C < 0.750 in accordance with C = 0.686.
Notes and references Theorem 6.3 is from Taylor [360]. Closely related results are given in a queueing setting in Kingman [231], Ross [308] and Rossberg & Siegel [309]. Some further references on variants and extensions of Lundberg's inequality are Kaas & Govaaerts [217], Willmot [382], Dickson [114] and Kalashnikov [218], [220], all of which also go into aspects of the heavy-tailed case.
7 Various approximations for the ruin probabil-
ity 7a The Beekman-Bowers approximation The idea is to write i (u) as F(M > u), fit a gamma distribution with parameters A, 6 to the distribution of M by matching the two first moments and use the approximation
0(u)
f u
Sa r(A)
xa - le-ax dx.
According to Corollary 3.5, this means that A, 8 are given by A/S = a1, 2A/52 = a2 (2) PIB3) ^ZP(B)2 __ PPB a2 al 2(1 - P)PB 3(1 - P)µ8 + 2(1 - p)2' i.e. S = 2a1 /a2, A = 2a21/a2. Notes and references The approximation was introduced by Beekman [60], with the present version suggested by Bowers in the discussion of [60].
7b De Vylder's approximation Given a risk process with parameters ,(3, B, p = 1, the idea is to approximate the ruin probability with the one for a different process with exponential claims, say with rate parameter S, arrival intensity a and premium rate p. In order to make the processes look so much as possible alike, we make the first three cumulants match, which according to Proposition 1.1 means -p=AUB-1=P-1,
(2) 6^=
2N
=OP
,
/3,4)
.
80
CHAPTER III. THE COMPOUND POISSON MODEL
These three equations have solutions 9/3µB2)3 30µa2)2 3µa2) (3) P+ (3) ' 0 - (3)2 P PB 2µB 2µB Letting /3* = /3/P, p* _ ,3* /S, the approximating risk process has ruin probability z,b(u) = p*e- (b-A*)", cf. Proposition 1.1.3 and Corollary 3.1, and hence the ruin probability approximation is b(u) e-(b-Aln)u.
Notes and references The approximation (7.2) was suggested by De Vylder [109]. Though of course it is based upon purely empirical grounds, numerical evidence (e.g. Grandell [171] pp. 19-24, [174]) shows that it may produce surprisingly good results.
7c The heavy traffic approximation The term heavy traffic comes from queueing theory, but has an obvious interpretation also in risk theory: on the average, the premiums exceed only slightly the expected claims. That is, heavy traffic conditions mean that the safety loading q is positive but small, or equivalently that /3 is only slightly smaller than /3max = 1/µ8. Mathematically, we shall represent this situation with a limit where /3 T fl but B is fixed. Proposition 7.1 As /3 f Nmax, (/3max - /3)M converges in distribution to the 2a exponential distribution with rate S = B' Proof Note first that 1 - p = (/3max -0)µB. Letting Bo be the stationary excess life distribution, we have according to the Pollaczeck-Khinchine formula in the form (3.7) that
Ee$(Amex -/j)M _
1-p
_
1-p
1 - PBo [s (/3max - /3)] 1 - p + p { 1 1-p ti 1 - P - Ps(/3max - )3 )PBo
_
Eo [s (0max
-
,3 )1 }
1-p 1 - p - s(/3max - /3)PBo
µB
PB - 8µBo -
S-s'
where 6 = µB/µBo = 2µa/µB2)
❑
7. VARIOUS APPROXIMATIONS FOR THE RUIN PROBABILITY
81
Corollary 7.2 If ,Q T /3max, u -* oo in such a way that (3max - /3)u -* v, then P(u) -4 e-6„ Proof Write z'(u) as P((/3max - l3)M > (/3max - /3)u). These results suggest the approximation Vi(u) e-6(0_.--0)u.
It is worth noting that this is essentially the same as the approximation (2) z/i(u) Ce- ryu ;ze a-2unµB laB (7.4)
suggested by the Cramer -Lundberg approximation and Proposition 6.2. This follows since rl = 1/p - 1 1 - p, and hence
6()3max _'3) =
2µ2B 1 - p _ 2rl11B PB p,B AB )
However , obviously Corollary 7. 2 provides the better mathematical foundation. Notes and references Heavy traffic limit theory for queues goes back to Kingman [230]. The present situation of Poisson arrivals is somewhat more elementary to deal with than the renewal case (see e .g. [APQ] Ch. VIII). We return to heavy traffic from a different point of view (diffusion approximations) in Chapter IV and give further references there . In the setting of risk theory, the first results of heavy traffic type seem to be due to Hadwiger [184]. Numerical evidence shows that the fit of (7.3) is reasonable for g being say 10-20% and u being small or moderate, while the approximation may be far off for large u.
7d The light traffic approximation As for heavy traffic , the term light traffic comes from queueing theory, but has an obvious interpretation also in risk theory: on the average , the premiums are much larger than the expected claims . That is , light traffic conditions mean that the safety loading rl is positive and large , or equivalently that 0 is small compared to µB . Mathematically, we shall represent this situation with a limit where 3 10 but B is fixed. Of course, in risk theory heavy traffic is most often argued to be the typical case rather than light traffic . However , light traffic is of some interest as a complement to heavy traffic , as well as it is needed for the interpolation approximation to be studied in the next subsection.
82 CHAPTER III. THE COMPOUND POISSON MODEL Proposition 7.3 As ,(3 10, 0(u)
/3
J
B(x)dx = /3E[U - u; U > 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'), ^IE)
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(')
f
BM (y) dy < f 00 Bi2i (y) dy
x
x
for all x; an equivalent characterization is f f dB(') < f f dB (2) for any nondecreasing convex function f. In the literature on risk theory, most often the term stop-loss ordering is used instead of increasing convex ordering because for a given distribution B, one can interpret f x°° B(y) dy as the net stop-loss premium in a stop-loss or excess-of-loss reinsurance arrangement with retention limit x, cf. XI.6. Finally, we have the convex ordering. Bill is said to be convexly smaller than B(2) (in symbols, B(' <, B(2)) if f fdB(1) < f fdB(2) for any convex function f. Rather than measuring difference in size, this ordering measures difference in variability. In particular (consider the convex functions x and -x) the definition implies that B(1) and B(2) must have the same mean, whereas (consider x2) B(2) has the larger variance. Proposition 8.1 If B(') r(2)(u) probabilities, the proof is complete. ❑ Of course, Proposition 8.1 is quite weak, and a particular deficit is that we cannot compare the risks of claim size distributions with the same mean: if BM
8. COMPARISONS OF CLAIM SIZE DISTRIBUTIONS
85
Proof Since the means are equal, say to p, we have Bol) (x) f ' B(1) (y) dy < -' f' B(2) (y) dy = Bo2) (x)• µ I.e., Bo1) <_d Bo2) which implies the same order relation for all convolution powers. Hence by the Pollaczeck-Khinchine formula 00
,,(1) (,u) = (1 _ P) E /3npnBo( 1):n(u) n=1
=
< (1- p ) E /3"µ"Bo2)* n(u) _ V(2) (u) n=1
Corollary 8.3 If B(1) <, B(2), then /'(')(u) < 0(2)(u) for all u. A general picture that emerges from these results and numerical studies like in Example 8.6 below is that (in a rough formulation) increased variation in B increases the risk (assuming that we fix the mean). The problem is to specify what 'variation' means. A first attempt would of course be to identify 'variation' with variance. The heavy traffic approximation (7.4) certainly supports this view: noting that, with fixed mean, larger variance is paramount to larger second moment, it is seen that asymptotically in heavy traffic larger claim size variance leads to larger ruin probabilities. Corollary 8.3 provides another instance of this, and here is one more result of the same flavor: Corollary 8.4 Let D refer to the distribution degenerate at 'LB . Then V, (D) (u) < O(B) (U ) for all u.
Proof If f is convex, we have by Jensen 's inequality that E f (U) > f ( EU). This ❑ implies that D <, B, from which the result immediately follows. A partial converse to Proposition 8.2 is the following: Proposition 8.5 If '0(1)(u) < p(2) (U) for all u and a, then B(1) <, B(2). Proof Consider the light traffic approximation in Proposition 7.1.
❑
We finally give a numerical example illustrating how differences in the claim size distribution B may lead to very different ruin probabilities even if we fix the mean p = PB. Example 8.6 Fix /3 at 1/1.1 and µB at 1 so that the safety loading 11 is 10%, and consider the following claim size distributions: B1: the standard exponential distribution with density a-y;
CHAPTER III. THE COMPOUND POISSON MODEL
86
B2: the hyperexponential distribution with density 0.lA,e-'\1x + 0.9A2e-'2r where A, = 0.1358, A2 = 3.4142; B3: the Erlang distribution with density 4xe-2x; B4: the Pareto distribution with density 3/(1 + 2x)5/2. Let ua denote the a fractile of the ruin function, i.e. 1/)(u,,) = a, and consider a = 5%, 1%, 0.1%, 0.01%. One then obtains the following table:
U005 U0.0' U0.001 u0.000,
B, 32 50 75
B2 B3 B4 35 181 24 282 37 70 245 425 56
100
568
74
1100
(the table was produced using simulation and the numbers are therefore subject to statistical uncertainty). Note to make the figures comparable, all distributions have mean 1. In terms of variances o2, we have 0r3 = 2 < or2 = 1 < 02 = 10 < 04 = 00 so that in this sense B4 is the most variable. However, in comparison to B2 the effect on the ua does not show before a = 0.01%, which appears to be smaller than the range of interest in insurance risk (certainly not in queueing applications!), and this is presumably a consequence of a heavier tail rather than larger variance. For B1i B2, B3 the comparison is as expected from the intutition concerning the variability of these distributions, with the hyperexponential distribution being more variable than the exponential distribution and the Erlang distribution less. 11 Notes and references Further relevant references are Goovaerts et al. [166], van Heerwarden [189], Kluppelberg [234], Pellerey [287] and (for the convex ordering) Makowski [ 252]. We return to ordering of ruin probabilities in a special problem in VI.4.
9 Sensitivity estimates In a broad setting, sensitivity analysis (or pertubation analysis) deals with the calculation of the derivative (the gradient in higher dimensions) of a performance measure s(O) of a stochastic or deterministic system, the behaviour of which is governed by a parameter 9. A standard example from queueing theory is
9. SENSITIVITY ESTIMATES 87 a queueing network, with 0 the vector of service rates at different nodes and routing probabilities, and s(9) the expected sojourn time of a customer in the network. In the present setting, s(9) is of course the ruin probability t' = Vi(u) (with u fixed) and 0 a set of parameters determining the arrival rate 0, the premium rate p and the claim size distribution B. For example, we may be interested in a'/ap for assesing the effects of a small change in the premium, or we may be interested in aV)/0/3 as a measure of the uncertainty on '0 if 0 is only approximatively known, say estimated from data. Example 9.1 Consider the case of claims which are exponential with rate 8 (the premium rate is one). Then ib = Pe-(6-13)u, and hence a _ e-(6-0)u + u e-(6-0)u =
( -i-
+
which is of the order of magnitude uV,(u) for large u.
Assume for example that 8 is known, while /3 = j3 is an estimate, obtained say in the natural way as the empirical arrival rate Nt/t in [0, t]. Then if t is large , the distribution of %3 -0 is approximatively normal N(0„ Q/t). Thus, if = a e-(6-A)u, it follows that -' is approximatively normal N(0, a2/t), where Q2 =
fl
(
l2
1113 /
_
Ou2v)2.
In particular , the standard deviation on the normalized estimate ^/1' (the relative error ) is approximatively ,01/2u, i.e. increasing in u. Similar conclusions will be found below. ❑ Proposition 9.2 Consider a risk process { Rt} with a general premium rate p. Then
ao = 00 -Qa/,
ap
where the partial derivatives are evaluated at p = 1. Proof This is an easy time transformation argument in a similar way as in Proposition 1.1.3. Let R(P) = Rtli,.
Then the arrival rate /3(P) for { R(P) }
is )31p, and hence the effect of changing p from 1 to 1 + Ap corresponds to changing /3 to /3/(1 + Op) /3(1 - Ap). Thus at p = 1,
a0
as ao
19P - 19P a/ - -
80
a/3
0
CHAPTER III. THE COMPOUND POISSON MODEL
88
As a consequence, it suffices to fix the premium at p = 1 and consider only the effects of changing ,3 or/and B. In the case of the claim size distribution B, various parametric families of claim size distributions could be considered, but we shall concentrate on a special structure covering a number of important cases, namely that of a two-parameter exponential family of the form Bo,((dx ) = exp {Ox + (t(x) - w(O, ()} p(dx) , x > 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 ^ue'ri`i7i( 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°`U =
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, ()}
CHAPTER III. THE COMPOUND POISSON MODEL
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)
CHAPTER III. THE COMPOUND POISSON MODEL
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 <
E
<
ry.
lcT(a ) 4 /c(a).
Then r.(ry - e) < 0 < r.(ry + e)
and hence KT(7 - E) < 0 < kT(7 + E)
for all sufficiently large T . I.e., 7T E (-y - e, -y + E) eventually, and the truth of this for all e > 0 implies ryT a-t 'y. Now write KT(7T) -
where
ryT
kT(7)
ryT
is some point between
= 4T(7T)( 7T
(10.4)
-7),
and ry. If ryT E (7 -
6"Y
+
E),
we have
KT(7 - E) < 4T(7T) < 4T(7 + E).
By the law of large numbers, NT
BT [a]
=
1
E Uie°U'
a$'
EUe "u = B'[a].
NT i =1
Hence r.'T(a)
n'(a) for all a so that for all sufficiently large T K7 - E ) < 4T(7T) < (7 +0'
which implies 'T(ry4) a$' r.'(-y). Combining ( 10.4) and Lemma 10.2, it follows that 7T-7
KT(7T) - KT(7) kT(7) K'(7) ,c'(7)
N (0' T (2(7) / N (0, °7IT)
.
Theorem 10.1 can be used to obtain error bounds on the ruin probabilities when the parameters ,Q, 0 are estimated from data . To this end , first note that e-7TU
N (e-7U u2e-27Uo'2/T) 7
96
CHAPTER III. THE COMPOUND POISSON MODEL
Thus an asymptotic upper a confidence bound for a-7' (and hence by Lundberg's inequality for 0(u)) is e-"TU + f- ue-ryuU ";T
VIT where r7ry.T = 3TKT ( 21T)IKT (^T)2 is the empirical estimate of vy and fc, satisfies b(- f,,) = a (e.g., ft, = 1.96 if a = 2.5%). Notes and references Theorem 10.1 is from Grandell [170]. A major restriction of the approach is the condition B[2ry] < oo which may be quite restrictive. For example , if B is exponential with rate 8 so that ry = 8 -,Q, it means 2 (8 -,0) < 5, i.e. 6 < 2,3 or equivalently p > 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) ul 0
Only the compound Poisson case is treated; generalizations to other models are either discussed in the Notes and References or in relevant chapters.
The notation is essentially as in Chapter III. In particular, the premium rate is 1, the Poisson intensity is 0 and the claim size distribution is B with m.g.f. B[•] and mean AB. The safety loading is q = 1/p - 1 where p = 13µB. Unless otherwise stated, it is assumed that i > 0 and that the adjustment coefficient (Lundberg exponent) -y, defined as solution of c(ry) = 0 where ic(s) _ /3(B[s] - 1) - s, exists. Further let 'Yo be the unique point in (0, 'y) where c(a) attains it minimum value. See Fig. 0.1 (the role of ryy will be explained in Section 4b).
97
98
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
Figure 0.1 The claims surplus is {St}, the time of ruin is T(u) and ^(u) = ST(t&) - u is the overshoot.
1 Exponential claims Proposition 1.1 In the compound Poisson model with exponential claims with rate S and safety loading 77 > 0, the conditional mean and variance of the time to ruin are given by /3u+1
E[-r(u) I T (u) < oo] Var [T ( u)
I T( u) < oo]
(1.1)
J -)3 _
2/3Su+/3+S
(1.2)
(S-)3)3
Proof Let as in Example 111.5 . 1 FL, EL refer to the exponentially tilted process with arrival intensity S and exponential claims with rate /3 (thus , PL = 6/0 = 1/p > 1). By the likelihood identity III.(4.9), we have for k = 1, 2 that E [T(u)k; 7- (U) < 00]
= ELT(u)ke-'YS.(,.) = e-7u ELe-'Y^(u) ELT(U)k
= e-'Yu b ELT(u)k = O(u)ELT(u)k,
using that the overshoot l; (u) is exponential with rate 0 w.r.t. FL and independent of T(u). In particular, E[T(u) I T(u) < 00
] = ELT (U),
Var[T(u) I T(u) < 00] =
VarL T( U)
.
1. EXPONENTIAL CLAIMS For (1 . 1), we have by Wald's identity that (note that ELSt = t(pL - 1)) ELST(u) ELT(u)
(PL - 1)ELT(u), u + ELe(u) _
PL - 1
/3u + 1 u + 1 //3 = 6-/3 6/0-1
For (1.2), Wald's second moment identity yields 2 EL (Sr(u) - (PL - 1)T(u))2 = UL where = s;"(ry) = 26//32. Since Sr (u) and (PL - 1)T(u) are independent with QL the same mean , the 1.h.s. is V1rLSr( u) +VarL ((PL - 1)T(u)) = VarLe(u) + (PL - 1)2VarLT(u)
Ca
+ 2
1I - 12 VarLT(u).
Thus the l.h.s. of (1.2) is aLELT( u) - 1//32
(6/)3 -1)2
26(/3u + 1)/(6 - /3) - 1 (6-)3)2
which is the same as the r.h.s. 0 Proposition 1.2 In the compound Poisson model with exponential claims with rate 6 and safety loading rl > 0, the Laplace transform of the time to ruin is given by Ee-a7( u) = E [e-aT (u). T(u) < oo]
= e-Bu I 1 - I (1.3)
fora > r.(-yo) = 2V - /3 - 6, where
B = 0(a) =
+ (6-/3-a)2+4a6 2
and hence that the value of ic(yo) Proof It is readily checked that yo = 6 - V/ is as asserted. Let 0 > -yo be determined by ^c(0 ) = a. This means that /3(6/(6 - 0) - 1) - B = a, which leads to the quadratic 02 + (/3 - 6 + a)0 - 6a = 0 with solution 0 (the
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
100
sign of the square root is + because 0 > 0). But by the fundamental likelihood ratio identity ( Theorem 111.4.3) we have E [e-«T(u ); T(u) < oo] =
EB [exp {-aT(u ) - 9ST(u)
= e- OuEee -04(u) = e-e u
+T(u)!c(0)} ; T(u) < oo]
be BB+B
where we used that PB(T(u) < oo ) = 1 because 0 > ryo and hence E9S1 = K'(0) ❑ > 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)31/2e^e)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.
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
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
StM
St
NJ
Figure 2.1 In formulas, Stv^ _
St+v - S„ 0
Define M(v,t)= {Stv) < SM,0<w
as the event that
1 - (0, T)
IS,(, ")
} is at a minimum at time t. Then
= P(Tr(0) > T) = P(M(0,T)) 1 fT P(M(v,T))dv E^T I(M(v,T))dv, T T o
where the second equality follows from II.(6.3) with A = (0, co ), and the third from the obvious fact (exchangeability properties of the Poisson process) that has the same distribution as St = { Si0)} so that P(M(v,T)) does not {Stv)} depend on v.
2. THE RUIN PROBABILITY WITH NO INITIAL RESERVE
105
Now consider the evaluation of fo I(M(v, T)) dv. Obviously, this integral is 0 if STv) . ST > 0. If ST < 0, there exist v such that M(v,T) occurs. For example, letting w = inf It > 0 : St_ = mino<w
{ST<St+ v-S,,, 0
T TE f I( M(v, T)) dv = TEST = T fP(ST < -x) dx T
T
NT
1 f P(ST < -x) dx = 1 f P Ui T - xdx. T T o i =1 Let f (•, t) denote the density of F(•, t). T
Theorem 2 .2 1-0(u,T) = F(u+T,T)-f(I -z /)(0,T-t))f(u+t,t)dt.
Proof The event {ST < u} = { Ei T Ui < u + T j can occur in two ways: either ruin does not occur in [0, T], or it occurs, in which case there is a last time o where St downcrosses level u, cf. Fig 2.2.
106
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
u
II T
Q
Figure 2.2 Here o, E [t, t + dt] occurs if and only if St E [u, u + dt] and there is no upcrossing of level u after time t, which occurs w.p. O(T - t). Hence
P(ST
J0 T (1-V,(0,T-t))P(StE[u,u+dt]),
which is the same as the assertion of the theorem.
❑
The following representation of T(0) will be used in the next section. The proof is combined with the proof of Theorem 111.2.2. Proposition 2.3 Define r_ (z) = inf It > 0 : St = -z}, z > 0. Let Z be a r.v. which is independent of St and has the stationary excess distribution B0. Then P(T(0) E • I T(0) < oo) = P(T_ (Z) E •). Proof of Theorem 111.2 . 2. For a fixed T > 0, define St = ST - ST_ t_ and let
A(z,T) =
{St < 0, 0 < t < T, ST_ _ -z} ,
C(z,T) =
{St > - z, 0 < t
C*(z,T) =
{S t > -z, 0 < t
2. THE RUIN PROBABILITY WITH NO INITIAL RESERVE
107
Then
P(r(0) E [T,T + dT], -ST(o)_ E [z, z + dz]) = P(A(z,T))f3B(z) dz dT. (2.1)
T
-z
-------------- -------
Figure 2.3 But by sample path inspection (cf. Fig. 2.3), A(z,T) = C*(z,T), and since {St}o
= 3R(z) dz JP(C(z,T))dT = Off(z) dz P(T_ (z ) < oo) = 3B(z) dz. Thus P(-Sr(o)_>x,ST(o) >y;T(0)
f
F(U > y + z U > z) P(Sr(o)_ E [z, z + dz], T(0) < oo) B(y
x
+ z) f3-B(z) dz = 3
B(z)
f °^
B(y + z) dz = f3
x
+
f
B(z) dz,
v
which is the assertion of Theorem 111.2.2.
❑
Proof of Proposition 2.3. It follows by division by P(ST(o)_ E [z, z + dz], T(0) < oo) = OR(z) dz in (2.1) that
P(T(0) E [T, T + dT] I S7(o)_ E [z, z + dz], 7-( 0) < oo) = P (C(z)) dT.
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
108 Hence
]P(7-(0) E [T,T + dT] T(0) < oo) dT f ' P(C(z))P(Sr( o)_ E [z, z + dz], T(0) < oo) 0 = dT f 0
=
P(C(z))P(Z E [z, z + dz], T(0) < oo)
dTP(T_(Z) E [T,T+dT]).
Notes and references For Theorems 2.1, 2.2, see in addition to Prabhu [293] also Seal [326], [329]. Theorem 2.1 and the present proof is in the spirit of Ballot theorems, cf. Tak'ecs [359]; a martingale proof is in Delbaen & Haezendonck [103]. Proposition 2.3 was noted by Asmussen & Kl(ippelberg [36], who instead of the present direct proof gave two arguments, one based upon a result of Asmussen & Schmidt [49] generalizing Theorem 11.6.5 and one upon excursion theory for Markov processes (see IX.5a). In the setting of general Levy processes, some relevant references are Shtatland [338] and Gusak & Korolyuk [181].
3 Laplace transforms Throughout in this section, r(a) denotes the solution < 'Yo of the equation -a = ic(r (a)) = ,(3(B[r( a)] - 1) - r(a),
(3.1)
where -a > r.(-yo). Let T_ (y) be defined as Proposition 2.3. Note that T_ (y) < oo a.s. because of77>0. Lemma 3.1 Eear-( y) = eyr(a).
Proof Optional stopping of the martingale I er (a) 9 -t,c(r(a)) l =
l er( a)se+at }
❑
yields 1 = e-yr(a)Eear-(y). I L Let ga(x) be the density of the measure E[ear(°); r(0) < oo, ^(0) E dx] (recall that ^(0) = Sr(o)) and write ga[b] = f OD ebxga(x) dx. Lemma 3 .2 ga(x) = Qe-xr(a) f "o eyr(a)B(dy) x
3. LAPLACE TRANSFORMS
109
Proof Let Z be the surplus - ST(o)_ just before ruin . Then by Proposition 2.3, E[ear (o) I T(0) < oo , Z = y] = EeaT- (v) = ev''(a). Further by Theorem 111.2.2
P(Z E [y, y + dy], £(0) E dx) = /3B(x + dy) dx and hence ga(x)
= f
e r)/3B(x + dy) _ /3 f
e(v- x)(a)
B(dy)•
x
Lemma 3 .3 ga[b] = c(b)
+ b + a - r(a) b - r(a)
Proof oo Q f
ex(b-r(a))dx f00eyr(a)B(dy)
0
x
Q f evraB(dy) e-(a))dx 0 Q cc ev(b-r (a)) - 1] evr(a)B(dy)[ b - r(a) = ab [B[b] - r(a) -B[r(a)]] . The result now follows by inserting /3B[s] = ic( s) +/3+ s and ic(r(a)) =-a.
❑
Corollary 3.4 E[eaT (o); rr(0) < oo) = 1_ r(a) Proof Let b = 0.
❑
Here is a classical result : the double m.g.f. (Laplace transform) of the ruin time T(u):
Corollary 3.5
eb"E[eaT(" ); T(u) < oo] du = fo 00
-a/r(a) - ic(b)/b x(b) + a
Proof Define Za(u) = E [eaT(" ); r(u) < oo). It is then easily seen that Za(u) is the solution of the renewal equation Za (u) = za (u) + fo Z. (u - x)ga (x) dx where za(u) = f,°° ga(x)dx. Hence
0 TO
eb"du E[eaT(");T(0) < oo] = 20[b] = za[b] (9a[b] -9a[0])/b 1 - ga [b] 1 - ga [b]
Using Lemma 3.3, the result follows after simple algebra.
❑
110
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
4 When does ruin occur? For the general compound Poisson model, the known results are even less explicit than for the exponential claims case, and take basically the form of approximations and inequalities. The first main result of the present section is that the value umL, where _
1
_
1
6B'[7J -1
ML w(ry)
-
1
-
C
,3LELU -1 1-p'
is in some appropriate sense critical as the most 'likely' time of ruin (here C is the Cramer-Lundberg constant). Later results then deal with more precise and refined versions of this statement. Theorem 4 .1 Assume 77 > 0. Then given r(u) < 00, T(u)/u mL as u -+ oo. That is, for any c > 0
P(
T(u)
u
- mL
> E
T(u) < 00 )
-40.
(4.1)
Further, for any m i,h(u, mu )
( 0 m < ML '(u) 1 m > rL.
For the proof, we need the following auxiliary result: Proposition 4.2 Assume ri < 0, i.e. P = /3µB > 1. Then as u -* oo, T(u) a.. u
1 ET(u) 1 p-1
7-(u) - mu D 2
-4 N(0,w )
u where
Pw2 = 311B)m3•
v/,UProof The assumption 11 < 0 ensures that P(T(u) < oo) = 1 and r(u) a4' oo. By Proposition 111.1.2, St/t 1 1/m, and hence a.s. t
T(u)
T(u)
t m = lim = lim = lim U-tioo u + Sr(u) u-+oo S, (u) t- 00 St
T(u)
= lim , uoo u
using e,(u ) = o(u) a.s., cf. Proposition A1.6. This proves the first assertion of (4.3). For the second , note that by Wald's identity
u + EC(u) = ES.r(u) = Er(u) • ES, = (p - 1)Er(u)
4. WHEN DOES RUIN OCCUR?
111
and that Ee(u)/u -a 0, cf. again Proposition A1.6. For (4 . 4), note first that ( Proposition 111.1.5) St - t/m D (2)
N (o, apB ) .
,^
According to Anscombe' s theorem (e.g. Theorem 7.3.2 of [86]) and (4.3), the same conclusion holds with t replaced by r(u). If Z - N(0,1), this can be rewritten as u + 1(u) -,r(u)/m T(u) ti
µB2) Z,
implying
T(u) - mu -m ,6µB2) Z v m (3µB2) Z, Tu)
T( u)
- mu
(2) '• m3/2 µB 6 7 - -7
-
11 Proof of Theorem 4.1 The l.h.s. of (4.1) is T (u) -
U
mL
I
> E, T(u) < oo f
P( T (u) <
/
00)
e-7uE L [e_7 (t1); 1'r(U) - mL U
>
E, T (u) < 00
J
0(u) e-7'PL
U u) - mL >E \I T
By Proposition 4.2, PL (•)-+ 0, proving (4.1), and (4.2) follows immediately from ❑ (4.1). Notes and references Theorem 4.1 is standard, though it is not easy to attribute priority to any particular author. Thus, the result comes out not only by the present direct proof but also from any of the results in the following subsections.
4a Segerdahl's normal approximation We shall now prove a classical result due to Segerdahl, which may be viewed both as a refinement of Theorem 4.1 (by considering 0(u, T) for T which are close to the critical value umL), and as a time-dependent version of the CramerLundberg approximation.
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
112
Corollary 4.3 (SEGERDAHL [333]) Let C be the Cramer-Lundberg constant and define wL = f3LELU2mL = f3B"[ry]mL where ML = 1/(pL-1) = 1/($B'[ry]1). Then for any y, e'°'/b (u,um,L+YWLV'U) - a C4'(y )•
( 4.5)
For the proof, we need the following auxiliary result: Proposition 4.4 (SIAM'S LEMMA) If 71 < 0, then e(u) and r(u) are asymptotically independent in the sense that, letting Z be a N(0,w2) r.v. with w2 as in (4.4), one has 9 (r(u)_rnu) Ef (^(u))
-* E.f ( (oo)) . E9(Z) (4.6)
whenever f, g are continuous and bounded on [0, oo ), resp . (-oo, oo). Proof Define u' = u - ul/4. Then the distribution of T(u) - T(u') given F,(u,) is readily seen to be degenerate at zero if ST(u•) > 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) <
'5(u) - z/)(u , y u) <
e -7v" , e-7v"
,
y <
^'(7)
y > k'(7) .
(4 .9) ( 4 . 10 )
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
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 ) = <
e-ayuEay [e-ay^ ( u)+T(U)K ( ay); yu < T (u) < oo
e-
ayuEav [eT ( u)K(ay); yu
1
l
11 < T(u) < oo
j
< e-ayu +Y UK(ay)
Remark 4.6 It may appear that the proof uses considerably less information on ay than is inherent in the definition (4.8). However, the point is that we want to select an a which produces the largest possible exponent in the inequalities. From the proof it is seen that this amounts to that a should maximize a-yic(a). ❑ Differentiating w.r.t. a, we arrive at the expression in (4.8). In view of Theorem 4.5, yy is sometimes called the time-dependent Lundberg exponent. An easy combination with the proof of Theorem 111.6.3 yields easily the following sharpening of (4.9): Proposition 4.7 i,b (u, yu) < C+(ay)e-7a„ where l C+(ay) = sup
00 eayR(xy)B(
.
f
dy)
Notes and references Theorem 4 . 5 is due to Gerber [156 ], who used a martingale argument. For a different proof, see Martin-LM [257] . Numerical comparisons are in Grandell [172 ]; the bound a-7y° turns out to be rather crude , which may be understood from Theorem 4.8 below , which shows that the correct rate of decay of tp(u, yu) is e -'Yyu/ .v"U-. f
Some urther discussion is given in XI.2, and generalizations to more general models are given in Chapter VI. Hoglund [203] treats the renewal case.
4. WHEN DOES RUIN OCCUR?
115
4c Arfwedson's saddlepoint approximation Our next objective is to strengthen the time-dependent Lundberg inequalities to approximations. As a motivation, it is instructive to reinspect the choice of the change of measure in the proof, i.e. the choice of ay. For any a > 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)
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
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
W Sz2
O
to
lP
4
sup W-i(t)• O
Since the r.h.s. has a continuous distribution, this implies
P
sup
12
0
Stu2 /µ2 > u
-4 P
(
sup W_1( t) > u O
But the l.h.s. is 1/ip (ua2 /IpI,Ta2 /p2), and the r.h.s. is IG(T; -1; u).
❑
For practical purposes , Corollary 5 .2 suggests the approximation 0(u,T) IG(Tp2/ a2); ulpI/a2). (5.5) Note that letting T --* oo in ( 5.5), we obtain formally the approximation V,(u) ti IG(oo; ulpl /a2) = e-2"1µl / or2.
(5.6)
This is the same as the heavy -traffic approximation derived in III.7c. However, since ti(u) has infinite horizon , the continuity argument above does not generalize immediately, and in fact some additional arguments are needed to justify (5.6) from Theorem 5.1. Because of the direct argument in Chapter III, we omit the details ; see Grandell [ 168], [169] or [APQ] pp. 196, 199.
120
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
Checks of the numerical fits of (5.5) and (5.6) are presented, e.g., in Asmussen [12]. The picture which emerges is that the approximations are not terribly precise, in particular for large u. In view of the excellent fit of the CramerLundberg approximation, (5.6) therefore does not appear to of much practical relevance for the compound Poisson model. However, for more general models it may be easier to generalize the diffusion approximation than the CramerLundberg approximation; as an example of such a generalization we mention the paper [129] by Emanuel et al. on the premium rule involving interest. In contrast, the simplicity of (5.5) combined with the fact that finite horizon ruin probabilities are so hard to deal with even for the compound Poisson model makes this approximation more appealing. However, in the next subsection we shall derive a refinement of (5.5) for the compound Poisson model which does not require much more computation, and which is much more precise. We conclude this section by giving a more general triangular array version of Theorem 5.1. The proof is a straightforward combination of the proof of Theorem 5.1 and Section VIII.6 of [APQ].
Theorem 5.3 Consider a family {Ste) } oc claim surplus processes indexed by a parameter 9, such that the Poisson rate Oe, the claim size distribution B9 and the premium rate p9 depends on 0. Assume further that 039µB6 < pe, that D
00 -4090, B0 * Boo, pt? -4 peo, pe - 00µB6 -+ 0, as 0 -* 00 and that the U2 are uniformly integrable w.r.t. the B9. Then as 0 _+ 90, we have ^A, 0) { 2 StQ2 /µ2
D { W_ i(t)}t>o t>o
2 where p = pe = pe - Po = 09µB6 - Pe, a2 = ae = 00µa6
Notes and references Diffusion approximations of random walks via Donsker's theorem is a classical topic of probability theory. See for example Billingsley [64]. The first application in risk theory is Iglehart [207], and two further standard references in the area are Grandell [168], [169]. All material of this section can be found in these references. For claims with infinite variance, Furrer, Michna & Weron [152] suggested an approximation by a stable Levy process rather than a Brownian motion. Further relevant references in this direction are Furrer [151] and Boxma & Cohen [75].
6. CORRECTED DIFFUSION APPROXIMATIONS
121
6 Corrected diffusion approximations The idea behind the simple diffusion approximation is to replace the risk process by a Brownian motion (by fitting the two first moments ) and use the Brownian first passage probabilities as approximation for the ruin probabilities. Since Brownian motion is skip -free, this idea ignores (among other things) the presence of the overshoot e(u), which we have seen to play an important role for example for the Cramer-Lundberg approximation . The objective of the corrected diffusion approximation is to take this and other deficits into consideration. The set-up is the exponential family of compound risk processes with parameters (3B, B9 constructed in III.4. However , whereas there we let the given risk process with safety loading 77 > 0 correspond to 9 = 0 , it is more convenient here to use some value 9o < 0 and let 9 = 0 correspond to n = 0 (zero drift); this is because in the regime of the diffusion approximation , 77 is close to zero, and we want to consider the limit 77 10 corresponding to Oo f 0. In terms of the given risk process with Poisson intensity ,6, claim size distribution B , ,c(s) = ,Q (B[s] - 1) - s and p = /3µB < 1, 77 = 1/p - 1 > 0, this means the following:
1. Determine yo > 0 by r.'(-yo) = 0 and let 90 = -'Yo. 2. Let PO refer to the risk process with parameters e-9oz
Qo = QB[-90], Bo(dx) = B[-eo]B(dx). Then EOU' = Boki[0] = Biki[-eo]/E[-9o] and "(s) = k(s-Bo)-k(-9o), 0(0) = 0. 3. For each 9, let P9 refer to the risk process with parameters
Q9
=
QoB0[9]
=
QB[9 -9o],
e9z keo)z
B9(dx) =Bale] Bo(dx)
= B[9
- 90] B(dx).
Then r.9(s) = Ico ( s + 9) - ao (0) _ /c(s + 9 - 90) - /c(9 - 90) and the given risk process corresponds to Poo where 90 = -'yo. In this set-up, P9(r (u) < oo) = 1 for 9 > 0, PB('r(u ) < oo) < 1 for 9 < 0, and we are studying b(u,T) = Peo(-r(u) < T) for 90 < 0, 9o T 0.
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
122
Recall that IG(x; C ; u) denotes the distribution function of the passage time of Brownian motion {W((t)} with unit variance and drift C from level 0 to level u > 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).
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
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
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
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
CHAPTER IV. PROBABILITY OF RUIN IN FINITE TIME
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}o o is just an independent copy of {St}t>o). Theorem 7 .1 Let {F(u)}u>0 be any family of events with F(u) E F, (u) and satifying PL(F(u)) -* 1, u -* oo. Then also P(u)(F(u)) -+ 1. Proof
P(u) (F(u)c) = F(flu)c; r(u) < oo) - EL[e-7S- (u); F(u)c] P(r(u) < oo)
?P(U)
< EL[e-7u; F(u)c] ti e-' ru]PL (F(u)`) --> 0. - vi(u) Ce-'Yu
Corollary 7.2 If B is exponential, then P(u) and FL coincide on
.FT(u)-
= o' (T(u
),
{St}0< t
Proof Write e-'rsr(u ) = e-'rue-'r£(u). In the exponential case, .F,(u)_ and ^(u) are independent , so in the in the proof, the numerator becomes e-'ruELe-7^ (u)PL(F( u)t) = e-7uCFL (F(u)°) when F(u) E .F,(u)_ and similarly the denominator is exactly equal to Ce-7u. Note that basically the difference between FT(u) and .T,(u)_ is that i;(u) is not .TT(u) _-measurable. In fact, ^(u) is exponential with rate 8 w.r.t. P(u) and rate = aL w.r.t. FL As example, we give a typical application of Theorem 7.1, stating roughly that under F(u), the Poisson rate changes from ,3 to ,3L and the claim size distribution from B to BL. Recall that 13L = (3B[ry] and BL(dx) = e'rxB(dx)/B[7], and let M(u) be the index of the claim leading to ruin (thus T(u) = Ti + T2 + ... + TMOO ).
7. HOW DOES RUIN OCCUR?
129
Corollary 7.3 M(u)
pcu)
M(u) >2 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.
This page is intentionally left blank
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
the number of arrivals before t, and M is the maximum of {St}, r(u) the time to ruin. The ruin probability corresponding to the zero-delayed case is denoted by 1/'(u), the one corresponding to the stationary case by 00)(u), and the one corresponding to T1 = s by 1/i8 (u). Proposition 1.1 Define p = !µB. Then no matter the distribution Al of T1i AA lim St = lim
t-*oo
ESt
t t-ioo t
= p - 1,
Var(St) = 11Ba2A + I4AaB lim t goo t PA
131
CHAPTER V. RENEWAL ARRIVALS
132 Furthermore for any a > 0,
lim E [St+a - St] = a(p - 1). t 4oo
Proof Obviously, Nt
UI Nt -t = ENt•pB - t.
ESt = E E
However , by the elementary renewal theorem (cf. A.1) ENt/t -+ 1/µA. From this ( 1.1) follows , and (1 . 3) follows similarly by Blackwell 's renewal theorem, stating that E[Nt+a - Nt] -* a/PA. For (1 . 2), we get similarly by using known facts about ENt and Var Nt that Nt Var(St) =
Nt
U; Nt + EVar U;
Var E
Nt
= Var(PBNt) + E(4Nt) Q2 2 0`2A tpB B + o(t). s + t
µA
PA
0 Of course, Proposition 1.1 gives the desired interpretation of the constant p as the expected claims per unit time. Thus, the definition 77 = 1/p - 1 of the safety loading appears reasonable here as well. The renewal model is often referedd to as the Sparre Andersen process, after E. Sparre Andersen whose 1959 paper [7] was the first to treat renewal assumptions in risk theory in more depth. The simplest case is of course the Poisson case where A and Al are both exponential with rate 0. This has a direct physical interpretation (a large portfolio with claims arising with small rates and independently). Here are two special cases of the renewal model with a similar direct interpretation: Example 1.2 (DETERMINISTIC ARRIVALS) If A is degenerate, say at a, one could imagine that the claims are recorded only at discrete epochs (say each week or month) and thus each U,a is really the accumulated claims over a period ❑ of length a. Example 1 .3 (SWITCHED POISSON ARRIVALS) Assume that the process has a random environment with two states ON, OFF, such that no arrivals occur in the off state, but the arrival rate in the ON state is ,0 > 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
❑
For later use, we note that the ruin probabilities for the delayed case T1 = s can be expressed as in terms of the ones for the zero-delayed case as u+8
z/i8(u) = B(u + s) + '( u + s - y)B(dy). (1.4)
fo
Indeed, the first term represents the probability F(U1 - s > u) of ruin at the time s of the first claim whereas the second is P(r(u) < oo, U1 - s < u), as follows easily by noting that the evolution of the risk process after time s is that of a renewal risk model with initial reserve U1 - s. For the stationary case, integrate (1.4) w.r.t. Ao.
CHAPTER V. RENEWAL ARRIVALS
134
2 Exponential claims. The compound Poisson model with negative claims We first consider a variant of the compound Poisson model obtained essentially by sign-reversion . That is , the claims and the premium rate are negative so that the risk reserve process , resp . the claim surplus process are given by Nt
Nt
Rt = u+^U; -t, St = t - Ut, b=1
!=1
where {Nt } is a Poisson process with rate ,3* (say ) and the U, are independent of {Nt} and i. i.d. with common distribution B* (say) concentrated on (0, 00). A typical sample path of {Rt } is illustrated in Fig. 2.1.
U
Figure 2.1 r* (u) One situation where this model is argued to be relevant is life annuities. The initial reserve is obtained by pre-payments from the policy holders, each of which receive a payment at constant rate during the lifetime . At the time of death , the remaining part of the pre-payment (if any ) is made available to the company. Using Lundberg conjugation , we shall be able to compute the ruin probabilities i(i* (u) for this model very quickly (,0* (u) = P (rr* (u) < oo) where rr* (u) = inf It > 0: Rt < 0} ) . A simple sample path comparison will then provide us with the ruin probabilities for the renewal model with exponential claim size distribution. Theorem 2 .1 If,3* pB. < 1, then 0 * (u) = 1 for all u > 0. If ,a*PB• > 1, then 0*(u) = e -'r" where ry > 0 is the unique solution of 0 = k*(-ry) = *(B*[-ry] - 1) +ry. (2.1)
135
2. EXPONENTIAL OR NEGATIVE CLAIMS [Note that r.* (a) = log Ee-'st I. Proof Define
St =u - Rt,
St=Rt-u=-St.
Then { St } is the claim surplus process of a standard compound Poisson risk process with parameters 0 *, B*. If I3*pB* < 1, then by Proposition 111.1.2 sup St = -inf St = 00 t>o t>o and hence -0* (u) = 1 follows. (a) is*(a)
(b) ,(a)
-7
Figure 2.2 Assume now ,3*,UB. > 1 . Then the function k* is defined on the whole of (-oo, 0) and has typically the shape on Fig. 2.2(a). Hence -y exists and is unique. Let
B(dx) = ^e-7x B*(dx), B* [-7] and let {St} be a compound Poisson risk process with parameters ,0, B. Then the c.g.f. of {St} is c(a) = is*(a-7), cf. Fig. 2.2(b), and the Lundberg conjugate of {St} is { St } and vice versa. Define T_ (u) = inf It > 0 : St = -u} , T_ (u) = inf { t > 0 : St = -u 'r* (u). Since ic'(0) < 0, the safety loading of { St} is > 0. Hence T_(u) < oo a.s., and thus 1 = P(T- (u ) < oo) = E {e-7sr_ (u); T_ ( u) < 001 e7"P(T_ (u) < oo) = e"V)* (u). 0 Now return to the renewal model.
CHAPTER V. RENEWAL ARRIVALS
136
Theorem 2 .2 If B is exponential, with rate S (say), and 5PA > 1, then ,)(u) _ 1r+e-7" where ry > 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
3. CHANGE OF MEASURE VIA EXPONENTIAL FAMILIES
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
CHAPTER V. RENEWAL ARRIVALS
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
h(Jt) = east - tK( a)e- ( a+r' (a))(Jt -s) h(s)
where c(a) is the solution of (3.2). Proposition 3.4 The probability measure Pa;s is the probability measure governing a renewal risk process with Jo = s and the interarrival distribution A and the service time distribution B changed to Aa, resp. Ba where
Aa (dt) - e-(«+k(a))t esy A(dt), Ba(dx) = -B(dx). A[-a - c(a)] B[a] Proof Pa;s(Jo = s) = 1 follows trivially from Lo = 1. Further, since JT, = J8 =
T2, Ease
AU1+6T2
[ AU1+6T2 = Ea a LT,
]
= E. [e1U1 + 6T2ea ( U1-s)-stc ( a)e-(a+K(a ))( T2-s)I
B[a +,13]A[b - a - rc(a)] = B
[a + /3] A[b - a - rc(a)] B[a]
A[-a - c(a)]
= Ba[13]Aa[5],
which shows that U1, T2 are independent with distributions Ba, resp. Aa as asserted . An easy extension of the argument shows that U1, . . . , U,,, T2, ... J n+1 ❑ are independent with distribution Aa for the Tk and Ba for the Uk. Remark 3 . 5 For the compound Poisson case where A is exponential with rate ,8, (3.2) means 1 = B[a]/3/(/3+a+rc (a)), i.e. rc(a) = 0 (B[a] - 1)-a in agreement ❑ with Chapter III. Note that the changed distributions of A and B are in general not the same for Pa;s and P(d). An important exception is, however, the determination of the adjustment coefficient ry where the defining equations rc(d) (ry) = 0 and rc(ry) = 0 are the same.
4. THE DUALITY WITH QUEUEING THEORY
141
The Markov additive point of view is relevant when studying problems which cannot be reduced to the imbedded random walk, say finite horizon ruin probabilities where the approach via the imbedded random walk yields results on the probability of ruin after N claims, not after time T. Using the Markov additive approach yields for example the following analogue of Theorem IV.4.5 Proposition 3.6 Let y < let ay > 0 be the solution of ic'(ay) = 1/y, and define yy = ay - yx(ay). Then "^ e-(ay+w(aY))8
Ys(u,yu )
e-7vu
A[-ay - rc( ay)]
=
e-(aa+-(-r ))sb[a ]e-7yu
L y1
In particular, for the zero-delayed case zp8(u, yu ) < e-7yu. Proof As in the proof of Theorem IV.4.5, it is easily seen that ic(ay ) > 0. Let M(u) be the number of claims leading to ruin . Then J(rr(u)) = TM(u)+1 and hence Ws(u, yu) =
F'ay;s e-aysr(")+r(u ) K(ay) h (s) ; T(u) < yu
h(JT(u)) < e-ayu+yuk(ay )
( Eia y Le-(a(+k(ay))s
e-(ay+w(ay))8 e - -y
j
v; - ay+ray))TM(,.)+1 e J
yuAa
y [ay +
K(ay)
,
which is the same as the asserted inequality for 0.(u, yu). The claim for the zero-delayed case follows by integration w.r.t. A(ds). ❑ Notes and references
The approach via the embedded random walk is standard, see e.g. [APQ] Ch. XII. For the approach via Markov additive processes, see in particular Dassios & Embrechts [98] and Asmussen & Rubinstein [45].
4 The duality with queueing theory We first review some basic facts about the GI/G/1 queue, defined as the single server queue with first in first out (FIFO; or FCFS = first come first served) queueing discipline and renewal interarrival times. Label the customers 1, 2, .. . and assume that T„ is the time between the arrivals of customers n - 1 and n, and U„ the service time of customer n. The actual waiting time Wn of customer n is defined as his time spent in queue (excluding the service time), that is, the time from he arrives to the queue till he starts service. The virtual waiting time Vt at time t is the residual amount of work at time t, that is, the amount of
142 CHAPTER V. RENEWAL ARRIVALS time the server will have to work until the system is empty provided no new customers arrive (for this reason often the term workload process is used) or, equivalently, the waiting time a customer would have if he arrived at time t. Thus, since customer n arrives at time on, we have Wn = Van-
(4.1)
(left limit). The traffic intensity of the queue is p = EU/ET.
The following result shows that {Wn} is a Lindley process in the sense of II.4: Proposition 4.1 Wn+1 = (Wn + U. - Tn)+ Proof The amount of residual work just before customer n arrives is VQ„ -. It then jumps to VQ„ - + Un, whereas in [On , an+1) = [on, on + Tn) the residual work decreases linearly until possibly zero is hit, in which case {V} remains at zero until time on+1. Thus Vos}1 _ = (Wn + Un - Tn)+, and combining with (4.1), the proposition follows. 0 Applying Theorem 11.4.1, we get:
Corollary 4.2
Let Mnd)
= maxk=o,...,n-1 (U1 +• • •+Uk -Tl - • • • Tk ). If W1 = 0,
then Wn v M. The next result summarizes the fundamental duality relations between the steady-state behaviour of the queue and the ruin probabilities (part (a) was essentially derived already in 11.4): Proposition 4.3 Assume rl > 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
143
4. THE DUALITY WITH QUEUEING THEORY
(T1,..., TN) with (TN,..., Ti) and similarly for the U,,. However, by an obvious reversibility argument this does not affect the distribution , and hence in particular ZT is distributed as the virtual waiting time just before the Nth arrival, i.e. as WN. It follows that P(WN > u) =,(N)(u) has the limit tp(u) for all u, which implies the convergence in distribution and (4.2). For part (b), we let T be deterministic . Then the arrivals of {Rt} in [0, T] form a stationary renewal process with interarrival distribution A, hence (since the residual lifetime at 0 and the age at T have the same distribution , cf. A.le) the same is true for the time-reversed point process which is the interarrival process for { Zt}o
P(s)(VT > u) = P(s)(r(u) < T), (4.4) Tlim F(s) (VT > u) = limo P(s) (r(u) < T) = '+^io) (u)• 0 It should be noted that this argument only establishes the convergence in
distribution subject to certain initial conditions, namely W1 = 0 in (a) and Vo = 0, T1 - Ao in (b). In fact , convergence in distribution hold for arbitrary initial conditions , but this requires some additional arguments (involving regeneration at 0 but not difficult) that we omit. Letting n oo in Corollary 4.2, we obtain: Corollary 4.4 The steady-state actual waiting time W has the same distribution as M(d). Corollary 4.5 (LINDLEY'S INTEGRAL EQUATION) Let F(x) = P(U1-T1 < x), K(x) = P(W < x). Then
K(x) =
J x00K(x - y)F(dy),
x > 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
CHAPTER V. RENEWAL ARRIVALS
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
0
and r(u) = inf It > 0: St > 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 ^ij = aid/pi, Qi = ,3i/pi. ❑ From now on, we assume again pi = 1 so that the claim surplus is Nt
St = ?Ui_t. 1
CHAPTER VI. MARKOVIAN ENVIRONMENT
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 a ., y Ni) i
Nti) t
a.. ^j
Nt
a'
7riNi,
t t iEE
Also, denoting the sizes of the claims arriving in state i by U(') 1 standard law of large numbers yields N
N 1: I(Ukik < x)
a$.
Bi(x),
U(')
the
N -+ oo
k=1
Hence 1
Nt
1
N`+)
Nits
Nt E I ( Ut <- x) = Nt E > I (Uk) X) Nt Bi(x) 1=1 iEE k=1 iEE 1: t5 Bi( x) = B*(x). iEE
13 A different interpretation of B* is as the Palm distribution of the claim size, cf. Example 11.6.4. The next result shows that we can think of the averaged compound Poisson risk model as the limit of the Markov-modulated one obtained by speeding up the Markov-modulation. Proposition 1.9 Consider a Markov-modulated risk process {St} with param-
eters
Ni,
Bi, A, and let
{St
°i} refer to the one with parameters
Pi,
Bi, aA,
{St} to the compound Poisson model with parameters 0 *, B*. Then {St-)} + {St*} in D[0, oo) as a -4 oo. In particular, zli( (u) ..
i,* (u) for all u.
Proof According to Proposition 1.7, the Fi-distribution of T1 in
{St(a
) } is phase-
type with representation (E, e;,aA - (/3i)aiag). By Proposition A5.2, this converges to the exponential distribution with rate 0* as a -* oo, and furthermore in the limit JT, has distribution (7ri)3i //3*)iEE and is independent of Ti. In particular, the limiting distribution of the first claim size U1 is B*. Conditioning
CHAPTER VI. MARKOVIAN ENVIRONMENT
150
upon FT, shows similarly that in the limit (T2, U2) are independent of .FT,, with T2 being exponential with rate ,l3* and U2 having distribution B*. Continuing in this manner shows that the limiting distribution of (T,,, U,,),,=1,2.... is as in {St }. From this the convergence in distribution follows by general facts on weak convergence in D[0, oo), which also yield O(a) (u, T) -+ ?P* (u; T) for all u and T. The fact that indeed 0(a) (U) -3 0* (u) follows, e.g., from Theorem 3.2.1 of [145]. 0 Example 1.10 Let
a -a ) \ a a
E_{1,2}, A= ( -
J
9 3 2 a1=2, B1=3E3+2E7,
5 5
132=2, B2=1E3+4E7, 5 5 where E5 denotes the exponential distribution with intensity parameter 5 and a > 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
CHAPTER VI. MARKOVIAN ENVIRONMENT
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 ^ifjµs; - 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)
CHAPTER VI. MARKOVIAN ENVIRONMENT
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 ^i3
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).
2. THE LADDER HEIGHT DISTRIBUTION
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
-
^O \ -T.
----------------------------
Figure 2.2
CHAPTER VI. MARKOVIAN ENVIRONMENT
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)
_ ^iF 7ri
(JJ =i,St EA,St <S*,u< t), (2.6)
2. THE LADDER HEIGHT DISTRIBUTION
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-.,0t)
= P.,,(Jt=j,Jo=i,StEA,S„<0,0
and this immediately yields (2.6).
It is convenient at this stage to rewrite the above results in terms of the matrix K = 0-'Q'A, where A is the diagonal matrix with 7r on the diagonal: Corollary 2.6 (a) R(dx) = e-Kxdx, x < 0;
(b) for z > 0, G+((z, oo)) = f o' eIXS((x + z, oo))dx; (c) the matrix K satisfies the non-linear matrix equation K = W(K) where W( K) = A -
fi
( i) diag +
J "O eKx S(dx); 0
+1) = cp (K( n)) defined by K(o) = A - (,Qi)diag, K( n (d) the sequence converges monotonically to K. ``
{K(n)}
[the W(•) here is of course not the same as in Proposition 2.4]. From Qe = 0, it is readily checked that 7r is a left eigenvector of K corresponding to the eigenvalue 0 (when p < 1), and we let k be the corresponding right eigenvector normalized by Irk = 1. Remark 2.7 It is instructive to see how Proposition 2.1 can be rederived using the more detailed form of G+ in Corollary 2.6(b): from 7rK = 0 we get 7rG+(dy)e = J W 7reKx(fiiBi(dy + x))diag dx • e
0
w(fiiB1(dy + x))col dx f 0 EirifiiBi(y)dy = fi*B*(y)dy• iEE 0
Though maybe Corollary 2.6 is hardly all that explicit in general, we shall see that nevertheless we have enough information to derive, e.g., the CramerLundberg approximation (Section 3), and to obtain a simple solution in the
158
CHAPTER VI. MARKOVIAN ENVIRONMENT
special case of phase-type claims (Chapter VIII). As preparation, we shall give at this place some simple consequences of Corollary 2.6. Lemma 2 .8 (I - IIG+II)e = (1 - p)k. Proof Using Corollary 2.6(b) with z = 0, we get
IIG+II = feIxsux, oo dx.
(2.7)
In particular, multiplying by K and integrating by parts yields 0
I)S(dx) KIIG+II = - (eKx T
= K - A + (,13i)diag -
Z
S(dx) = K -A.
2.8)
0 OO
Let L = (kir - K)-'. Then (k7r - K) k = k implies Lk = k. Now using (2.7), (2.8) and ireKx = ir, we get
kirIIG +IIe =
ao k f
0 KIIG+IIe = Ke,
7rS((x , oo))e = k (lri(3ips, ) rowe = pk,
(kir-K)(I - IIG+II)e = k-Ke-pk+Ke = ( 1-p)k. Multiplying by L to the left, the proof is complete.
❑
Here is an alternative algorithm to the iteration scheme in Corollary 2.6 for computing K. Let IAI denote the determinant of the matrix A and d the number of states in E. Proposition 2.9 The following assertions are equivalent: (a) all d eigenvalues of K are distinct; (b) there exist d distinct solutions 8 1 ,- .. , sd E {s E C : its < 0} of (A + (131(Bi[s] - 1))diag - sIl = 0. (2.9) I n that case , then Si, ... , sd are precisely the eigenvalues of K, and the corresponding left row eigenvectors al, ... , ad can be computed by
ai (A -
(fi(Bi[Si]
-
1))d iag - siI) = 0.
(2.10)
159
2. THE LADDER HEIGHT DISTRIBUTION Thus, al seal K=
(2.11)
ad sdad Proof Since K is similar to the subintensity matrix Q, all eigenvalues must indeed be in Is E C : 2s < 0}. Assume aK = sa. Then multiplying K = W(K) by a to the left, we get sa = a
(
- (f3i)diag +
eS(dx)
= a (A - (/3i) diag + (/3iEi[s])diag)
f A It follows that if (a) holds, then so does (b), and the eigenvalues and eigenvectors
can be computed as asserted. The proof that (b) implies (a) is more involved and omitted; see Asmussen ❑ [16]. In the computation of the Cramer-Lundberg constant C, we shall also need some formulas which are only valid if p > 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
3. CHANGE OF MEASURE VIA EXPONENTIAL FAMILIES
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
CHAPTER VI. MARKOVIAN ENVIRONMENT
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) - <
hi e--fu. min2EE h9
Assuming it has been shown that C = limo, 0 EL;i[e-7^(u)/hj,(„j exists and is independent of i (which is not too difficult, cf. the proof of Lemma 3.8 below), it also follows immediately that 0j(u) - hiCe-7u. However, the calculation of C is non-trivial. Recall the definition of G+, K, k from Section 2. Theorem 3 .7 (THE CRAMER-LUNDBERG APPROXIMATION) In the light-tailed case, 0j(u) - hiCe-7u, where
C (PL -1) "Lk.
(3.4)
To calculate C, we need two lemmas . For the first, recall the definition of (+, M+ in Lemma 2.10. Lemma 3 .8 As u -4 oo, (^(u), JT(u)) converges in distribution w.r.t. PL;i, with the density gj(x) (say) of the limit (e(oo), JT(,,,,)) at b(oo) = x, JT(oo) = j being independent of i and given by gi (x) = L 1 L E CL;'GL (e,.1; (x, oo)) S+M+e LEE
Proof We shall need to invoke the concept of semi-regeneration , see A.1f. Interpreting the ladder points as semi-regeneration points (the types being the environmental states in which they occur), {e(u),JJ(u))} is semi-regenerative with the first semi-regeneration point being (^(0), JT(o)) _ (S,+, J,+). The formula for gj (x) now follows immediately from Proposition A1.7, noting that the ❑ non-lattice property is obvious because all GL (j, j; •) have densities.
3. CHANGE OF MEASURE VIA EXPONENTIAL FAMILIES Lemma 3 .9 KL = 0-1K0 - ryI, G+[-ry] _
163
-111G+IIA, G+['y]h = h.
Proof Appealing to the occupation measure interpretation of K, cf. Corollary 2.6, we get for x < 0 that
fPs(StE dx,J =j,r > t)dt
ete-Kxej dx =
= hie-7x f O PL;i(St E dx, Jt = j, T+ > t) dt hj o
= ht e-7xe^e-K`xej dx, which is equivalent to the first statement of the lemma. The proof of the second is a similar but easier application of the basic likelihood ratio identity Proposition 3.2. In the same way we get G+['y] = AIIG+IIT-1, and since IIG+ IIe = e, it follows that
G +[ry l h
= oIIG+IIo -1h = AIIG+ IIe =
De
= h.
Proof of Theorem 3.7 Using Lemma 3.8, we get EL (e-'W- ); JT(.) = jl = f 00 e- 7xgj (x) dx L J o 1
°°
f e-7^G+( t, j; (x, oo)) dx S+M+e LEE °
-
-
1 (+;l f S +M +e LEE 0
0 1(1 - e-7 x ) G+(1,j; dx)
E(+(IIG+(e,j)II-G+[t,j;
1
rr ry S +M +e LEE
In matrix formulation, this means that
C =
E L;i
e-7f(-) - L
hj,r(_)
YC+M+e 'y(PL - 1)
L
ryC M e
c+
1
L
(IIG+II - G +[- 7]) 0-le
L (-ir KL) (I - G+[- y]) 0-le,
CHAPTER VI. MARKOVIAN ENVIRONMENT
164
using Lemma 2.10 for the two last equalities. Inserting first Lemma 3.9 and next Lemma 2.8, this becomes 1 7r LA -1(-YI - K)(I - IIG+II)e 'Y(PL - 1) = 1 P 7r LA -1(yI - K) k = 1-P 7rLO-1k. Y(PL - 1) (PL - 1 ) Thus, to complete the proof it only remains to check that irL = vL A. The normalization vLhL = 1 ensures vLOe = 1. Finally, VLOAL = vLAA-'K['Y]A = 0
❑
since by definition vLK[y] = k(y)vL = 0.
3b Ramifications of Lundberg 's inequality We consider first the time-dependent version of Lundberg 's inequality, cf. IV.4. The idea is as there to substitute T = yu in 'Pi (u, T) and to replace the Lundberg exponent y by yy = ay - yk(ay ), where ay is the unique solution of rc(ay)= 1 Y Graphically, the situation is just as in Fig. 0.1 of Chapter IV. Thus, one has always yy > y, whereas ay > -y, k( ay) > 0 when y < 1/k'(y), and ay < y, k(ay) < 0 when y > 1/k'(-y). 1
Theorem 3 .10 Let C+°) (y) _
Then
miniEE hiav)
Vi(u,yu) Pi(u) -
V,i(u,yu)
C+°)(y)hiav)
< C+)(y)hiar )e -'Yvu,
Proof Consider first the case y <
1
e-7vu,
y>
(y) (3.7)
Then, since k (ay) > 0, (3 .1) yields
'12(u,yu)
hiav)]E'iav,i
h(ay ) J*(u)
(3.6)
y< (y)
exp {-ayST(,L ) +r(u)k( ay)}; T(u) < yu
165
3. CHANGE OF MEASURE VIA EXPONENTIAL FAMILIES
hiav)e _avuE, av
exp {-e() + r(u))} ; r(u) yu
'i [h,
[e*(u)K(av); r(u) < yu] hiay)C+o)(y)e-avuEav;i h=av)C+o) (y)e-ayu+yuw(av). 1
Similarly, if y > 1lk'(ry), we have ic(ay) < 0 and get 'i(u) - V)i(u, yu) f
h(av)e
-avuE«v;i I (a)v exp {-aye(u) + r(u)r.(ay)}; yu < r(u) < 00 h 4(u)
< h(av)C+o)(y)e-avuEav ;i [eT(u)K(av ); yu < r(u) < 00] < hiav)C+o)( y)e-avu+yuw(av) 0 Note that the proof appears to use less information than is inherent in the definition (3.5). However, as in the classical case (3.5) will produce the maximal ryy for which the argument works. Our next objective is to improve upon the constant in front of a-7u in Lundberg's inequality as well as to supplement with a lower bound:
Theorem 3.11 Let Bj (x) C_ = min 1 • inf jEE hj x>o f2° e'r( v-x)Bj(dy) '
C+ _
mE
1 Bj(x) J Y -x)Bj (dy). hj P .00
su
e7(
(
3.8 )
Then for all i E E and all u > 0, C-hie -ryu < Vi(u ) < C+hie -7u.
(3.9)
For the proof, we shall need the matrices G+ and R of Section 2. We further write G(u) for the vector with ith component Gi(u) = EiEE G+(i,j; (u, oo)) and, for a vector
jEE
166
CHAPTER VI. MARKOVIAN ENVIRONMENT
Lemma 3 .12 Assume sup1,u IMP:°) (u) I < oo, and define W(n+1) (u) = G(u) + (G+ * tp(n))(u). Then cpin)(u) sit (u) as n -+ oo. Proof Write UN = EN G+ , U = U". = Eo G+ G. Then iterating the defining equation ip(n+1) = G + G+ * V(n) we get W(N+1) = UN * G + G+N+1) * ^(o) However, if r+ (n) is the nth ladder epoch, we have G *(N +1) * ^,(0) ] (u) < sup
Jt
t,u
Iv
2°)(u)I Pi(rr+(N + 1) < oo) --+ 0.
Hence lim cp(n) exists and equals U * G. _ To see that the ith component of U * G(u) equals ?Pi (u), just note that the recursion oo. Lemma 3 .13 For all i and u, 00 f C_ hj f e(Y)G+(i, j; dy) : 1(u) < C+ > 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)
3. CHANGE OF MEASURE VIA EXPONENTIAL FAMILIES
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<
Vi (u )
-
0i(u,
T) < C+(')' o)hi7u)e-7ou8T .
(3.12)
Proof We first note that just as in the proof of Theorem 3.11, it follows that Vi(u) < C_(yo)
h=70)e-7ou.
(3.13)
Hence, letting MT = maxo u) - Pi(MT > u) = Pi(MT < u,M > u) = Pi(STu) = Ei [VGJT (u - ST); MT < u, ST < u] < C+(yo)e-7ouEi [h^7o)e70ST1 l T J
=
C h(7o)e-7ou8T . +i
168
CHAPTER VI. MARKOVIAN ENVIRONMENT
Notes and references The results and proofs are from Asmussen and Rolski [44]. Further related discussion is given in Grigelionis [176], [177].
4 Comparisons with the compound Poisson model 4a Ordering of the ruin functions For two risk functions 0', ", we define the stochastic ordering by 0' < s.o. V)" if z/i'(u) <'c"" (u), u > 0. (4.1) Obviously, this correponds to the usual stochastic ordering of the maxima M', M" of the corresponding two claim surplus proceses (note that 0'(u) _ P(M' > 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.o. B2 <_s.o.... <s.o. Bp.
(4.3)
The Markov process {Jt} is stochastically monotone (4.4) To avoid trivialities, we also assume that there exist i # j such that either /3i <,33 or Bi 0 Bj. Occasionally we strengthen (4.3) to B = Bi does not depend on i.
(4.5)
Note that whereas (4.2) alone just amounts to an ordering of the states, this is not the case for (4.3). For the notion of monotone Markov processes, we refer to
4. COMPARISONS WITH THE COMPOUND POISSON MODEL
169
Stoyan [352], Section 4.2; note that (4.4) is automatic in some simple examples like birth-death processes or p = 2 . Conditions (4.2)-(4.4) say basically that if i < j , then j is the more risky state , and it is in fact easy to show that Vii(u ) < t/j(u) (this is used in the derivation of (4.9 ) below). Theorem 4 .1 Assume that conditions (4.2)-(4.4) hold. Then V,* For the proof, we need two lemmas. The first is a standard result going back to Chebycheff and appearing in a more general form in Esary, Proschan & Walkup [140], the second follows from an extension of Theorem I1.6.5 (cf. also Proposition 2.1) which with basically the same proof can be found in Asmussen & Schmidt [49]. Lemma 4 . 2 If al < ... < a,,, b1 < ... < bp and 7ri > 0 (i = 1, ... , p), ^i 7ri = 1, then P
P
P 7rjbj.
1:7riaibi > E 7riai i=1 i=1 j=1
The equality holds if and only if a1 = ... = aP or b1 = ... = b,,. Lemma 4 . 3 (a) P,r (JT(o) = i, 7-(0) < oo) = pirf+), where 7r2+) = QiµBilri/p, (b) P,r (Sr(o) E dx Jr(o) = i, T(0) < oo) = Bi(x) dx/tcai . Proof of Theorem 4.1. Conditioning upon the first ladder epoch, we obtain (cf. Proposition 2.1 for the first term in (4.7) and Lemma 4.3 for the second) *(u)
_
/3 *B* (u) +,13*
J0 u 0*(u - x)B*(x) dx, (4.6)
fl*B*(u) + p> 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
CHAPTER VI. MARKOVIAN ENVIRONMENT
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 /^ ^i/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
CHAPTER VI. MARKOVIAN ENVIRONMENT
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
CHAPTER VI. MARKOVIAN ENVIRONMENT
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
CHAPTER VI. MARKOVIAN ENVIRONMENT
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)
Jt = (s + t) mod 1
P(8) - a.s .. (6.4)
At a first sight this point of view may appear quite artificial, but it turns out to have obvious benefits in terms of guidelining the analysis of the model as a parallel of the analysis for the Markovian environment risk process. Notes and references The model has been studied in risk theory by, e.g., Daykin et.al. [101] , Dassios & Embrechts [98] and Asmussen & Rolski [43], [44] (the literature in the mathematical equivalent setting of queueing theory is somewhat more extensive, see the Notes to Section 7). The exposition of the present chapter is basically an extract from [44], with some variants in the proofs.
6b Lundberg conjugation Motivated by the discussion in Chapter II.5 (see in particular Remark 11.5.8), we start by deriving formulas giving the m.g.f. of the claim surplus process. To this end, let
f 8+1 tc *(a) _ (B* [a] - 1) -a =
J8
,3(v)(B(vl [a] - 1) dv - a
be the c.g.f. of the averaged compound Poisson model (the last expression is independent of s by periodicity), and define h(s;a)
= exp { - ^8 [,Q(v) (B(„) [a] - 1) - a - tc* (a)] dv
then h (.; a) is periodic on R. J Theorem 6 . 3 E(8)eaSt = h(s; a) etw*(a) h(s+t;a) Proof Conditioning upon whether a claim occurs in [t, t + dt] or not, we obtain E.(8) [eaSt+dt
I7t]
=
(1 - (3(s + t)dt)e«St -adt + /3(s + t)dt - east B(8+t) [a]
=
east - (1 - adt +,3(s + t)dt[B(8+t)[a] - 1]) ,
179
6. RISK THEORY IN A PERIODIC ENVIRONMENT E(8)east+ dt
=
E(8)east (1 - adt +,0(s + t)dt[B(8+t)[a] - 1]) ,
Et.(8)east
=
E (8)east (-a +,3(s +
dt log E(8)east
=
-a + f3(s + t) [B(8+t) [a] - 1],
l og E(8) et
=
-at +
d
t)[D(8
+t)[a] - 1]) ,
+ v)(B([a] - 1)dv f
=
log h(s + t; a) - log h(s; a),
where
atetk•(a) h(t; a) = exp I f t3(v)(kv)[a) - 1)dv -
o
h(t; a)
Thus E(8)east = h(s + t; a) = h(s; a) et,.* (a) h(s; a) h(s + t; a)
Corollary 6.4 For each 0 such that the integrals in the definition of h(t ; 0) exist and are finite, h(s + t; B) eoSt -t,c* (e) {Le,t}t>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
CHAPTER VI. MARKOVIAN ENVIRONMENT
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:
CHAPTER VI. MARKOVIAN ENVIRONMENT
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
6. RISK THEORY IN A PERIODIC ENVIRONMENT
183
Thus, e.g., in our basic example with A = 1, we obtain Co) = 1.42 so that tp(8) (u) < 1.42 • exp {J_ 1 cos 27rs - 47r sin 27rs + 167r cos 47rs - 167r I Cu. (6.12) As for the Markovian environment model, Lundberg's inequality can be con-
siderably sharpened and extended. We state the results below; the proofs are basically the same as in Section 3 and we refer to [44] for details. Consider first the time-dependent version of Lundberg's inequality. Just as in IV.4, we substitute T = yu in 0(u, T) and replace the Lundberg exponent ry by ryy = ay - yr. (ay), where ay is the unique solution of
(6.13)
W(ay) =y•
Elementary convexity arguments show that we always have ryy > -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
,(8) (u, yu) 000 (u) - 0(8) (u+ yu)
< C+)(y)h(s)
e-7yu,
(6.15)
The next result improves upon the constant C+) in front of e-ryu in Theorem 6.11 as well as it supplements with a lower bound. Theorem 6.13 Let = 1 B(t) C o J, e7 ( y-x)B(t)(dy)
(6.16
Then for all $ E [0, 1 ) and all u > 0, C_h(s)e-7u < V,(s)(u) < C+h(s)e-7".
(6.17)
In order to apply Theorem 6.13 to our basic example, we first note that the function u{w • 3e-3x + ( 1 - w) .7e - 7x j dx
fu° ex-u {w • 3e - 3x + (1 - w ) • 7e
_7x }
_ 6w + 6(1 - w)e-4u dx 9w + 7(1 - w)e-4u
184
CHAPTER VI. MARKOVIAN ENVIRONMENT
attains its minimum 2 /3 for u = oo and its maximum 6 /(7 + 2w) for u = 0. Thus C_ = 2 inf ex cos 2irs - 1 sin 27rs + 1 cos 47rs - 9 3 0<8<1 p 27r 47r 167r 161r
2 0.013,\ _ _e3 C+ = sup
6 exp { -A (- cos 27rs - -L sin 27rs + 1 I cos 47rs - 19 }
0 <8<1
8 + cos 21rs
Thus e.g. for A = 1 (where 3 e-0.013,\ = 0 .66, C+ = 1.20), 1/i18 1 s (u) > 0.66. exp 2^ cos 21rs - 4^ sin 2irs + 16^ cos 41rs - 16- I e-u, ,181 s(u) <
1.20 •exp { 2n cos 27rs - 1 sin 2irs + 16_ cos 47rs - 19
I
e-u.
Finally, we have the following result: Theorem 6 . 14 Let C+('yo) be as in (6.16) with 'y replaced by -yo and h(t) by h(t; -yo), and let 8 = er' (Y0). Then -7oudT . 0 <'p(8)(u ) -,(8)(u,T) < C+('Yo)h( s;'Yo)e
(6.18)
Notes and references The material is from Asmussen & Rolski [44]. Some of the present proofs are more elementary by avoiding the general point process machinery of [44], but thereby also slightly longer.
6c Markov-modulated approximations A periodic risk model may be seen as a varying environment model, where the environment at time t is (s + t) mod 1 E [0, 1), with s the initial season. Of course, such a deterministic periodic environment may be seen as a special case of a Markovian one (allowing a continuous state space E = [0, 1) for the environment), and in fact, much of the analysis of the preceding section is modelled after the techniques developed in the preceding sections for the case of a finite E. This observation motivates to look for a more formal connection between the periodic model and the one evolving in a finite Markovian environment. The idea is basically to approximate the (deterministic) continuous clock by a discrete (random) Markovian one with n 'months'. Thus, the nth Markovian environmental process {Jt} moves cyclically on {1, ... , n}, completing a cycle
7. DUAL QUEUEING MODELS
185
within one unit of time on the average , so that the intensity matrix is A(n) given by -n n 0 ••• 0 0 -n n ••• 0
(6.19)
A(n) _
n
0 0
••• -n
Arrivals occur at rate /3ni and their claim sizes are distributed according to Bni if the governing Markov process is in state i. We want to choose the /3ni and Bni in order to achieve good convergence to the periodic model. To this end, one simple choice is Oni = 0(
i - 1 ((i 1)/n) ) and Bni = B ,
(6.20)
but others are also possible. We let {Stn)}
be the claim surplus process of t>o the nth approximating Markov-modulated model, M(n) = Supt>o Stn), and the ruin probability corresponding to the initial state i of the environment is then Y'yn)(t) = F (M(n) > t),
(6.21)
which serves as an approximation to 0(1)(u) whenever n is large and i/n s. Notes and references
See Rolski [306].
7 Dual queueing models The essence of the results of the present section is that the ruin probabilities i/ (u), z/'i (u, T) can be expressed in a simple way in terms of the waiting time probabilities of a queueing system with the input being the time-reversed input of the risk process. This queue is commonly denoted as the Markov-modulated M/G/1 queue and has received considerable attention in the last decade. Thus, since the settings are equivalent from a mathematical point of view, it is desirable to have formulas permitting freely to translate from one setting into the other. Let 0j, Bi, A be the parameters defining the risk process in a random environment and consider a queueing system governed by a Markov process {Jt } ('Markov-modulated') as follows: • The intensity matrix for {Jt } is the time-reversed intensity matrix At _ A Aii'r?/7ri())i,jEE of the risk process, AE= • The arrival intensity is /3i when Jt = i;
CHAPTER VI. MARKOVIAN ENVIRONMENT
186
• Customers arriving when Jt = i have service time distribution Bi; • The queueing discipline is FIFO. The actual waiting time process 1W-1.=1 , 2 .... and the virtual waiting time (workload) process {Vt}too are defined exactly as for the renewal model in Chapter V. Proposition 7.1 Assume V0 = 0. Then (VT > u, JJ = i). (7.1) Pi(T(u) < T, JT = j) = LjPj 7ri In particular,
,0i (u , T) =
1 P.n(VT > u, JT =
7ri
i)
= 'P. (VT > u I JT = 2), (7.2)
Oi(u) = -1- P(V > u, J* = i) = P.T(V > u I J* = i), (7.3) 7ri
where (V, J*) is the steady-state limit of (Vt, Jt ). Proof Consider stationary versions of {Jt}o u, Jo = j, JT = i} coincide. Taking probabilities and using the stationarity yields 7riPi(T(u) < T, JT = j) = 7rjPj(VT > u, JT = Z),
and (7.1) follows. For (7.2), just sum (7.1) over j, and for (7.3), let T - oo in ❑ (7.2) and use that limF (VT > u, JT = i) = P(V > u, J* = i) for all j. Now let In denote the environment when customer n arrives and I* the steady-state limit. Proposition 7.2 The relation between the steady-state distributions of the actual and the virtual waiting time distribution is given by F(W > u, I* )3i P(V > u, J* = i), (7.4) where 0* = >jEE 7rj/3j. In particular, P(W > u, I* = i).
ii (u) = it /3
7. DUAL QUEUEING MODELS
187
Proof Identifying the distribution of (W, I*) with the time-average , we have N
1: I(W, >u,I,,=i) a4. P(W >u,I *=i), N -* oo. n=1
However, if T is large, on average 0*T customers arrive in [0, T], and of these, on average /32TP(V > u, J* = i) see W > u, I* = i. Taking the ratio yields (7.4), and (7.5) follows from (7.4) and (7.3).
❑
Notes and references One of the earliest papers drawing attention to the Markovmodulated M/G/1 queue is Burman & Smith [84]. The first comprehensive solution of the waiting time problem is Regterschot & de Smit [301], a paper relying heavily on classical complex plane methods. A more probabilistic treatment was given by Asmussen [17], and further references (to which we add Prabhu & Zhu [296]) can be found there. Proposition 7.1 is from Asmussen [16], with (7.3) improving somewhat upon (2.7) of that paper. The relation (7.4) can be found in Regterschot & de Smit [301]; a general formalism allowing this type of conclusion is 'conditional PASTA', see Regterschot & van Doorn [123]. In the setting of the periodic model of Section 6, the dual queueing model is a periodic M/G/1 queue with arrival rate 0(-t) and service time distribution B(-') at time t of the year (assuming w.l.o.g. that /3(t), B(t) have been periodically extended to negative t). With {Vt} denoting the workload process of the periodic queue, p < 1 then ensures that V(*) = limN-loo VN+9 exists in distribution, and one has PI'>(rr(u) < T) = P(-'_T)(VT > u),
(7.6)
P(- T)(T(u) u),
(7.7)
P(1-')(r(u) < oo) = P(')(00) > u).
(7.8)
For treatments of periodic M/G/1 queue, see in particular Harrison & Lemoine [186], Lemoine [242], [243], and Rolski [306].
This page is intentionally left blank
Chapter VII
Premiums depending on the current reserve 1 Introduction We assume as in Chapter III that the claim arrival process {Nt} is Poisson with rate ,6, and that the claim sizes U1, U2, ... are i.i.d. with common distribution B and independent of {Nt}. Thus, the aggregate claims in [0, t] are Nt
At = Ui
(1.1)
(other terms are accumulated claims or total claims). However , the premium charged is assumed to depend upon the current reserve Rt so that the premium rate is p(r) when Rt = r. Thus in between jumps, {Rt} moves according to the differential equation R = p(R), and the evolution of the reserve may be described by the equation
Rt = u - At + p(R8) ds. Zt
(1.2)
As earlier, z/i(u) = F IinffRt< 0IRo=u 1
tk(u,T) = FloinfTRt< OIRo=u1
denote the ruin probabilities with/initial reserve u and infinite, resp . finite horizon, and T(u) = inf {t > 0 : Rt < u} is the time to ruin starting from Ro = u so that '(u) = F(T(u) < oo), i&(u,T) = F(T(u) < T). 189
190
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
The following examples provide some main motivation for studying the model: Example 1 .1 Assume that the company reduces the premium rate from pi to p2 when the reserve comes above some critical value v. That is, pi > p2 and p(r) =
One reason could be competition, where one would try to attract new customers as soon as the business has become reasonably safe. Another could be the payout of dividends: here the premium paid by the policy holders is the same for all r, but when the reserve comes above v, dividends are paid out at rate pi - p2. Example 1.2 (INTEREST) If the company charges a constant premium rate p ❑ but invests its money at interest rate e, we get p(r) = p + er. Example 1.3 (ABSOLUTE RUIN) Consider the same situation as in Example 1.2, but assume now that the company borrows the deficit in the bank when the reserve goes negative, say at interest rate b. Thus at deficit x > 0 (meaning Rt = -x), the payout rate of interest is Sx and absolute ruin occurs when this exceeds the premium inflow p, i.e. when x > p/S, rather than when the reserve itself becomes negative. In this situation, we can put Rt = Rt + p/S, P(r) _ p + e(r - p/S) r > p/S p-5(p/5-r) 0
❑
Now return to the general model. Proposition 1.4 Either i,i(u) = 1 for all u, or o(u) < 1 for all u. Proof Obviously '(u) < ilb(v) when u > v. Assume 0(u) < 1 for some u. If Ro = v < u, there is positive probability, say e, that {Rt} will reach level u before the first claim arrives. Hence in terms of survival probabilities, 1 - Vi(v) ❑ > e(1 - '(u)) > 0 so that V'(v) < 1. A basic question is thus which premium rules p(r) ensure that 'O(u) < 1. No tractable necessary and sufficient condition is known in complete generality of the model. However, it seems reasonable to assume monotonicity (p(r) is
191
1. INTRODUCTION
decreasing in Example 1.1 and increasing in Example 1.2) for r sufficiently large so that p(oo) = limr.+ p(r) exists. This is basically covered by the following result (but note that the case p(r) .I3IB requires a more detailed analysis and that µB < oo is not always necessary for O(u) < 1 when p(r) -4 oo, cf. [APQ] pp. 296-297): Theorem 1.5 (a) If p(r) < /.3µB for all sufficiently large r, then ?(u) = 1 for all u; (b) If p(r) > /3µB + e for all sufficiently large r and some e > 0, then l/i(u) < 1 for all u, and P(Rt -+ oo) > 0. Proof This follows by a simple comparison with the compound Poisson model. Let Op(u) refer to the compound Poisson model with the same 0, B and (constant) premium rate p. In case (a), let uo be chosen such that p(r) < p = /3µB for r > uo. Starting from Ro = uo, the probability that Rt < uo for some t is at least tp(0) = 1 (cf. Proposition I1I.1.2(d)), hence Rt < uo also for a whole sequence of is converging to oo. However, obviously infu 0, and hence by a geometric trials argument,o(uo) = 1 so that t/'(u) = 1 for all u by Proposition 1.4. In case (b), let uo be chosen such that p(r) > p = 0I-LB + e for r > uo. Then if u > no, we have z/i(u) u} coincide. In particular, ,b(u,T) = P(VT > u),
(1.5)
and the process {Vt} has a proper limit in distribution , say V, if and only if V)(u) < 1 for all u. Then
0(u) = P(V > u).
(1.6)
192
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
In order to make Theorem 1.6 applicable, we thus need to look more into the stationary distribution G, say, for the storage process {Vt}. It is intuitively obvious and not too hard to prove that G is a mixture of two components, one having an atom at 0 of size 'yo, say, and the other being given by a density g(x) on (0, oo). It follows in particular that 0(u) = fg(Y)dy. (1.7) Proposition 1.7 p(x)g(x) = -tofB (x) + a f
(x - y)g(y) dy.
(1.8)
Proof In stationarity, the flow of mass from [0, x] to (x, oo) must be the same as the flow the other way. In view of the path structure of {V t }, this means that the rate of upcrossings of level x must be the same as the rate of downcrossings. Now obviously, the l.h.s. of (1.8) is the rate of downcrossings (the event of an arrival in [t, t + dt] can be neglected so that a path of {Vt} corresponds to a downcrossing in [t, t + dt] if and only if Vt E [x, x + p(x)dt]). An attempt of an upcrossing occurs as result of an arrival, say when {Vt} is in state y, and is succesful if the jump size is larger than x - y. Considering the cases y = 0 and 0 < y < x separately, we arrive at the desired interpretation of the r.h.s. of (1.8) as the rate of upcrossings. ❑ Define
^x 1 w(x) Jo p(t) dt.
Then w(x) is the time it takes for the reserve to reach level x provided it starts with Ro = 0 and no claims arrive. Note that it may happen that w (x) = oo for all x > 0, say if p(r) goes to 0 at rate 1 /r or faster as r j 0. Corollary 1.8 Assume that B is exponential with rate b, B(x) = e- 6x and that w(x) < oo for all x > 0. Then the ruin probability is tp (u) = f' g(y)dy, where g(x) = p( ^ ^ exp {,Qw(x) - Sx}, yo
1 + oo Q exp {,6w(x) - Sx} dx. Jo AX) (1.9)
Proof We may rewrite (1.8) as g(x) = p 1
{yo13e_6x +, Oe-ax f x e'Yg (y) dy } = p) e-axa(x)
193
1. INTRODUCTION where c(x) = 1o + fo elyg(y) dy so that (x) = eaxg(x) _
1 p(x)
nkx).
Thus log rc(x) = log rc(0) + Jo X L dt = log rc(0) + /3w(x), p(t) c(x) = rc (0)em"lxl = Yoes"lxl, g(x) = e-axK' (x) = e-6x ,Yo)3w'(x)e'6"lxl which is the same as the expression in (1.9). That 'Yo has the asserted value is ❑ a consequence of 1 = I I G I I = yo + f g• Remark 1.9 The exponential case in Corollary 1.8 is the only one in which explicit formulas are known (or almost so; see further the notes to Section 2), and thus it becomes important to develop algorithms for computing the ruin probabilities. We next outline one possible approach based upon the integral equation (1.8) (another one is based upon numerical solution of a system of differential equations which can be derived under phase-type assumptions, see further VIII.7). A Volterra integral equation has the general form x g(x) = h(x) + f K(x, y)9(y) dy, 0
(1.10)
where g(x) is an unknown function (x > 0), h(x) is known and K(x,y) is a suitable kernel. Dividing (1.8) by p(x) and letting K(x, y) _ ,QB(x - y) _ 'YoIB(x) p(x) , h(x) p(x) we see that for fixed -to, the function g(x) in (1.8) satisfies (1.10). For the purpose of explicit computation of g(x) (and thereby -%(u)), the general theory of Volterra equations does not seem to lead beyond the exponential case already treated in Corollary 1.8. However, one might try instead a numerical solution. We consider the simplest possible approach based upon the most basic numerical integration procedure, the trapezoidal rule dx = 2hfxN() [f ( xo) + 2f (xi) + 2f ( x2) + ... + 2f (XN-1) + f (xN)1 p
194
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
where xk = x0 + kh. Fixing h > 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
1 - q(u) + q ( u)z,b(v) p1(v)
u
=
v
1 + pi (v ) - '02 (0) pi (u) + (0, (u - v) - pi (u)) z/i(v )
v < u < oo.
Proof Let w = inf{ t > 0 1 Rt= v or Rt < 0} and let Q1 (u) = Pu(RC,, = v) be the probability of upcrossing level v before ruin given the process starts at u < v. If we for a moment consider the process under level v, Rt , only, we get Vil (u ) = 1 - q, (u ) + g1(u),O1( v). Solving for ql (u), it follows that q1 (u) = q(u). With this interpretation of q(u) is follows that if u < v then the probability of ruin will be the sum of the probability of being ruined before upcrossing v, 1 - q(u), and the probability of ruin given we hit v first , q(u)z'(v). Similarly, if u > 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
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
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
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
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)
3. THE LOCAL ADJUSTMENT COEFFICIENT
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).
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
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
^oo
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},
3. THE LOCAL ADJUSTMENT COEFFICIENT
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
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
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.
3. THE LOCAL ADJUSTMENT COEFFICIENT
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 Su - c3 logu a= 1
a dx -
c4ul -1/° 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).
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
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) <'Yo(x)• Proof That 7(x) is non-decreasing follows easily by inspection of (3.4). The assumption implies that ru(t) - u is a non-decreasing function of u. Hence for u Ee7o(u)(ul+v-r»(Ti)).
By convexity of the m . g.f. of U1 + v - r„(Ti), this is only possible if 7o(v) 2 7o(u)•
3. THE LOCAL ADJUSTMENT COEFFICIENT
209
For (b), note that the assumption implies that ru(t) - u > tp(u). Hence 1
= Ee-Yo(u)(U1+u-ru(T1)) < E,e70(u)(U1-P(u)T1)
0 + 7o(u)p(u)'
0
<_ 00['Yo( u)] - 1) - 7o (u)p(u)•
Since (3.4) considered as function of 7 is convex and 0 for -y = 0, this is only possible if -yo(u) > 7(u). ❑ We prove Theorem 3.2 in terms of 7o; the case of 7 then follows immediately by Proposition 3.10(b): Theorem 3.11 Assume that p(x) is a non-decreasing function of x. Then (u) < efo Yo(x)dx.
(3.16)
Proof Define 411(n)(u) = P('r(u) < on) as the ruin probability after at most n claims (on = TI + • • • + Tn). We shall show(' by induction that Y'(n) (u) <
e-
fo
'Yo(x)dx
(3.17)
from which the theorem follows by letting n -+ oo. The case n = 0 is clear since here To = 0 so that ik(°)(u) = 0. Assume (3.17) shown for n and let Fu(x) = P(U1 + u - ru(T1) < x). Separating after whether ruin occurs at the first claim or not, we obtain „I,(n+l) (u)
1 - Fu(u ) +
J ✓
^(n)(u - x)Fu(dx) 00
U efo J
o (y) dYF
(dx)
)+f
I
=
11
/' / 00 e f oFu
J
fu dx) + of u :7o(Y)dYFu(dx)
u
00
l`
1
Considering the cases 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
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
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
uk_l,n
u k}1,n AX),
and, in accordance with the notation i/iE (u), op*. (u), let Op*.;E (u) denote the ruin probability for the classical model with 0 replaced by ,3/e and U; by €U=. Lemma 3.12 lim sup4^o -f log O, (u) < I(u). Proof For ruin to occur, {RtE)} (starting from u = un,n) must first downcross un-l,n. The probability of this is at least n n;E (u/n), the probability that ruin occurs in the Cramer-Lundberg model with p* = pn,n (starting from u/n) without that 2u/n is upcrossed before ruin. Further, given downcrossing occurs, the value of {R(E)} at the time of downcrossing is < un-l,n so that n,n;E (u/ n) Y'E (un - I,n) pn niE (u /n)
n_1 n;E ( u/n) ^•e. (un-2,n)
n
>
II v ^k n;E (u/n)
k =1
Now as e . 0,
-Y*u /E, 0;.;e (u) = v';. W O ,., C*e-
where the first equality follows by an easy scaling argument and the approximation by (3.3). Let Ck,n, ryk,nbe C*, resp. y* evaluated for p* = Pk,n; in
3. THE LOCAL ADJUSTMENT COEFFICIENT
211
particular, since ry' is an increasing function of p', also sup ?'(x).
ryk,n =
uk_1,n <X
Clearly, *p•;E (u/n) -Op•;E (urn)
<
+^p•;F (2u/n),
\
, /' (u/n) -'T nk,n cE (2u/n)
*I,n;E (u/n) OP-
Ck ne-7k,nu /fn( 1 Ck
-
ne-7k,nu/en(1 +
e- 7k,nu /En) o(1)),
where o(1) refers to the limit e .. 0 with n and u fixed. It follows that n
log Ypk,,i;! (u/n)
-log V'C (u) k =1
n
n
m 7k,n - log Ck,n + 0(1),
limsup-elogv), (u) CIO
<
k=1 k=1 n u _ nE7 k,nk=1
Letting n -4 oo and using a Riemann sum approximation completes the proof. 11 Theorem 3. 3 now follows easily in case (a). Indeed , in obvious notation one has -tC (x) = y(x)/e, so that Theorem 3.2 gives 7PE (u) < e-Ii"i/f =
lim inf -Clog 0E (u) > I (u). 40
Combining with the upper bound of Lemma 3.12 completes the proof. In case (b), we need the following condition: Condition 3.13 There exists a r. v. V < oo such that (i) for any u < oo there exist Cu < oo and a (u) > supy <„ 7(x) such that P(V > x) < Cue- a( u)z; (3.18) (ii) the family of claim overshoot distributions is stochastically dominated by V, i.e. for all x , y > 0 it holds that
F(U>x
+yIU>x)
B(x + y) < F (V > y). B(x)
(3.19)
212 CHAPTER VII. RESERVE-DEPENDENT PREMIUMS To complete the proof, let v < u and define ^(E) (u, v ) = v - R<) (u v).
T(E) (u, v ) = inf { t > 0 : R(c ) < v R) = u } , Then Y'E (u)
,,,^''
(R,( . ) (u
l
<
oo]
E
=
E [OE (u/n - ^(E) (u, u/n)) ; T() (u, u/n) < oo]
=
E [OE (u/n - E(E) (u, u/n)) I T(E) (u, u/n) < oo] . P (T(E) (u, u/n) < oo)
<
EV), (u/n - eV) • P (T(E) (u, u/n) < oo) .
[
,,
u
/n)) ; T(E) (u, u /n)
=
Write EO, (u/n - EV) = El + E2, where El is the contribution from the event that the process does not reach level 2u/n before ruin and E2 is the rest. Then the standard Lundberg inequality yields (u/En - V)
El < E?;1 n;E (u/n - EV) = EiI 1 ,n <
e-ry1,nu /EnE [e71,n V; V < u/En] + P(V > u/En)
= e-71 nu/Eno(l)
(using (3.18) for the last equality). For E2, we first note that the number of downcrossings of 2u/n starting from RoE) = 2u/n is bounded by a geometric r.v. N with
EN < 1 = infx>2u/nA(x) = 0(1), infx>2u /n P(x) - ,QEU 1 - of:>2 in n(x);E (0) cf. (3.5) and the standard formula for b(0). The probability of ruin in between two downcrossings is bounded by Epp ;E (2u/n - EV) = e- 2-y 1 ' . u /en 0(i) _n
so that E2 < e-2ryl nu/En0(1),
Ei + E2 < e-71,nu/En0(1)
3. THE LOCAL ADJUSTMENT COEFFICIENT
213
Hence lim inf -e log Ali, (u) 40
lim inf -e log(Ei { +E2) + logP (r(`) (u, u/n) < oo 40 )I U nryl n+liminf-elogP (T(')(u,u/n) < oo) CI -
> u
n
n ryi n' i=1
❑
Another Riemann sum approximation completes the proof.
Notes and references With the exception of Theorem 3.1, the results are from Asmussen & Nielsen [39]; they also discuss simulation based upon 'local exponential change of measure' for which the likelihood ratio is (
/'t
/'t
Ns
Lt = exp S - J y(Rs-)dR,) = exp - J -r(Rs)p(R,)ds + -Y(R2;-)Ui } . l o JJJ o ;=1 J An approximation similar to (3.7) for ruin probabilities in the presence of an upper barrier b appears in Cottrell et al. [89], where the key mathematical tool is the deep Wentzell-Freidlin theory of slow Markov walks (see e .g. Bucklew [81]). Djehiche [122] gives an approximation for tp(u,T) = P „(info
(3.20)
(with ic(x, s) as in (3.4) and the prime meaning differentiation w.r.t. s), whereas the most probable path leading to ruin is the solution of r(x) _ -k (x,7(x))
(3.21)
(the initial condition is r(0) = u in both cases). Whereas the result of [122] is given in terms of an action integral which does not look very explicit, one can in fact arrive at the optimal path by showing that the approximation for 0(u, T) is maximized over T by taking T as the time for (3.21) to pass from u to 0; the approximation (3.7) then comes out (at least heuristically) by analytical manipulations with the action integral. Similarly, it might be possible to show that the limits e . 0 and b T 00 are interchangeable in the setting of [89]. Typically, the rigorous implementation of these ideas via large deviations techniques would require slightly stronger smoothness conditions on p(x) than ours and conditions somewhat different from Condition 3.13,
214
CHAPTER VII. RESERVE-DEPENDENT PREMIUMS
the simplest being to require b[s] to be defined for all s > 0 (thus excluding , e.g., the exponential distribution ). We should like, however, to point out as a maybe much more important fact that the present approach is far more elementary and self-contained than that using large deviations theory. For different types of applications of large deviations to ruin probabilities , see XI.3.
Chapter VIII
Matrix-analytic methods 1 Definition and basic properties of phase-type distributions Phase-type distributions are the computational vehicle of much of modern applied probability. Typically, if a problem can be solved explicitly when the relevant distributions are exponentials, then the problem may admit an algorithmic solution involving a reasonable degree of computational effort if one allows for the more general assumption of phase-type structure, and not in other cases. A proper knowledge of phase-type distributions seems therefore a must for anyone working in an applied probability area like risk theory. A distribution B on (0, oo) is said to be of phase-type if B is the distribution of the lifetime of a terminating Markov process {Jt}t>o with finitely many states and time homogeneous transition rates. More precisely, a terminating Markov process {Jt} with state space E and intensity matrix T is defined as the restriction to E of a Markov process {Jt}o
0
.
We often write p for the number of elements of E. Note that since (1.1) is 'Here as usual , P; refers to the case Jo = i; if v = (vi)iEE is a probability distribution, we write Pv for the case where Jo has distribution v so that Pv = KER viPi•
215
216
CHAPTER VIII. MATRIX-ANALYTIC METHODS
the intensity matrix of a non-terminating Markov process, the rows sum to one which in matrix notation can be rewritten as t + Te = 0 where e is the column E-vector with all components equal to one. In particular, T is a subintensity matrix2, and we have t = -Te.
(1.2)
The interpretation of the column vector t is as the exit rate vector, i.e. the ith component ti gives the intensity in state i for leaving E and going to the absorbing state A. We now say that B is of phase-type with representation (E, a, T) (or sometimes just (a,T)) if B is the Pa-distribution of the absorption time C = inf{t > 0 : it = A}, i.e. B(t) = Fa(^ < t ). Equivalently, C is the lifetime sup It > 0 : Jt E E} of {Jt}. A convenient graphical representation is the phase diagram in terms of the entrance probabilities ai, the exit rates ti and the transition rates (intensities) tij:
tj
3 aj ai i
tjk
ti tk
FkJ ak
Figure 1.1 The phase diagram of a phase-type distribution with 3 phases, E = {i, j, k}. The initial vector a is written as a row vector. Here are some important special cases: Example 1 .1 Suppose that p = 1 and write ,0 = -t11. Then a = a1 = 1, t1 = /3, and the phase-type distribution is the lifetime of a particle with constant failure rate /3, that is, an exponential distribution with rate parameter ,3. Thus the phase-type distributions with p = 1 is exactly the class of exponential distributions. 0 2this means that tii < 0, tij > 0 for i 54 j and EjEE tij < 0
1. PHASE-TYPE DISTRIBUTIONS 217 Example 1.2 The Erlang distribution EP with p phases is defined Gamma distribution with integer parameter p and density bp XP-1 -6x (p- 1)!e
Since this corresponds to a convolution of p exponential densities with the same rate S, the EP distribution may be represented by the phase diagram (p = 3)
Figure 1.2 corresponding to E = {1, . . . , p}, a = (1 0 0 ... 00))
-S s o ... 0 0 0 0 -S 6 ... 0 0 0 T=
t= 0 ••• -S S 0 0 0 0 0 0 ... 0 -S 6
Example 1.3 The hyperexponential distribution HP with p parallel channels is defined as a mixture of p exponential distributions with rates 51, ... , 6, so that the density is P E ai6ie-6,x i=1
Thus E _ -Si 0
0 -S2
0 0
... •..
0 0
0 0
0
0
0
•••
-Sp-1
0
0
0
0 •..
T
t=
and the phase diagram is (p = 2)
0
-SP
CHAPTER VIII. MATRIX-ANALYTIC METHODS
218
Figure 1.3 0 Example 1 .4 (COXIAN DISTRIBUTIONS) This class of distributions is popular in much of the applied literature, and is defined as the class of phase-type distributions with a phase diagram of the following form: 2
1
b2- t2
617 ti
yt bP- 1 tP-1
t2
1
Figure 1.4 For example, the Erlang distribution is a special case of a Coxian distribution. The basic analytical properties of phase -type distributions are given by the following result . Recall that the matrix-exponential eK is defined by the standard series expansion Eo K"/n! 3. Theorem 1 . 5 Let B be phase-type with representation (E, a, T). Then: (a) the c.d.f is B (x) = 1 - aeTxe; (b) the density is b(x ) = B'(x) = aeTxt; (c) the m.g.f. B[s] = f0°O esxB (dx) is a(-sI -T)-lt (d) the nth moment f0°O xnB(dx) is (- 1)"n! aT-"e. Proof Let P8 = (p ^) be the s-step EA x EA transition matrix for {Jt } and P8 the s-step E x E-transition matrix for {Jt} , i.e. the restriction of P8 to E. Then for i , j E E, the backwards equation for {Jt} (e.g. [APQ ] p. 36) yields s . p: dp; d-. . ds^
=
ds'
= ttlaj +
tikpkj. E t ikp kj = kEE kEE
3For a number of additional important properties of matrix-exponentials and discussion of computational aspects , see A.3
1. PHASE-TYPE DISTRIBUTIONS
219
That is, d8 P8 = TP8, and since obviously P° = I, the solution is P8 = eT8. Since
1 - B(x) = 1'a (( > x) = P., (Jx E E) =
1: aipF. = aPxe, i,jEE
this proves (a), and (b) then follows from B'(x) _ -cx Pxe = -aeTxTe = aeTxt (since T and eTx commute). For (c), the rule (A.12) for integrating matrixexponentials yields
B[s] =
esxaeTxt dx = a ( f°°e(81+T)dx ) t
J
a(-sI - T) -1t. Alternatively, define hi = Eie8S. Then h -tit ti + ti3 h j - tii -tii - s j# -tii i
(1.5)
Indeed , - tii is the rate of the exponential holding time of state i and hence (-tii)/(-tii - s) is the m .g.f. of the initial sojourn in state i. After that, we i w.p. tij / - tii and have an additional time to absorption either go to state j which has m .g.f. hj , or w.p. ti/ - tii we go to A, in which case the time to absorption is 0 with m .g.f. 1. Rewriting ( 1.5) as hi(tii + s)
=
-ti
-
t ij hj, j#i
tijhj + his = -ti,
jEE
this means in vector notation that (T + sI)h = -t, i.e. h = -(T + sI)-1t, and since b[s] = ah, we arrive once more at the stated expression for B[s]. Part (d) follows by differentiating the m.g.f.,
d" dsn a
(- s I - T) -'t
=
B(n)[0] = _
(- 1 ) n +l n ! a (s I + T ) - n -lt , (-1) n+1n!aT - n-1t = (-1)nn!aT-n-1Te (-1)nn! aT-ne.
Alternatively, for n = 1 we may put ki = Ei( and get as in (1.5)
-L j-
ki = 1 + tii
j:Ai
-tii
(1.6)
220 CHAPTER VIII. MATRIX-ANALYTIC METHODS which is solved as above to get k = -aT-le.
0
Example 1.6 Though typically the evaluation of matrix-exponentials is most conveniently carried out on a computer, there are some examples where it is appealing to write T on diagonal form, making the problem trivial. One obvious instance is the hyperexponential distribution, another the case p = 2 where explicit diagonalization formulas are always available, see the Appendix. Consider for example
3 9 a= (2 2), T= 2 111 so that 2 2
Then (cf. Example A3.7) the diagonal form of T is
T
9
9
10
70
6
1
9
10
70
7
1
7
10
10
0
9 110
where the two matrices on the r.h.s. are idempotent. This implies that we can compute the nth moment as
(-1)"n! aT -"e
9 9 10 70
1"n! 1 1 22
7 1 10 10 1 9
+6- "n!
(
( l 2 2 ) 17 9 0 \ 1 / 10 10
32 n! 35 +n!353 6" Similarly, we get the density as
aeTyt = e x
9 9 6 (1 1) 10 7 1 0 10
2
221
1. PHASE-TYPE DISTRIBUTIONS
+e -6x (1
2
1 10
11 2
7 10
9 6 70 7 9 10
2
35e-x + 18e-6x 35
The following result becomes basic in Sections 4, 5 and serves at this stage to introduce Kronecker notation and calculus (see A.4b for definitions and basic rules): Proposition 1.7 If B is phase-type with representation (v,T), then the matrix m.g.f. B[Q] of B is
f3[Q] =
J
e'1zB(dx) _ (v (9 I)(-T ® Q)-1(t ® I). (1.7)
Proof According to (A.29) and Proposition A4.4, 00
B[Q] =
Jf0
veTxteQx dx
(v (& I)
(
= (v ® I) (
(T ®Q)xdx fo o" e
f° eT
x edx I (t I)
t ® I) _ (v ® I)(-T ® Q)-1(t ® I). )( 0
Sometimes it is relevant also to consider phase-type distributions, where the initial vector a is substochastic, hail = E=EE a; < 1. There are two ways to interpret this: • The phase-type distribution B is defective, i.e 11BIJ = 1laDD < 1; a random variable U having a defective phase-type distribution with representation (a, T) is then defined to be oo on a set of probability 1- 11aDD, or one just lets U be undefined on this additional set. • The phase-type distribution B is zero-modified, i.e a mixture of a phasetype distribution with representation (a/llall,T) with weight hall and an atom at zero with weight 1 - hall. This is the traditional choice in the literature, and in fact one also most often there allows a to have a component ao at A.
222
CHAPTER VIII. MATRIX-ANALYTIC METHODS
la Asymptotic exponentiality Writing T on the Jordan canonical form, it is easily seen that the asymptotic form of the tail of a general phase-type distribution has the form
B(x) _ Cxke-nx, where C, 77 > 0 and k = 0, 1, 2.. .. The Erlang distribution gives an example where k > 0 (in fact, here k = p-1), but in many practical cases, one has k = 0. Here is a sufficient condition: Proposition 1.8 Let B be phase-type with representation (a, T), assume that T is irreducible , let -,q be the eigenvalue of largest real part of T, let v, h be the corresponding left and right eigenvectors normalized by vh = 1 and define C = ah • ve . Then the tail B(x) is asymptotically exponential, B(x) - Ce-7'.
(1.8)
Proof By Perron-Frobenius theory (A.4c), i is real and positive, v, h can be chosen with strictly positive component, and we have eTx - hve-7x, x -* oo. Using B(x) = aeTxe , the result follows (with C = (ah)(ve)). 0 Of course, the conditions of Proposition 1.8 are far from necessary ( a mixture of phase-type distributions with the respective T(') irreducible has obviously an asymptotically exponential tail, but the relevant T is not irreducible, cf. Example A5.8). In Proposition A5.1 of the Appendix, we give a criterion for asymptotical exponentiality of a phase-type distribution B, not only in the tail but in the whole distribution. Notes and references The idea behind using phase-type distributions goes back to Erlang, but todays interest in the topic was largely initiated by M.F. Neuts, see his book [269] (a historical important intermediate step is Jensen [214]). Other expositions of the basic theory of phase-type distributions can be found in [APQ], Lipsky [247], Rolski, Schmidli, Schmidt & Teugels [307] and Wolff [384]. All material of the present section is standard; the text is essentially identical to Section 2 of Asmussen [26]. In older literature, distributions with a rational m.g.f. (or Laplace transform) are often used where one would now work instead with phase-type distributions. See in particular the notes to Section 6. O'Cinneide [276] gave a necessary and sufficient for a distribution B with a rational m.g.f. B[s] = p(s)/q(s) to be phase-type: the density b(x) should be strictly positive for x > 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
Jt={Jt?ul}, U1
is Markov and has two types of jumps , the jumps of the j(k) and the it } k) to the next J( k+l) A jump jumps corresponding to a transition from one Jt Then {
4Here the empty sum U1 +... + U0 is 0
224 CHAPTER VIII. MATRIX-ANALYTIC METHODS of the last type from i to j occurs at rate tiaj , and the jumps of the first type are governed by T. Hence the intensity matrix is T + ta, and the distribution of Jx is ae ( T+t«)x. The renewal density at x is now just the rate of jumps of the second type, which is ti in state i. Hence ( 2.1) follows by the law of total probability. ❑ The argument goes through without change if the renewal process is terminating, i.e. B is defective , and hence ( 2.1) remains valid for that case. However, the phase-type assumptions also yield the distribution of a further quantity of fundamental importance in later parts of this chapter , the lifetime of the renewal process. This is defined as U1 + ... + Uit_1 where s; is the first k with Uk = 00, that is, as the time of the last renewal; since Uk = oo with probability 1 - IIBII which is > 0 in the defective case, this is well-defined. Corollary 2.2 Consider a terminating renewal process with interarrivals which are defective phase-type with representation (a,T), i.e. IIafl < 1. Then the lifetime is zero-modified phase -type with representation (a,T + ta). Proof Just note that { it } is a governing phase process for the lifetime.
❑
Returning to non-terminating renewal processes , define the excess life e(t) at time t as the time until the next renewal following t, see Fig. 2.1. fi(t)
U2 U1 - U1
U3
U2
U3
U4
Figure 2.1 Corollary 2.3 Consider a renewal process with interarrivals which are phasetype with representation (a, T), and let µB = -aT-le be the mean of B. Then: (a) the excess life t(t) at time t is phase-type with representation ( vt,T) where vt = ae (T+ta)t .,
(b) £(t) has a limiting distribution as t -* oo, which is phase -type with representation (v,T) where v = -aT-1 /µB. Equivalently, the density is veTxt = B(x)/µB.
225
2. RENEWAL THEORY
Proof Consider again the process { Jt } in the proof of Theorem 2.1. The time of the next renewal after t is the time of the next jump of the second type, hence e(t) is phase-type with representation (vt,T) where vt is the distribution of it which is obviously given by the expression in (a). Hence in (b) it is immediate that v exists and is the stationary limiting distribution of it, i.e. the unique positive solution of ve = 1, (2.2) v(T + ta) = 0. Here are two different arguments that this yields the asserted expression: (i) Just check that -aT-1/µB satisfies (2.2): -aT-1 e = AB = 1 µB µB -a + aT-'Tea -aT-1(T + ta) µB
PB -a + aea -a + a µB
=0.
µB
(ii) First check the asserted identity for the density: since T, T-1 and eTx commute, we get B(x) aeTxe aT-1eTxTe µB µB PB
= veTxt.
Next appeal to the standard fact from renewal theory that the limiting distribution of e(x) has density B(x)/µB, cf. Al.e. ❑ Example 2 .4 Consider a non-terminating renewal process with two phases. The formulas involve the matrix-exponential of the intensity matrix Q = T + to =
( tll + tlal t12 + t2al
tlz + tlaz _ -q1 ql t22 + t2a2
(say).
q2 -q2
According to Example A3.6, we first compute the stationary distribution of Q, = qz ql (x1 xz) = ql + qz ql + q ' and the non-zero eigenvalue A = -ql - q2. The renewal density is then aeQtt = (al a2) ( 7i
7"2.) ( t2 )
CHAPTER VIII. MATRIX-ANALYTIC METHODS
226
+
e;`t (al a2)
J
(7r1 7r2) ( t2 7rltl +
7r2t2
+
C
11 172 -ir12 / \ t 2 ) r1
+ eAt (al a2) ( 71(t2 - tl)
eat (a17r2
- a27rl) (tl - t2)
1 + eat (a17r2 - a27r1) (t1 - t2) . t1B 0
Example 2 .5 Let B be Erlang(2). Then
Q=
0
55
)+(1o)=( j
ad ).
Hence 7r = (1/2 1/2), A = -25, and Example 2.4 yields the renewal density as u(t) = 2 (1 - e-2bt) 13
Example 2 .6 Let B be hyperexponential. Then _ Q
51 0
0 -52
+
51 52
_
)
-5152
51a2
52a1
-62a1
(al a2)
Hence
Slat + 52a1
51a2 51a2+52a1
A = -51a2 - 52a1, and Example 2.4 yields the renewal density as
u(t) = 5152
e- (biaz + aza, )t (51 - 52) 25152
51x2+5251 51a2+5251
Notes and references Renewal theory for phase-type distributions is treated in Neuts [268] and Kao [221]. The present treatment is somewhat more probabilistic.
3. THE COMPOUND POISSON MODEL 227
3 The compound Poisson model 3a Phase-type claims Consider the compound Poisson (Cramer-Lundberg) model in the notation of Section 1, with 0 denoting the Poisson intensity, B the claim size distribution, r(u) the time of ruin with initial reserve u, {St} the claim surplus process, G+(.) = F(ST(o) E •, T(0) < oo) the ladder height distribution and M = supt>o St. We asssume that B is phase-type with representation (a, T). Corollary 3.1 Assume that the claim size distribution B is phase-type with representation (a, T). Then: (a) G+ is defective phase-type with representation (a+, T) where a+ is given by a+ = - f3aT-1, and M is zero-modified phase-type with representation (a+, T + to+). (b) V,(u) = a+e(T+tQ+)u
Note in particular that p = IIG+II = a+e. Proof The result follows immediately by combining the Pollaczeck-Khinchine formula by general results on phase-type distributions: for (a), use the phasetype representation of Bo, cf. Corollary 2.3. For (b), represent the maximum M as the lifetime of a terminating renewal process and use Corollary 2.2. Since the results is so basic, we shall, however, add a more self-contained explanation of why of the phase-type structure is preserved. The essence is contained in Fig. 3.1 on the next page. Here we have taken the terminating Markov process underlying B with two states, marked by thin and thick lines on the figure. Then each claim (jump) corresponds to one (finite) sample path of the Markov process. The stars represent the ladder points ST+(k). Considering the first, we see that the ladder height Sr+ is just the residual lifetime of the Markov process corresponding to the claim causing upcrossing of level 0, i.e. itself phasetype with the same phase generator T and the initial vector a+ being the distribution of the upcrossing Markov process at time -ST+_. Next, the Markov processes representing ladder steps can be pieced together to one {my}. Within ladder steps, the transitions are governed by T whereas termination of ladder steps may lead to some additional ones: a transition from i to j occurs if the ladder step terminates in state i, which occurs at rate ti, and if there is a subsequent ladder step starting in j whic occurs w.p. a+j. Thus the total rate is tip + tia+.i, and rewriting in matrix form yields the phase generator of {my} as T + ta+. Now just observe that the initial vector of {mx} is a+ and that the lifelength is M.
228 CHAPTER VIII. MATRIX-ANALYTIC METHODS t -- M---------------------------------------
--------
S
ST+-(2-) -
{mx}
.,.t t
d kkt
--S.-------
Figure 3.1 This derivation is a complete proof except for the identification of a+ with -,QaT-1. This is in fact a simple consequence of the form of the excess distribution B0, see Corollary 2.3. 0 Example 3.2 Assume that ,Q = 3 and
b(x) = - 1 , 3e-3x + - . 7e-7x 2 2 Thus b is hyperexponential (a mixture of exponential distributions) with a (2 2 ), T = (-3 - 7)diag so that
a+
=
T+ta+ =
-QaT 1 = -3
(
3 2 2)
0
3 0 07/+( 7I \ 2 14
3 9 2 14 7
11
2
2
229
4. THE RENEWAL MODEL This is the same matrix as is Example 1.6, so that as there 1
9 9 e(T+ta+)u
9
10 70
e_u 10 70
7
1
7
9
10
) + e6'410 ( 10
10
Thus ,^(u) = a+e( T+ta+)ue = 24e-u + 1 e-6u 35 35 0
Notes and references Corollary 3.1 can be found in Neuts [269] (in the setting of M/G/1 queues, cf. the duality result given in Corollary 11.4.6), but that such a simple and general solution exists does not appear to have been well known to the risk theoretic community. The result carries over to B being matrix-exponential, see Section 6. In the next sections, we encounter similar expressions for the ruin probabilities in the renewal- and Markov-modulated models, but there the vector a+ is not explicit but needs to be calculated (typically by an iteration).
The parameters of Example 3.2 are taken from Gerber [157]; his derivation of +'(u) is different. For further more or less explicit computations of ruin probabilities, see Shin [340]. It is notable that the phase-type assumption does not seem to simplify the computation of finite horizon ruin probabilities substantially. For an attempt, see Stanford & Stroinski [351] .
4 The renewal model We consider the renewal model in the notation of Chapter V, with A denoting the interarrival distribution and B the service time distribution. We assume p = PB/µA < 1 and that B is phase-type with representation (a, T). We shall derive phase-type representations of the ruin probabilities V) (u), 0(8) (u) (recall that z/i(u) refers to the zero-delayed case and iY(8) (u) to the stationary case). For the compound Poisson model, this was obtained in Section 3, and the argument for the renewal case starts in just the same way (cf. the discussion around Fig. 3.1 which does not use that A is exponential) by noting that the distribution G+ of the ascending ladder height ST+ is necessarily (defective) phase-type with representation (a+, T) for some vector a+ = (a+;j). That is, if we define {mz} just as for the Poisson case (cf. Fig. 3.1): Proposition 4.1 In the zero-delayed case, (a) G+ is of phase-type with representation (a+,T), where a+ is the (defective)
CHAPTER VIII. MATRIX-ANALYTIC METHODS
230 distribution of mo;
(b) The maximum claim surplus M is the lifetime of {mx}; (c) {mx } is a (terminating) Markov process on E, with intensity matrix Q given by Q = T + to+.
The key difference from the Poisson case is that it is more difficult to evaluate a+. In fact, the form in which we derive a+ for the renewal model is as the unique solution of a fixpoint problem a+ = cp(a+), which for numerical purposes can be solved by iteration. Nevertheless, the calculation of the first ladder height is simple in the stationary case:
Proposition 4.2 The distribution G(s) of the first ladder height of the claim surplus process {Ste) } for the stationary case is phase -type with representation (a(8),T), where a(8) = -aT-1/PA. Proof Obviously, the Palm distribution of the claim size is just B. Hence by Theorem 11.6.5, G(') = pBo, where B0 is the stationary excess life distribution corresponding to B. But by Corollary 2.3, B0 is phase-type with representation ❑
(-aT-1/µa,T)• Proposition 4.3 a+ satisfies a+ = V(a+), where
w(a +) = aA[T + to+) = a
J0
e(T+t-+)1A(dy).
(4.1)
Proof We condition upon T1 = y and define {m.*} from {St+y - Sy-} in the same way as {mx} is defined from {St}, cf. Fig. 4.1. Then {m,*'} is Markov with the same transition intensities as {mx}, but with initial distribution a rather than a+. Also, obviously mo = m. Since the conditional distribution of my given T1 = y is ae4y, it follows by integrating y out that the distribution a+ ❑ of mo is given by the final expression in (4.1). We have now almost collected all pieces of the main result of this section:
Theorem 4 .4 Consider the renewal model with interarrival distribution A and the claim size distribution B being of phase-type with representation (a,T). Then
4. THE RENEWAL MODEL
231
I
i
- M----------------------------- •..
{mx} ------------------- ----------
y
^--
T1= y -`•r---------------
Figure 4.1 ,^(u) = a+e ( T+ta+)xe,
,^(8)(u) = a ( 8)e(T+ta +) xe,
(4.2)
where a+ satisfies (4.1) and a(8) _ -aT- 1/pA. Furthermore , a+ can be computed by iteration of (4.1), i.e. by a+ = lim a +n)
where a+°) - 0, a+l ) = cp (a+°)) , a+2) = ^p (a+l)) , .... (4.3)
Proof The first expression in (4.2 ) follows from Proposition 4.1 by noting that the distribution of mo is a+. The second follows in a similar way by noting that only the first ladder step has a different distribution in the stationary case, and that this is given by Proposition 4.2; thus , the maximum claim surplus for the stationary case has a similar representation as in Proposition 4.1(b), only with initial distribution a(*) for mo. It remains to prove convergence of the iteration scheme (4.3). The term tf3 in cp(i3) represents feedback with rate vector t and feedback probability vector (3. Hence ^p(,3) (defined on the domain of subprobability vectors ,0) is an increasing function of /3. In particular , a+) > 0 = a+o) implies a+) _ (a+)
> W (a+)) = a+)
CHAPTER VIII. MATRIX-ANALYTIC METHODS
232
and (by induction ) that { a+ n) } is an increasing sequence such that limn,. a+ ) exists . Similarly, 0 = a+) < a+ yields a+) _ (a+0))
(a+) = a+
(n and by induction that a(n) < a+ for all n . Thus , limn-4oo a ) < a+. To prove the converse inequality, we use an argument similar to the proof of Proposition VI.2.4. Let Fn = {T1 + • • • + Tn+1 > r+}be the event that {my} has at most n arrivals in [T1, 7-+ ], and let &+".) = P(mTl = i; Fn ). Obviously, &+n) T a+, so to complete the proof it suffices to show that &+n) < a+) for all n. For n = 0, both quantities are just 0 . Assume the assertion shown for n - 1. Then each subexcursion of {St+Tl - ST,-} can contain at most n - 1 arrivals (n arrivals are excluded because of the initial arrival at time T1 ). It follows that on Fn the feedback to {mz} after each ladder step cannot exceed &+n-1) so that a+ n) < a f ^ e(T+ t&+ -1))YA(dy) o
<
a is e(T+t«+-1')YA(dy) _ w (a+-1 )) = a+n). 0
0 We next give an alternative algorithm, which links together the phase-type setting and the classical complex plane approach to the renewal model (see further the notes). To this end, let F be the distribution of U1 - T1. Then F[s] = a(-sI - T)-'t • A[-s] (4.4) whenever EeR(S)U < oo. However, (4.4) makes sense and provides an analytic continuation of F[•] as long as -s ¢ sp(T). Theorem 4.5 Let s be some complex number with k(s) > 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
CHAPTER VIII. MATRIX-ANALYTIC METHODS
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 =
5. MARKOV-MODULATED INPUT
(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
CHAPTER VIII. MATRIX-ANALYTIC METHODS
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
5. MARKOV-MODULATED INPUT
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
CHAPTER VIII. MATRIX-ANALYTIC METHODS
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
CHAPTER VIII. MATRIX-ANALYTIC METHODS
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
CHAPTER VIII. MATRIX-ANALYTIC METHODS
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), O<- tl < t2 < u. Define further vi(u) as the probability that the risk process starting from RD = u downcrosses level u for the first time in phase i. Note that in general >iEE 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
CHAPTER VIII. MATRIX-ANALYTIC METHODS
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<7 I
where o, denotes the time of downcrossing level u . Then P(B,) is the tail of a (defective) random variable so that P(Bv) -+ 0 as v -4 oo, and similarly P"(Bv) - ^ 0. Since the processes Rt and Rt coincide under level B,,, then P(A n Bv) _ P"(A n BV'). Now since both P(A n Bv) -3 0 and P"(A n Bv) -- 0 as v -+ 00 we have
P(A) -P"(A) = P(AnBv)+P(AnBv) -P"(AnB,,) -P"(AnBv) = P(AnB,)-P"(AnB,)
-+ 0 ❑
as v -+ oo. From Section 3, we have p(r) = p =
vi (u) -0aTe;, P
which implies that v, (v) is given by the r.h.s. of (7.5). Thus, we can first for a given v solve (7.4) backwards for {va (t)}v>t>o, starting from v"(v) = -,i7rT-1/p. This yields v, (u) for any values of u and v such that u < v.
248
CHAPTER VIII. MATRIX-ANALYTIC METHODS
Next consider a sequence of solutions obtained from a sequence of initial values {v; (u)},, where, say, v = u, 2u, 3u etc. Thus we obtain a convergent sequence of solutions that converges to {vi(t)}u>t>o• Notes and references The exposition is based upon Asmussen & Bladt [30] which also contains numerical illustrations. The algorithm based upon numerical solution of a Volterra integral equation (Remark VII.1.9, numerically implemented in Schock Petersen [288]) and the present one based upon differential equations require both discretization along a discrete grid 0, 1/n, 2/n,.... However, typically the complexity in n is 0(n2) for integral equations but 0(n) for integral equations. The precision depends on the particular quadrature rule being employed. The trapezoidal rule used in [288] gives a precision of 0(n 3), while the fourth-order Runge-Kutta method implemented in [30] gives 0(n-5).
7b Two-step premium rules We now assume the premium function to be constant in two levels as in VII.1a, p(r)
P, r v. (7.6)
We may think of process Rt as pieced together of two standard risk processes RI and Rte with constant premiums p1, p2, such that Rt coincide with RI under level v and with Rt above level v. Let ii'( u) = a+'ie(T+ta +^)"e denote the ruin probability for R't where a+ = a+i) = -laT-1/pi, cf. Corollary 3.1. We recall from Propositon VII.1.10 that in addition to the O'(•), the evaluation of Vi(u) requires q(u) = 1 - zp1(u)/(1 - z51(v)), 0 < u < v, which is available since the z/i'(.) are so, as well p1(u), the probability of ruin between a and the next upcrossing of v, where v = inf It > 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)
CHAPTER VIII. MATRIX-ANALYTIC METHODS
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<0<1. For further examples, see I.2b. The definition b[s] = oo for all s > 0 of heavy tails is too general to allow for a general non-trivial results on ruin probabilities, and instead we shall work within the class S of subexponential distributions . For the definition , we require that B is concentrated on (0, oo ) and say then that B is subexponential (B E S) if 251
252 CHAPTER IX. HEAVY TAILS
B*2\ 2, B(x)
Here B*2 is the convolution square, that is, the distribution of independent r.v.'s X1, X2 with distribution B. In terms of r.v.'s, (1.1) then means P(X1 +X2 > 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-^oo
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 <=v) f (x - y)Ai(dy) v Ai o - ajB(x)Ai(v) = ajB( x)(1+o„(1))
(j = 3 - i). Since P(X1+X2 > 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 <,3 < 1. Then b(x) = Ox0-le-xp, a(x) = ax0-1. Thus A(x) is everywhere decreasing, and exa(x)b(x) = (3x0-1e-(1-0)x9 is integrable. Thus, the DFR Weibull distri❑ bution is subexponential. Example 1.15 In the lognormal distribution, x - e-009x-v)2/2a2/(x 2irv2) logx ( ) 't (-(logx - U) /or)
v 2x
This yields easily that ex,`(x)b(x) is integrable. Further, elementary but tedious calculations (which we omit) show that A(x) is ultimately decreasing. Thus, the ❑ lognormal distribution is subexponential. In the regularly varying case, subexponentiality has alrady been proved in Corollary 1.12. To illustrate how Proposition 1.13 works in this setting, we first quote Karamata's theorem (Bingham, Goldie & Teugels [66]): Proposition 1.16 For L(x) slowly varying and a > 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-.
CHAPTER IX. HEAVY TAILS
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
CHAPTER IX. HEAVY TAILS
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^ <
0}
be the first descending ladder epoch, G_(A) = P(S,y_ E
A) the descending ladder height distribution (IIG -II = 1 because of PB < P A) and let PG_ be the mean of G_. Lemma 3 .3 G+ (x) - FI(x) /IPG_I, x -+
oo.
Proof Let R+(A) = E E'+ -' I(S,(,d)) E A) denote the pre-19+ occupation measure and let and U_ = Eo G'_" be the renewal measure corresponding to G_. Then 0 0 F( x - y) R+(dy ) _ j (x_y)U_(dY) G+ (x) =
J
00
00
(the first identity is obvious and the second follows since an easy time reversion argument shows that R+ = U_, cf. A.2). The heuristics is now that because of (b), the contribution from the interval (- N, 0] to the integral is O(F(x)) = o(FI(x)), whereas for large y , U_ (dy) is close to Lebesgue measure on (- oo, 0] normalized by IPG_ I so that we should have to G+(x) - 1
IPG_ I
/
F(x - y) dy = 1 Pi (X) oo
IPG_ I
263
3. THE RENEWAL MODEL
We now make this precise. If G_ is non-lattice, then by Blackwell 's renewal theorem U_ (-n - 1, -n] -+ 1/I µG_ I. In the lattice case, we can assume that the span is 1 and then the same conclusion holds since then U-(-n - 1, -n] is just the probability of a renewal at n. Given e, choose N such that F(n - 1)/F(n) < 1 + e for n > 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
CHAPTER IX. HEAVY TAILS
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 `one 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 <- X2 < ... such that
{SXo+t
-
SXo}0
- Sxi}0
(viewed as random elements of the space of D-functions with finite lifelengths) are i.i.d. and the distribution of {Sxk+t - Sxk}o
Figure 4.1 Note that no specific sample path structure of {St} (like in Fig. 4.1) is assumed. We return to this point in Example 4.4 below. Define
S.
= Sx, +1,
M,1
=
max
S,
M* =
max
S;,,
M = sup St.
k=0,...,n n=0,1,... t>0
The idea is now to observe that in the zero-delayed case, {Sn}n=o,1,... (corresponding to the filled circles on Fig. 4.1 except for the first one) is a random walk. Thus the assumption ,F*(X) = P0(Si > x) ,., G(x)
(4.1)
CHAPTER IX. HEAVY TAILS
266
for some G such that both G E S and Go E S makes (3.3) applicable so that F(M* > u) 141 F*(u), u -p 00. (4.2) Imposing suitable conditions on the behaviour of {St} within a cycle will then ensure that M and M* are sufficiently close to be tail equivalent. The one we focus on is Fo (Mix) > x) ,,, Fo(Si > X), (4.3) where
Mnx) = sup
Sxn +t - Sxn =
o
sup Sxn+t - S.* -i o
Since clearly M(x) > Sl , the assumption means that Mix) and Sl are not too far away. See Fig. 4.2.
--------------N N Xi=0
N Figure 4.2 Theorem 4.1 Assume that (4.1) and (4.3) hold. Then '00 (u) = Fo(M > u) ,,, jF11 F* (U). Proof Since M > M*, it suffices by (4.2) to show F(M* > u) > 1. (4.4) liminf u->oo F(M > u)
4. MODELS WITH DEPENDENT INPUT
267
Define 79* (u) = inf {n = 1 , 2, ...: Sn > u} , /3(u)
= inf{n=1,2,...: S;,+Mn+1>u}
(note that {M> u} = {3(u) < oo}). Let a > 0 be fixed. We shall use the estimate
Po(M > u) Miu^+ 1 < a) = o(Po (M > u)) (4.5) which follows since Po (M > u, MW O(u)+1 < a) IN
(
U
A1; E (u - a, u)}
n=1
< P(M* E (u - a, u))/P(M* = 0) = o(Po(M* > u)). Given e > 0, choose a such that Po(Si > x ) > (1 - e)Po (MMX> > x), x > a. Then by Lemma 3.4, Po(M* > u)
- Po (M* > u, S;(u) - S;( u)-1 > a) 00 1: Po(MnaV(u-Sn*)) n=1
00 > (1-E)EPo(MnaV(u
-S;,
))
n=1 00
> (1 - E) Po ( n
max St u, Mn+l > a V (u - Sn 0
( 1 - e)Po (M > u, M^xu)+l > a) - (1 - e)Po (M > U). Letting first u -+ oo and next e . 0 yields (4.4).
❑
Under suitable conditions , Theorem 4.1 can be rewritten as (4.6)
00 (U)
1 p pBo(u)
where B is the Palm distribution of claims and p - 1 = limti00 St/t. To this end, assume the path structure Nt
St = EUi-t+Zt i=1
CHAPTER IX. HEAVY TAILS
268 with {Zt} continuous, independent of {>
N` U;} and satisfying Zt/t
a4'
0. Then
the Palm distribution of claims is N,
B(x) = E N Eo
I( U1 < x) .
x
0
(4.8)
i=1
Write ,Q = EoNx/EoX. Corollary 4.2 Assume that {St} is regenerative and satisfies (4.7). Assume further that (i) both B and Bo are subexponential; (ii) EozNX < oo for some z > 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
Thus Theorem 4.1 is in force, and the rest is just rewriting of constants: since p = 1+tlim St = 1+ . oX (see Proposition A1.4), we get 00 (u)
1
J
Po(Sl > 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 supo
x-+oo
B2(x)
G(x)
= ci
for some distribution G such that both G and the integrated tail fx°O G(y) dy are subexponential , and for some constants ci < oo such that cl + • • • + c, > 0. The average arrival rate /3 and the Palm distribution B of the claim sizes are given by P
Q
ir i/i,
=
P
B = >2 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.
CHAPTER IX. HEAVY TAILS
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
where w(Fc) = inf It > 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
w(0, oo) = r(0)
x=
St
y (a)
Figure 5.1 The sample path in (a) is the excursion of {St} conditioned to start in x = 0 and to end in y > 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)
CHAPTER IX. HEAVY TAILS
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<)-1 = Y)
S S and Qx y( ipil ... in) = Qx,J (i.in-1 ... 2p). To show Q y x (i0 i 1 ... 2n) = Qx,TI( 2n2n _1 ... 2p) when 20, 21 , ... , in E
in = x, note first that Pt'
(R l = il, ... , Rn = in = x; Rn+1 E FC) TioilTili2...rin_1in
E Txj jEFC
m21 s2120 m2252221
m in Ssn n-1
mjSjx
m2p mil min-1 jEF`
1 Sinin _ 1
...
Si l io
MY
Mx
E mjSjx.
jEF^
Thus
Sxin-1 ... Silt' Qx(ioii ... in) = oo E Sxik_1 ... .gilt' k=1 ii .....ik-1EF
Similarly but easier Sxin_1 ... Si11 S 1 ... i0) Q x,y(inin _
E SYj jEF`
00 Sxik _1 ... Silt' E SO k=1 i1, ...,ik_1EF Sxin_1 ... Si1y 00
E E 5xik_ 1 ... Si1y k=1 i1 ,...,ik_1EF
jEF°
F,
20 =
y,
5. FINITE-HORIZON RUIN PROBABILITIES 275
5b The time to ruin Our approach to the study of the asymptotic distribution of the ruin time is to decompose the path of { St} in ladder segments . To clarify the ideas we first consider the case where ruin occurs already in the first ladder segment , that is, the case r (O) < oo, ST(o) > 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
CHAPTER IX. HEAVY TAILS
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) .
CHAPTER IX. HEAVY TAILS
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) - /3Ea B(u). The heuristic motivation is the usual in the heavy-tailed area, that MQ becomes large as consequence of one big jump. The form of the result then follows by noting that the process has mean time Ea to make this big jump and that it then occurs with intensity /3B(u). More precisely, one expects the level y form which the big jump occurs to be 0(1); the probability that is exceeds u is then B(u - y) - B(u). The rigorous proof is, however, non-trivial and we refer to Asmussen [22]. Proof of Theorem 6.1. We will show that the stationary density f (x) of {Vt} satisfies f (x) /B(x) r(x) We then get V,(u) = P(V > u) = f f (y) dy - (3 u
and the result follows.
J
u
B(y) dy , p(Y)
CHAPTER IX. HEAVY TAILS
280
Define D(u) as the steady-state rate of downcrossings of {Vt} of level u and Da (u) as the expected number of downcrossings of level u during a cycle. Then D(u) = f(u)p(u) and, by regenerative process theory, D(u) = DQ(u)/µ. Further the conditional distribution of the number of downcrossings of u during a cycle given Mo > 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
= I('r(u) < T).
The situation is more intricate for the infinite horizon ruin probability 0(u). The difficulty in the naive choice Z = I(T(u) < oo) is that Z can not be simulated in finite time: no finite segment of {St} can tell whether ruin will ultimately occur or not. Sections 2-4 deal with alternative representations of Vi(u) allowing to overcome this difficulty.
lb Variance reduction techniques The purpose of the techniques we study is to reduce the variance on a CMC estimator Z of z, typically by modifying Z to an alternative estimator Z' with EZ' = EZ = z and (hopefully) Var(Z') < Var(Z). This is a classical area of the simulation literature, and many sophisticated ideas have been developed. Typically variance reduction involves both some theoretical idea (in some cases also a mathematical calculation), an added programming effort, and a longer CPU time to produce one replication. Therefore, one can argue that unless Var(Z') is considerable smaller than Var(Z), variance reduction is hardly worthwhile. Say that Var(Z') = Var(Z)/2. Then replacing the number of replications N by 2N will give the same precision for the CMC method as when simulating N' = N replications of Z', and in most cases this modest increase of N is totally unproblematic. We survey two methods which are used below to study ruin probabilities, conditional Monte Carlo and importance sampling. However, there are others which are widely used in other areas and potentially useful also for ruin probabilities. We mention in particular ( regression adjusted) control variates and common random numbers. Conditional Monte Carlo Let Z be a CMC estimator and Y some other r . v. generated at the same time as Z. Letting Z' = E[Z I Y], we then have EZ = EZ = z, so that Z' is a candidate for a Monte Carlo estimator of z. Further, writing
Var(Z) = Var(E [Z I Y]) + E(Var[Z I Y])
283
1. GENERALITIES
and ignoring the last term shows that Var(Z') < Var(Z) so that conditional Monte Carlo always leads to variance reduction. Importance sampling The idea is to compute z = EZ by simulating from a probability measure P different from the given probability measure F and having the property that there exists a r.v. L such that z = EZ = E[LZ]. (1.3) Thus, using the CMC method one generates (Z1, L1),. .. , (ZN, LN) from P and uses the estimator N
zrs = N > L:Zj i=1
and the confidence interval zrs f
1.96 sis v^
N 2 1 where srs =N
j(LiZi
2 1 N 2 2 2
- zrs) =
i=1
N > Lt Zi
- zrs.
i=1
In order to achieve (1.3), the obvious possibility is to take F and P mutually equivalent and L = dP/dP as the likelihood ratio.
Variance reduction may or may not be obtained: it depends on the choice of the alternative measure P, and the problem is to make an efficient choice. To this end, a crucial observation is that there is an optimal choice of P: define P by dP/dP = Z/EZ = Z/z, i.e. L = z/Z (the event {Z = 0} is not a concern because P(Z = 0) = 0). Then z Var(LZ) = E(LZ)2 - [E(LZ)] = E Z2 Zz - E [Z Z]2 = z2 - z2 = 0. Thus, it appears that we have produced an estimator with variance zero. However, the argument cheats because we are simulating since z is not avaliable analytically. Thus we cannot compute L = Z/z (further, it may often be impossible to describe P in such a way that it is straightforward to simulate from
P). Nevertheless, even if the optimal change of measure is not practical, it gives a guidance: choose P such that dP/dP is as proportional to Z as possible. This may also be difficult to assess , but tentatively, one would try to choose P to make large values of Z more likely.
CHAPTER X. SIMULATION METHODOLOGY
284
1c Rare events simulation The problem is to estimate z = P(A) when z is small , say of the order 10-3 or less. I.e., Z = I(A) and A is a rare event. In ruin probability theory, A = {T(u) < T} or A = {r(u) < oo} and the rare events assumption amount to u being large, as is the case of typical interest. The CMC method leads to a variance of oZ = z(1 - z) which tends to zero as z ^ 0. However, the issue is not so much that the precision is good as that relative precision is bad: oZ z(1 - z) 1 -> 00.
Z
z
V5
In other words , a confidence interval of width 10 -4 may look small, but if the point estimate z is of the order 10-5, it does not help telling whether z is of the magnitude 10-4, 10 - 5 or even much smaller . Another way to illustrate the problem is in terms of the sample size N needed to acquire a given relative precision , say 10%, in terms of the half-width of the confidence interval. This leads to the equation 1.96oz /(zV) = 0.1, i.e.
N - 100 - 1.96 2Z ( 1 - z) 100-1.96 2 z2
z
increases like z-1 as z .0. Thus, if z is small, large sample sizes are required. We shall focuse on importance sampling as a potential (though not the only) way to overcome this problem. The optimal change of measure ( as discussed above) is given by
P(B) = E
[Z i;B ]
= iP(AB) = P(BIA). z
I.e., the optimal P is the conditional distribution given A. However, just the same problem as for importance sampling in general comes up: we do not know z which is needed to compute the likelihood ratio and thereby the importance sampling estimator, and further it is usually not practicable to simulate from P(•IA). Again, we may try to make P look as much like P(•IA) as possible. An example where this works out nicely is given in Section 3. Two established efficiency criteria in rare events simulation are bounded relative error and logarithmic efficiency. To introduce these, assume that the rare event A = A(u) depends on a parameter u (say A = {r(u) < oo}). For each u, let z(u) = P(A(u)), assume that the A(u) are rare in the sense that z(u) -* 0, u -+ oo, and let Z(u) be a Monte Carlo estimator of z(u). We then
2. SIMULATION VIA THE POLLACZECK-KHINCHINE FORMULA
285
say that {Z(u)} has bounded relative error if Var(Z(u))/z(u)2 remains bounded as u -3 oo. According to the above discussion, this means that the sample size N = NE(u) required to obtain a given fixed relative precision (say a =10%) remains bounded. Logarithmic efficiency is defined by the slightly weaker requirement that one can get as close to the power 2 as desired: Var(Z(u)) should go to 0 as least as fast as z(u)2-E, i.e. Var(Z(u)) hm sup U-+00 z (u) 2-E
<
oo
(1.4)
for any e > 0. This allows Var(Z(u)) to decrease slightly slower than z(u)2, so that NE (u) may go to infinity. However, the mathematical definition puts certain restrictions on this growth rate, and in practice, logarithmic efficiency is almost as good as bounded relative error. The term logarithmic comes from the equivalent form - log Var(Z(u)) lim inf > 2 u-+oo - log z(u)
of (1.4). Notes and references For surveys on rare events simulation, see Asmussen & Rubinstein [45] and Heidelberger [190].
2 Simulation via the Pollaczeck-Khinchine formula For the compound Poisson model, the Pollaczeck-Khinchine formula III.(2.1) may be written as V) (u) = P(M > u), where M = X1 + • • • + XK, where X1, X2, ... are i.i.d. with common density bo(x) = B(x)/µB and K is geometric with parameter p, P(K = k) = (1 - p)pk. Thus, O (u) = z = EZ, where Z = I(M > u) may be generated as follows: 1. Generate K as geometric, F(K = k) = (1 - p)pk.
2. Generate X1, ... , XK from the density bo(x). Let M - X1 + + XK. 3. If M > u, let Z +- 1. Otherwise, let Z +- 0. The algorithm gives a solution to the infinite horizon problem , but as a CMC method , it is not efficient for large u . Therefore , it is appealing to combine with some variance reduction method . We shall here present an algorithm developed by Asmussen & Binswanger [ 271, which gives a logarithmically efficient estimator
286
CHAPTER X. SIMULATION METHODOLOGY
when the claim size distribution B (and hence Bo) has a regularly varying tail. So, assume in the following that Bo(x) - L(x)/x`' with a > 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
CHAPTER X. SIMULATION METHODOLOGY
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) <
yu]
-27yu .
Hence by (4.1), lim inf u--oo
- log Var(Z(u)) _ - log Var(Z(u)) l im of .yy> 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 =
CHAPTER X. SIMULATION METHODOLOGY
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)Ei`av re-av^(u)+(T(++)(U)
yu - o.yu1/2 <1 T(u) < yu
r > 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
'%(u) = P I info Rt < 0) = P(VV > u), (5.1) where the identity for Vi(u) requires that Vt has a limit in distribution V. In most of the simulation literature (say in queueing applications), the object of interest is {Vt} rather than {Rt}, and (5.1) is used to study Voo by simulating {Rt} (for example, the algorithm in Section 3 produces simulation estimates for the tail P(W > u) of the GI/G/1 waiting time W). However, we believe that there are examples also in risk theory where (5.1) may be useful. One main example is {Vt} being regenerative (see A.1): then by Proposition A1.3, zi(u) = INV. > u) = -E f I(VV > u) dt 0
(5.2)
293
5. REGENERATIVE SIMULATION
where w is the generic cycle for {Vt}. The method of regenerative simulation, which we survey below , provides estimates for F ( V,, > u) (and more general expectations Eg(V... )). Thus the method provides one answer on to how to avoid to simulate { Rt} for an infinitely long time period. For details , consider first the case of independent cycles . Simulate a zerodelayed version of {V t } until a large number N of cycles have been completed.
For the ith cycle, record Zi'i = (Z1'),
Z2'>) 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
CHAPTER X. SIMULATION METHODOLOGY
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
CHAPTER X. SIMULATION METHODOLOGY
Taking expectation, differentiating w.r.t. j3 and letting flo = 0, we get
1 M(u) 00(u)
= E
(_Ti)I(T(U)<)
E [(M(u) - T(u)) I(T(u) < co)
]
.
To resolve the infinite horizon problem , change the measure to FL as when simulating tp(u). We then arrive at the estimator
ZZ(u) =
(M(u) - T(u)) e-7ue--rVu)
for ?P,3 (u) (to generate Zp (u), the risk process should be simulated with parameters ,3L, BL). We recall (Proposition 111.9 . 4) that V5,3 (u) is of the order of magnitude ue-7u. Thus, the estimation of z(ip(u) is subject to the same problem concerning relative precision as in rare events simulation . However, since ELZp(u)2
we have
\ 1
< (M(U) _T(u)
VarL(ZQ(u)) ZO(u)2
)
2 a-2ryu = O(u2)e-27u,
O(u2)e-2
-yu - 0(1)
u2e-2ryu
so that in fact the estimator Zf(u) has bounded relative error. 0 Notes and references A survey of IPA and references is given by Glasserman [161] (see also Suri [358] for a tutorial), whereas for the SF method we refer to Rubinstein & Shapiro [312]. Example 6.1 is from Asmussen & Rubinstein [46] who also work out a number of similar sensitivity estimators, in part for different measures of risk than ruin probabilities, for different models and for the sensitivities w.r.t. different parameters.
There have been much work on resolving the difficulties associated with IPA pointed out above. In the setting of ruin probabilities, a relevant reference is VazquezAbad [374].
Chapter XI
Miscellaneous topics 1 The ruin problem for Bernoulli random walk and Brownian motion. The two-barrier ruin problem The two-barrier ruin probability 0,,(u) is defined as the probability of being ruined (starting from u) before the reserve reaches level a > 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]
= z°P
(RT ( u,a)
= 0) + zap ( R,r(u,a)
= z°Va(u) + za(1 -
Y,a(u)),
1. RANDOM WALK; BROWNIAN MOTION; TWO BARRIERS 299 and solving for 4/la(u) yields t/ia(u) = (za - zu)/(za - 1).
If 9 = 1/2, (1.2) is trivial (z = 1). However, {R,,} is then itself a martingale and we get in a similar manner
u = ER° = ER ra( u) = 0 •
Y'a (u)
+ all -
a-u Y'a( u)),
pa( u)
_
u
Corollary 1.2 For a Bernoulli random walk with 9 > 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<0, thenz1 (u)=1.
(1.5)
CHAPTER XI. MISCELLANEOUS TOPICS
300
❑
Proof Let a -* oo in (1.4).
The reason that the calculations work out so smoothly for Bernoulli random walks and Brownian motion is the skip-free nature of the paths, implying R(u,a) = a on {r (u,a) = -r+ (a)} and similarly for the boundary 0. For most standard risk processes , the paths are upwards skip-free but not downwards, and thus one encounters the problem of controlling the undershoot under level 0. Here is one more case where this is feasible: Example 1.5 Consider the compound Poisson model with exponential claims (with rate, say, 5). Here the undershoot under 0 is exponential with rate 5, and hence e-7u =
=
Ee-7Ro E [e-7R(,, , a) I R(u a ) < 0] P (R(u ,a) < 0) + e-7°P (R(u,a) = a)
5 y =
P (R (u,a ) < 0) + e -7aF ( R (u,a) = a )
Ic 5-ry 'pa(u)
+ e -' ° ( 1 - 7/la(u)).
Using y = 6 - 3, we obtain 'Oa a-7u - e-7a (u) = 6
/0 - e-7a
Again , letting a -* oo yields the standard expression pe-7u for .0 (u) (where ❑ p =,616), valid if p < 1 (otherwise , 7/'(u) = 1). However, passing to even more general cases the method quickly becomes unfeasible (see, however, VIII.5a). It may then be easier to first compute the one-barrier ruin probability O(u): Proposition 1.6 If the paths of {Rt} are upwards skipfree and 7//(a) < 1, 0. (u) _ O(u) - 0(a) 0 < u < a. 1 - vi(a) Proof By the upwards skip-free property, 7O(u) = 7/la(u) + (1 - +^a(u))^(a) If 7k(a) < 1, this immediately yields (1.7).
(1.7)
301
1. RANDOM WALK; BROWNIAN MOTION; TWO BARRIERS
Note thas this argument has already been used in VII.1a for computing ruin probabilities for a two-step premium function. We now return to Bernoulli random walk and Brownian motion to consider finite horizon ruin probabilities. For the symmetric (drift 0) case these are easily computable by means of the reflection principle: Proposition 1.7 For Brownian motion with drift 0, 0(u, T ) = P(T(u) < T ) = 241, (i). (1.8) Proof In terms of the claim surplus process { St} = {u - Rt}, we have ili(u,T) P(MT > u) where MT = maxo u in time T - r(u). Hence P(MT>u,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
CHAPTER XI. MISCELLANEOUS TOPICS
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
CHAPTER XI. MISCELLANEOUS TOPICS
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)
<
e-7yu,
1 y < k (y), (2.3)
< e -7yu, y > - (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)
<
00]
.
Let H(dt, dr) denote the conditional distribution of (T(u), RT(u)_) given r(u) < oo. A claim leading to ruin at time t has c.d.f. (B(y) - B(r))/B(r), y > 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))•
CHAPTER XI. MISCELLANEOUS TOPICS
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
CHAPTER XI. MISCELLANEOUS TOPICS
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 u} and o(u) = P('r(u) < oo). Assume that there exists 'y, e > 0 such that (i) Kn (0) = log Ee°Sn is well-defined and finite for 'y - e < 8 < -y + e; (ii) lim supn.,,. Ee9X n < oo for -e < 0 < e; (iii) #c (8) = limn, n Icn(0) exists and is finite for ry - e < 8 < y + e; (iv) tc(ry) = 0 and r. is differentiable at ry with 0 < K'(-y) < 00. Then i/.'(u) )Ng a-"u.
For the proof, we introduce a change of measure for X1, ... , Xn given by Fn(dxl,...,dxn) = 05n-Kn(7)Fn(dx1,...,dxn)
where Fn is the distribution of (X1i ... , Xn) and sn = x1 + • • • + xn (note that the r.h.s. integrates to 1 by the definition of Icn). We further write µ = tc'(ry). We shall need: Lemma 3 .3 For each i > 0, there exists z E (0, 1) and no such that Sn
n for n n0.
- p > 7 < zn,
>7 < zn
Pn Sn-1 - /^
3. LARGE DEVIATIONS
309
Proof Let 0 < 9 < e where a is as in Theorem 3.2. Clearly, P n(Sn/n > {c+77) < e
no(µ
+7)- r (7) +n)Enees n n = e- ne(µ +n)elcn(B
limsup 1 log Pn (Sn/n > µ + 17) < ic(9 + ry) - Bµ - 077 n-^oo n
and by Taylor expansion and (iv ), the r . h.s. is of order - 91) + o(O ) as 0 J. 0, in particular the r. h.s. can be chosen strictly negative by taking 9 small enough. This proves the existence of z < 1 and no such that Pn (Sn/n > µ.+r7) < zn for n > no. The corresponding claim for Pn(Sn/n < µ - 77) follows by symmetry (note that the argument did not use µ > 0). This establishes the first claim of the lemma , for Sn. For Sn-1i we have Fn(Sn -1/n > µ+r7) <
e-ne(µ+ 1?)EneeS„-1 = e-ne ( µ+n)EneeSn-eX„ e-no(µ +n) Ee(e+7)Sn -ex„ -wn (7)
< e- n e(µ +o)-w"(7) [Eep(B
+7)Sn]1 /p
[Ee-goX,.]1/q
= e- ne(p+ 17)- Kn(7)e'n (p(O +7))/p I Ee -geXn]1/q where we used Holder's inequality with 1/p+ 1/q = 1 and p chosen so close to 1 and 0 so close to 0 that j p(0 +,y) - 71 < e and jq9j < e. Since I Ee-qOX „ ] 1/q is bounded for large n by (ii), we get lim sup 1 log Pn (Sn-1 /n > 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
CHAPTER XI. MISCELLANEOUS TOPICS
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)
(3.6)
Pn \
for some z < 1 and all n > 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 \
<
1u(1-6)/µ1 00 1 zn < e-7u E Z n/2 < e--(U xn/2 E
e--Yu =
1 - zl/z
n=n(6)+1 n=0
[u(1 +6)/µJ
13
1u (1 +6) /µJ ekn(7)
E
C"
<
< e'
Lu(1-6)/µJ+l1
< e-7U
C
26u
+1 I
en6
Yu l
u(1-6)/lij+1
e6u(1+6)/µ
(3.10)
`p /
Finally, 00
I4
<
E F(Sn_1 < u, S. > u) Lu(1+6) /µJ +l 00 )^n 'YSn+kn (7) ; Sn-1 C U, Sn > U] [ e(u(1+6)/µJ+l
< e--Yu
e-n('Y ) fPn (u(1+6)/µJ+1
-7u r 00 e L^
1 n x n / 2x
(I Sn n 1 - ' 1 + b) e-7u x 1 /2
(3.11)
[u(1+6)/µJ+1 1 -
Thus an upper bound for z/'(u) is n(6) e-'Yu
eKn (7) + 2
1-
n=1
zl /2
+ (28U + 1) e6u(1+6)/µ Fi
and using (i), we get lim sup log u-/00
Letbl0.
O (U) < -y + b(1 + b) U
❑
CHAPTER XI. MISCELLANEOUS TOPICS
312
The following corollary shows that given that ruin occurs, the typical time is u/rc'(7) just as for the compound Poisson model, cf. IV.4. Corollary 3.4 Under the assumptions of Theorem 3.2, it holds for each b > 0 that 0(u) 1' g F(T(u) E (u(1 - b)/i(7), u(1 + b)/i(7))
Proof Since V,(u) = I1+I2+I3+I4'^ e-ry( u), 13 = P(T (u) E (u(1 -b)l^ (7),u(1+b)/rc'(7)), it suffices to show that for j = 1, 2, 4 there is an aj > 0 and a cj < oo such that Ij < c3e- 7' a-"ju. For 14, this is straightforward since the last inequality in (3.11 ) can be sharpened to x
[u(1+6)/µJ /2
4
1 - z 1/z
For I1, I2, we need to redefine n(b) as L,3ui where ,Q is so small that w = 1 - 4/3rc'(-y) > 0. For 12, the last steps of (3.9) can then be sharpened to x LQuJ /2
I2
< e-7u
1 - xl/2
to give the desired conclusion. For I,, we replace the bound P(Sn > u ) < 1 used in (3.8) by P(S,,
> u)
<
e-"'
E eIsn
=
e-ctueKn (a+'Y)-Kn(7)
where 0 < a < e and a is so small that r. (7 + a) < 2arc'(7). Then for n large, say n n1, we have rcn (a + 7) < 2n^c(7 + a) < 4narc' (7). Letting c11 = maxn
Il
Lou] E exp {-( 7 + a)u + Kn(a +7)} n=1
Lou] exp {-(-y + a)u} { 111 + exp {4narc'(7)} n=1
exp {-('y + a)u} c1 exp {4/3uarc'(7)} = clewhere a1 = aw.
ryue-«iu
, ❑
3.,-LARGE DEVIATIONS
313
Example 3 .5 Assume the Xn form a stationary Gaussian sequence with mean p < 0. It is then well-known and easy to prove that Sn has a normal distribution with mean np and a variance wn satisfying 00
i lim -wn = wz = Var(X1 ) + 2 E Cov(Xl, Xk+l)
n-aoo n
k=1
provided the sum converges absolutely. Hence z
z\
2
z
nr-n(9) _ n Cn0p+BZn/ -* ,(O) = 9µ+02 for all 9 E R, and we conclude that Theorem 3 . 2 is in force with -y = -2p/wz. 11
Inspection of the proof of Theorem 3.2 shows that the discrete time structure is used in an essential way. Obviously many of the most interesting examples have a continuous time scale. If {St}t> 0 is the claims surplus process, the key condition similar to (iii), (iv) becomes existence of a limit tc(9) of tct(9) _ log Ee8S° It and a y > 0 with a(y) = 0, r.'(-y) > 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
(3.13)
One would expect this to hold in considerable generality, and in fact, criteria are given in Duffield & O'Connell [124]. To verify these in concrete examples may well present considerable difficulties, but nevertheless, we shall give two continuous time examples and tacitly assume that this can be done. The reader not satisfied by this gap in the argument can easily construct a discrete time version of the models! The following formula (3.14) is needed in both examples . Let {Nt}t>0 be a possibly inhomogeneous Poisson process with arrival rate ,3(s) at time s. An event occuring at time s is rewarded by a r.v. V(s) with m.g.f. 09(9). Thus the total reward in the interval [0, t] is
Rt = E V (Un) n: o„
CHAPTER XI. MISCELLANEOUS TOPICS
314 where the
an
are the event times. Then
rt
logEeOR° =
J0 /3(s)(^8(9) - 1) ds
(3.14)
(to see this , derive , e.g., a differential equation in t). Example 3.6 We assume that claims arrive according to a homogeneous Poisson process with intensity 0 , but that a claim is not settled immediately. More precisely, if the nth claim arrives at time a,, , then the payments from the company in [on, O'n +S] is a r .v. Un(s). Thus, assuming a continuous premium inflow at unit rate, we have
S, = U„ ( t - Q„) - t, n: o,
which is a shot-noise process. We further assume that the processes {U1(s)}8>0 are i. i.d., non-decreasing and with finite limits Un as s T oo ( thus, Un represents the total payment for the nth claim). We let ic (9) = 3(EeWU° - 1) - 0 and assume there are -y, e > 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
CHAPTER XI. MISCELLANEOUS TOPICS
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
CHAPTER XI. MISCELLANEOUS TOPICS
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)
4. THE DISTRIBUTION OF THE AGGREGATE CLAIMS
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)
CHAPTER XI. MISCELLANEOUS TOPICS
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)^o(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
^e-Q
,3n-i /3
= -Pn-1 n! n (n - 1)! n
4. THE DISTRIBUTION OF THE AGGREGATE CLAIMS
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
CHAPTER XI. MISCELLANEOUS TOPICS
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 ^I 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
CHAPTER XI. MISCELLANEOUS TOPICS
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
CHAPTER XI. MISCELLANEOUS TOPICS
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) <_ F2 (x) = P(Y2 < x) x < b 1=P(Y2<x) 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
CHAPTER XI. MISCELLANEOUS TOPICS
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
CHAPTER XI. MISCELLANEOUS TOPICS
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}o0 being independent of To, T1, ... , Tk and {Xt }o
334
APPENDIX
Proposition A1.3 Consider a regenerative process such that the cycle length distribution is non-lattice with p < oo. Then Xt -Di X,,,, where the distribution of X,,,, is given by Eg(Xoo) = 1 E0 f Ylg (Xt)dt. µ 0
(A.6)
If F is spread-out, then Xt - + X,, in total variation.
id Cumulative processes Let {Tn} be a renewal process with i.i.d. cycles (we allow a different distribution of the first cycle). Then {Zt}t^,0 is called cumulative w.r.t. {Tn} if the processes {ZT +t - ZT }0
then Zt /t a$• EU1/µ; (b) If in addition Var(Ul ) < oo, then (Zt - tEU1/µ)/f has a limiting normal distribution with mean 0 and variance Var(Ui) + (!)2Var (Yi)_ 2EU1 Cov(U1, Y1)
le Residual and past lifetime Consider a renewal process and define e ( t) as the residual lifetime of the renewal interval straddling t, i.e. fi (t) = inf {Tk - t : t < Tk}, and q(t) = sup It - Tk : t < Tk} as the age. Then {e(t)}, {i7(t)} are Markov with state spaces (0, oo), resp . [0, oo). If p = oo, then e (t) - oo (i.e. P(C ( t) < a) -4 0 for any a < oo) and ij (t) * oo. Otherwise , under the condition of Blackwell's renewal theorem, C(t) and ij (t) both have a limiting stationary distribution F0 given by the density F (x)/p. We denote the limiting r.v.'s by e, r,. Then it (ii, C), and we have: holds more generally that (rl(t), e(t ))
APPENDIX
335
Theorem A1.5 Under the condition of Blackwell's renewal theorem, the joint distribution of (rl, ^) is given by the following four equivalent statements: (a) P (77
> x, ^ > y) = 1 f
(z)dz; +Y
(b) the joint distribution of (ri, l:) is the same as the distribution of (VW, (1 V)W) where V, W are independent, V is uniform on (0, 1) and W has distribution Fw given by dFw/dF(x) = x/pF; (c) the marginal distribution of q is FO, and the conditional distribution of given 17 = y is the overshoot distribution R0(Y) given by FO(Y) (z) = Fo (y+z)/Fo(y); (d) the marginal distribution of ^ is FO, and the conditional distribution of ri given l; = z is Foz)
The proof of (a) is straightforward by viewing {(r,(t),^(t))} as a regenerative process, and the equivalence of (a) with (b)-(d) is an easy exercise. In IV.4, we used: Proposition A1.6 Consider a renewal process with µ < oo. Then fi(t)/t a4' 0 and, if in addition EYo < oo, EC(t)/t -+ 0. Proof The number Nt of renewal before t satisfies Nt/t a4' p. Hence for t large enough, we can bound e(t) by M(t) = max {Yk : k < 2t/p}. Since the maximum Mn of n i.i.d. r.v.'s with finite mean satisfies Mn/n a$• 0 (BorelCantelli), the first statement follows. For the second, assume first the renewal process is zero-delayed. Then Eo^(t) satisfies a renewal equation with z(t) _ E[Y1 - t; Yl > 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: Sn<0}, G+(x) = P(S,+ < x, -r+ < oo), G_(x) = P(ST_ < x,T_ < oo).
We call r+ (T_) the strict ascending (weak descending) ladder epoch and G+ (G_) the corresponding ladder height distributions.
APPENDIX
337
Probabilistic Wiener-Hopf theory deals with the relation between F, G+, G_, the renewal measures 00
00
n=0
n=0
U+=>G+, 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<j
u ,r- =n w=m i
Figure A.1
3
8
APPENDIX
Reversing the time points 0,1, ... , m it follows (see Fig. A.1) that P(Sj -Sn,.>0, 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
n=1 0 -
00 00 1: F(A - x)P(Sk < 0, 0 < k < n, Sn-1 E dx)
f0 f0
n=1
F(A - x)R+(dx),
-
APPENDIX
339
and the proof of (c) is similar. For (d), consider a fixed n and let Xk = Xn_k+l, Sk = X1 + • • • + Xk = Sn - Sn_k. Then for A C (-oo, 0], P(SnEA ,T+> n) = P(Sk < O,O
is the probability that n is a weak descending ladder point with Sn E A. Summing over n yields R+ (A) = U_ (A), and the proof of (e) is similar. ❑ Remark A2.2 In terms of m.g.f.'s, we can rewrite (a) as 1 - F[s] = (1 - 0+[s])(1 - G_[s]) (A.9) whenever F[s], 6+ [s], G_ [s] are defined at the same time; this holds always on the line its = 0, and sometimes in a larger strip. Since G+ is concentrated on (0, oo), H+ (s) = 1-G+[s] is defined and bounded in the half-plane Is : ERs < 0} and non-zero in Is: Rs < 01 (because IIG+lI _< 1), and similarly H_ (s) = 1 - G_ [s] is defined and bounded in the half-plane is : ERs > 01 and non-zero in Is : ERs > 0}. The classical analytical form of the Wiener-Hopf problem is to write 1 -.P as a product H+H_ of functions with such properties. ❑
Notes and references In its above discrete time version, Wiener-Hopf theory is only used at a few places in this book. However, it serves as model and motivation for a number of results and arguments in continuous time. E.g., the derivation of the form of G+ for the compound Poisson model (Theorem 11.6.1), which is basic for the Pollaczeck-Khinchine formula, is based upon representing G+ as in (b), and using time-reversion as in (d) to obtain the explicit form of R+ (Lebesgue measure). In continuous time, the analogue of a random walk is a process with stationary independent increments (a Levy process, cf. 11.4). In this generality of, there is no direct analogue of Theorem A2.1. For example, if {St} is Brownian motion, then T+ = inf It > 0 : St = 0} is 0 a.s., and G+, G_ are trivial, being concentrated at 0. Nevertheless, a number of related identities can be derived; see for example Bingham [65]. Another main extension of the theory deals with Markov dependence. In discrete time, there are direct analogues of Theorem A2.1; see e.g. the survey [15] by the author and the extensive list of references there. Again, such developments motivate the approach in Chapter VI on the Markovian environment model. The present proof of Theorem A2.1(a) is from Kennedy [228].
APPENDIX
340
3 Matrix-exponentials T e exponential eA of a p x p matrix A is defined by the usual series expansion 00 An eA
n=0
n!
he series is always convergent because A' = O(nk Ialn) for some integer k < p, ere A is the eigenvalue of largest absolute value, JAI = max {Jjt : µ E sp(A)} and sp(A) is the set of all eigenvalues of A (the spectrum). Some fundamental properties are the following: sp(eA) = {e' : A E sp(A)} (A.10) d dteAt = AeAt = eAtA
(A.11)
A f eAtdt = eA, _I 0
(A.12)
eA-'AO = A-le AA (A.13) henever A is a diagonal matrix with all diagonal elements non-zero. It is seen from Theorem VIII. 1.5 that when handling phase -type distributi ons, one needs to compute matrix -inverses Q-1 and matrix -exponentials eQt ( r just eQ ). Here it is standard to compute matrix-inverses by Gauss-Jordan el imination with full pivoting , whereas there is no similar single established a proach in the case of matrix -exponentials. Here are, however , three of the c rrently most widely used ones: xample A3.1 (SCALING AND SQUARING) The difficulty in directly applying t e series expansion eQ = Eo Q"/n! arises when the elements of Q are large.
hen the elements of Q"/n! do not decrease very rapidly to zero and may contribute a non-negligible amount to eQ even when n is quite large and very any terms of the series may be needed (one may even experience floating point overflow when computing Qn). To circumvent this, write eQ = (eK)m where = Q/m for some suitable integer m (this is the scaling step). Thus, if m is s fficiently large, Eo Kn/n! converges rapidly and can be evaluated without p oblems, and eQ can then be computed as the mth power (by squaring if
=
2).
0
APPENDIX
341
Example A3.2 (UNIFORMIZATION) Formally, the procedure consists in choosing some suitable i > 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 , J^O)_ = 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 =
41) 0 U1 + U2.
Then {Jt} has lifetime U1 + U2 , initial distribution a and phase generator T. 11 Example A5.7 (THE NEGATIVE BINOMIAL DISTRIBUTION) The most trivial special case of Example A5.6 is the Erlang distribution Er which is the convolution of r exponential distributions. The discrete counterpart is the negative binomial distribution with point probabilities
bk k1) (1 k = r,r + 1,.... r - 1 This corresponds to a convolution of r geometric distributions with the same parameter p, and hence the negative binomial distribution is discrete phase❑ type, as is seen by minor modifications of Example A5.6.
354
APPENDIX
Example A5.8 (FINITE MIXTURES) Let B1, B2 be phase-type with representations (E(1),a(1),T(1)), resp. (E(2),a(2),T(2)). Then the mixture B = 9B1 + (1 - O)B2 (0 < 0 < 1) 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 o'i
=IT
Oa;'), i E E(1) T 0 I (A.37) (1) (1 - 0)ai2), i E E(2) 0 T(2)
(in block-partitioned notation, this means that a = (Oa(1) (1 - 0)a(2))). A reduced phase diagram is
0a(1)
E(1) A
- 0)a(2)
E(2)
Figure A.3 In exactly the same way, a mixture of more than two phase-type distributions is seen to be phase-type. In risk theory, one obvious interpretation of the claim ❑ size distribution B to be a mixture is several types of claims. Example A5.9 (INFINITE MIXTURES WITH T FIXED) Assume that a = a(°) depends on a parameter a E A whereas E and T are the same for all a. Let B(") be the corresponding phase-type distribution, and consider B(") = fA B(a) v(da) where v is a probability measure on A. Then it is trivial to see that B(") is ❑ phase-type with representation (a("),T,E) where a(°) = fAa(a)v(da). Example A5.10 (GEOMETRIC COMPOUNDS) Let B be phase-type with representation (E, a, T) and C = EO°_1(1 - p)pn-1B*n. Equivalently, if U1, U2,... are i.i.d. with common distribution and N is independent of the Uk and geometrically distributed with parameter p, P(N = n) = (1 - p)pn-1, then C is the distribution of Ul + • • • + UN. To obtain a phase process for C, we need to restart the phase process for B w.p. p at each termination. Thus, a reduced phase diagram is
f
a
E
t
Figure A.4
APPENDIX
355
and C is phase-type with representation (E, a, T + pta). Minor modifications of the argument show that 1. If U1 has a different initial vector, say v, but the same T, then U1 +• is phase-type with representation (E, v,T + pta);
+UN
2. if B is defective and N + 1 is the first n with U„ = oo, then U1 + • • + UN is zero-modified phase-type with representation (a, T + ta, E). Note that this was exactly the structure of the lifetime of a terminating renewal ❑ process, cf. Corollary VIII.2.2. Example A5.11 (OVERSHOOTS) The overshoot of U over x is defined as the distribution of (U - x)+. It is zero-modified phase-type with representation (E,aeTx,T) if U is phase-type with representation (E, a, T). Indeed, if {Jt} is a phase process for U, then Jy has distribution aeTx. If we replace x by a r.v. X independent of U, say with distribution F, it follows by mixing (Example A5.9) that (U - X)+ is zero-modified phase-type with representation (E,aF[T],T) where
F[T] =
J0 "o eTx F(dx)
is the matrix m.g.f. of F, cf. Proposition VIII.1.7.
❑
Example A5 . 12 (PHASE-TYPE COMPOUNDS ) Let fl, f2.... be the point probabilities of a discrete phase-type distribution with representation (E, a, P), let B be a continuous phase-type distribution with representation (F, v, T) and C = F,,°,°_1 f„ B*?l. Equivalently, if U1, U2, ... are i. i.d. with common distribution B and N is independent of the Uk with P(N = n) = f,,, then C is the distribution of U1 + • • • + UN. To obtain a phase representation for C , let the phase space be E x F = {i j : i E E, j E F}, let the initial vector be a ® v and ❑ let the phase generator be I ® T + P ® (ta). Example A5 . 13 (MINIMA AND MAXIMA ) Let U1, U2 be random variables with distributions B1, B2 of phase-type with representations (E('),a(1),TWWW), resp. (E(2), a(2), T(2) ). Then the minimum U1 A U2 and the maximum U1 V U2 are again phase-type. To see this, let {Jtl)}, { Jt2) } be independent with lifetimes U1, resp. U2. For U1 A U2, we then let the governing phase process be {Jt} _ {(411 Jt2))} 2) interpreting exit of either of {4 M }, { 4 } as exit of {Jt}. Thus the representation is (E(1) x E(2), a(1) ® a(2 ), T(1) ® T(2)).
356
APPENDIX
For U1 V U2, we need to allow { Jt,2) } to go on (on E(2)) when { i 1) } exits, and vice versa. Thus the state space is E(1 ) x E(2) U E(1) U E( 2), the initial vector is (a(1) (& a (2) 0 0), and the phase generator is T(1)
®T(2) T(1) ®t(2) t(1) ® T(2) 0 T(1) 0 0 0 T(2)
Notes and references The results of the present section are standard , see Neuts [269] (where the proof, however, relies more on matrix algebra than the probabilistic interpretation exploited here).
5d Phase-type approximation A fundamental property of phase-type distributions is denseness . That is, any distribution B on (0, oo) can be approximated 'arbitrarily close' by a phase-type distribution B: Theorem A5.14 To a given distribution B on (0, oo), there is a sequence {B,,} of phase-type distributions such that Bn 3 B as n -+ oo. Proof Assume first that B is a one-point distribution, say degenerate at b, and let Bn be the Erlang distribution E,,(Sn) with Sn = n/b. The mean of B„ is n/Sn = b and the variance is n/Sn = b2/n. Hence it is immediate that Bn 4 B. The general case now follows easily from this, the fact that any distribution B can be approximated arbitrarily close by a distribution with finite support, and the closedness of the class of phase-type distributions under the formation of finite mixtures, cf. Example A5.8. Here are the details at two somewhat different levels of abstraction: (diagonal argument , elementary) Let {bk} be any dense sequence of continuity points for B(x). Then we must find phase-type distributions Bn with B,(bk) -+ B(bk) for all k. Now we can find first a sequence {Dm} of distributions with finite support such that D,,(bk) -+ B(bk) for all k as n -* oo. By the diagonal argument (subsequent thinnings), we can assume that ID.(bk)'- B(bk) I < 1/n for n > k. Let the support of Dn be {xl(n),...,xq(n)(n)}, with weight pi(n) for xi(n). Then from above, q(n)
q(n)
C,-,n = I:pi(n)Er v ( __ n) ) ) a= 1
pi(n)a ,(n) = D, r # oo. i= 1
APPENDIX
357
Hence we can choose r(n) in such a way that ICr( n),n (bk) - D(bk)I < n, k < n. Then
2
ICr( n ),n( b k ) - B(bk )I < - , k < n, and we can take Bn = Cr(n),n.
❑
(abstract topological ) The essence of the argument above is that the closure (w.r.t. the topology for weak convergence) PET of the class PET of phase-type distributions contains all one-point distributions. Since PET is closed under the continuous operation of formation of finite mixtures, PIT contains all finite mixtures of one-point distributions, i.e. the class CO of all discrete distributions. But To is the class G of all distributions on [0, oo). Hence G C PET and L = PIT. ❑ Theorem A5.14 is fundamental and can motivate phase-type assumptions, say on the claim size distribution B in risk theory, in at least two ways: insensitivity Suppose we are able to verify a specific result when B is of phasetype say that two functionals Cpl (B) and W2 (B) coincide. If Cpl (B) and ^02(B) are weakly continuous, then it is immediate that WI(B) = p2(B) for all distributions B on [0, oo) approximation Assume that we can compute a functional W(B) when B is phase-type, and that cp is known to be continuous. For a general Bo, we can then approximate Bo by a phase-type B, compute W(B) and use this quantity as an approximation to cp(B0). In particular, if information on Bo is given in terms of observations (i.i.d. replications), one would use the B given by some statistical fitting procedure (see below). It should be noted, however, that this procedure should be used with care if ^p(B) is the ruin probability O(u) and u is large. Let E be the class of functions f : [0, oo) -* [0, oo) such that f (x) = O(e«x), x -4 oo, for some a < oo. Corollary A5.15 To a given distribution B on (0 , oo) and any fl, f2.... E E, there is a sequence {Bn} of phase -type distributions such that Bn Di B as n -4 oo and f ' f,(x)Bf,( dx) -* f r f{(x)B(dx), i = 1, 2,....
APPENDIX
358 Proof By Fatou' s lemma,
B,, -
B implies that
00 liminf
J
00
fi(x)B.(dx) >
n-,oo
o
J
fi(x)B(dx),
o
for each i, and hence it is sufficient to show that we can obtain limsup n-4oo
(A.38)
fi(x)Bn(dx) < f fi( x)B(dx ), i=1,2 ..... JTO o 0
We first show that for each f E E, n B=az, Bn=En
z
f f (x)Bn(dx) -fof (x)B(dx) = ° (A.39)
Indeed, if f (x ) = e°x, then
cc f (x)Bn ( dx) = (?!c )
e'= .f (z) = f 1 1 = 11-n/ o
.f (x)B(dx),
and the case of a general f then follows from the definition of the class E and a uniform integrability argument. Now returning to the proof of (A.38 ), we may assume that in the proof of Theorem A5.14 Dn has been chosen such that 1
00
f fi(x)D n(dx ) <
o
1++ '-
°°
f fi(x)B(dx),
i = 1, ..., n.
\ n o
By (A.39), f00 fi(x)Cr,n(dx) -+ f 0
fi(x)Dn(dx),
and hence we may choose r(n) such that
L
9l) f ' f (x)B(dx), i = 1, ... , n. f (x)Cr(n),n(dx) < 1+-
\\
0
Corollary A5.16 To a given distribution B on (0 , oo), there is a sequence {Bn} of phase -type distributions such that Bn -Di B as n -+ oo and all moments converge, f° xtBn(dx ) -* f °° x`B( dx), i = 1, 2, ....
APPENDIX
359
In compound Poisson risk processes with arrival intensity /3 and claim size distribution B satisfying ,l3µb < 1, the adjustment coefficient 'y = 7(B,/3) is defined as the unique solution > 0 of B[-y] = l+y/j3. The adjustment coefficient is a fundamental quantity, and therefore the following result is highly relevant as support for phase-type assumptions in risk theory: Corollary A5.17 To a given /3 > 0 and a given distribution B on (0, oo) with B[-y +e] < oo for some e > y = 7(B,/3), there is a sequence {B,,} of phase-type distributions such that Bfz + B as n -* oo and -Yn -4 ry where ryn = y(Bn,,3). Proof Let fi(x) = el'r+E;> y for some sequence {ei} with ei E (0, e ) and ei J. 0 as i -* oo. If ei > 0, then Bn['Y + ei] -* B[y + ei] > 1 +
7 Q
implies that 'yn < ry + ei for all sufficiently large n . I.e., lim sup ryn < 7. lim inf > is proved similarly. O We state without proof the following result: Corollary A5.18 In the setting of Corollary A5.16, one can obtain 7(Bn, /3) = ry for all n. Notes and references Theorem A5.14 is classical; the remaining results may be slightly stronger than those given in the literature, but are certainly not unexpected.
5e Phase-type fitting As has been mentioned a number of times already, there is substantial advantage in assuming the claim sizes to be phase-type when one wants to compute ruin probabilities. For practical purposes, the problem thus arises of how to fit a phase-type distribution B to a given set of data (1, . . . , (N. The present section is a survey of some of the available approaches and software for inplementing this. We shall formulate the problem in the slightly broader setting of fitting a phase-type distribution B to a given set of data (1i . . . , (N or a given distribution Bo. This is motivated in part from the fact that a number of non-phase-type distributions like the lognormal, the loggamma or the Weibull have been argued to provide adequate descriptions of claim size distributions, and in part from the fact that many of the algorithms that we describe below have been formulated within the set-up of fitting distributions. However, from a more conceptual
360
APPENDIX
point of view the two sets of problems are hardly different : an equivalent representation of a set of data (1 , ... , (N is the empirical distribution Be, giving mass 1 /N to each S=. Of course, one could argue that the results of the preceding section concerning phase-type approximation contains a solution to our problem : given Bo (or Be), we have constructed a sequence { B,,} of phase-type distribution such that Bo, and as fitted distribution we may take B,, for some suitable large n. B„ The problem is that the constructions of {B„} are not economical : the number of phases grows rapidly, and in practice this sets a limitation to the usefulness (the curse of dimensionality ; we do not not want to perform matrix calculus in hundreds or thousands dimensions). A number of approaches restrict the phase -type distribution to a suitable class of mixtures of Erlang distributions . The earliest such reference is Bux & Herzog [85] who assumed that the Erlang distributions have the same rate parameter, and used a non-linear programming approach . The constraints were the exact fit of the two first moments and the objective function to be minimized involved the deviation of the empirical and fitted c.d.f. at a a number of selected points . In a series of papers (e.g. [216] ), Johnson & Taaffe considered a mixture of two Erlangs (with different rates ) and matched (when possible ) the first three moments . Schmickler (the MEDA package; e .g. [317] ) has considered an extension of this set-up, where more than two Erlangs are allowed and in addition to the exact matching of the first three moments a more general deviation measure is minimized (e.g. the L1 distance between the c . d.f.'s). The characteristics of all of these methods is that even the number of parameters may be low (e.g . three for a mixture of two Erlangs ), the number of phases required for a good fit will typically be much larger, and this is what matters when using phase-type distributions as computational vehicle in say renewal theory, risk theory, reliability or queueing theory. It seems therefore a key issue to develop methods allowing for a more general phase diagram, and we next describe two such approaches which also have the feature of being based upon the traditional statistical tool of like maximum likelihood. A method developed by Bobbio and co-workers (see e. g. [70]) restrict attention to acyclic phase -type distributions , defined by the absence of loops in the phase diagram . The likelihood function is maximized by a local linearization method allowing to use linear programming techniques. Asmussen & Nerman [38] implemented maximum likelihood in the full class of phase-type distributions via the EM algorithm ; a program package written in C for the SUN workstation or the PC is available as shareware, cf. [202]. The observation is that the statistical problem would be straightforward if the whole ( EA-valued) phase process { Jtk)}
o
associated with each observa-
APPENDIX
361
tion Sk was available. In fact, then the estimators would be of simple occurenceexposure type,
EN
1 I (-(k) = i)
tii=i iEE, jEEA,
ai = N where
N Ti = I(J= i) dt, Nii = = , = j) f k=1 k =1 tE[0,(k] (Ti is the total time spent in state i and Nii is the total number of jumps from i to j). The general idea of the EM algorithm ([106]) is to replace such unobserved quantities by the conditional expectation given the observations; since this is parameter-dependent, one is lead to an iterative scheme, e.g. (n+1) _ Ea (n),T(n) (Nik IC1, ... , (N) tJk
Ea ( n),T(n) (Ti ^^ 1, ... , (N
) (^
54
k )+
and similarly for the cn+1) The crux is the computation of the conditional expectations. E.g., it is easy to see that N
Ea(n),T (n)(TiI(1,...,(N) =
(k
E Ea(n),T(n) k=1
I (Jti) dt o \f
N f:i a(n)eT(n)xei . eieT(n)((k- x)t(n) 1
a(n)eT(n )(kt(n)
and this and similar expressions are then computed by numerical solution of a set of differential equations. In practice, the methods of [70] and [38] appear to produce almost identical results. Thus, it seems open whether the restriction to the acyclic case is a severe loss of generality.
This page is intentionally left blank
Bibliography [1] J. Abate, G.L. Choudhury & W. Whitt (1994) Waiting-time tail probabilities in queues with long-tail service -time distributions . Queueing Systems 16, 311-338.
[2] J. Abate & W. Whitt ( 1992) The Fourier-series method for inverting transforms of probability distributions . Queueing Systems 10, 5-87. [3] J. Abate & W. Whitt (1998) Explicit M/G/1 waiting-time distributions for a class of long-tail service time distributions . Preprint, AT&T. [4] M. Abramowitz & I. Stegun ( 1972 ) Handbook of Mathematical Functions (10th ed.). Dover, New York.
[5] G. Alsmeyer (1991) Erneuerungstheorie. B.G. Teubner, Stuttgart. [6] V. Anantharam (1988 ) How large delays build up in a GI/GI11 queue. Queueing Systems 5 , 345-368. [7] E. Sparre Andersen (1957) On the collective theory of risk in the case of contagion between the claims . Transactions XVth International Congress of Actuaries, New York, II, 219-229. [8] G. Arfwedson ( 1954) Research in collective risk theory. Skand. Aktuar Tidskr. 37, 191-223.
[9] G. Arfwedson ( 1955) Research in collective risk theory. The case of equal risk sums. Skand. Aktuar Tidskr. 38, 53-100. [10] K. Arndt (1984) On the distribution of the supremum of a random walk on a Markov chain . In: Limit Theorems and Related Problems (A.A. Borokov, ed.), pp. 253-267. Optimizations Software, New York. [11) S. Asmussen ( 1982 ) Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI /G/1 queue. Adv. Appl. Probab. 14, 143-170. [12] S. Asmussen (1984) Approximations for the probability of ruin within finite time . Scand. Act. J. 1984 , 31-57 ; ibid. 1985, 57. [13] S. Asmussen (1985) Conjugate processes and the simulation of ruin problems. Stoch. Proc. Appl. 20, 213-229. [14] S. Asmussen (1987) Applied Probability and Queues. John Wiley & Sons, Chichester New York. [15] S. Asmussen (1989a) Aspects of matrix Wiener-Hopf factorisation in applied probability. The Mathematical Scientist 14, 101-116.
363
364
BIBLIOGRAPHY
[16] S. Asmussen (1989b) Risk theory in a Markovian environment. Scand. Act. J. 1989 , 69-100. [17] S. Asmussen (1991) Ladder heights and the Markov-modulated M/G/1 queue. Stoch. Proc. Appl. 37, 313-326. [18] S. Asmussen (1992a) Phase-type representations in random walk and queueing problems. Ann. Probab. 20, 772-789. [19] S. Asmussen (1992b) Light traffic equivalence in single server queues. Ann. Appl. Probab. 2, 555-574. [20] S. Asmussen (1992c) Stationary distributions for fluid flow models and Markovmodulated reflected Brownian motion. Stochastic Models 11, 21-49. [21] S. Asmussen (1995) Stationary distributions via first passage times. Advances in Queueing: Models, Methods V Problems (J. Dshalalow ed.), 79-102. CRC Press, Boca Raton, Florida. [22] S. Asmussen (1998a) Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8, 354-374. [23] S. Asmussen (1998b) Extreme value theory for queues via cycle maxima. Extremes 1 , 137-168. [24] S. Asmussen (1998c) A probabilistic look at the Wiener-Hopf equation. SIAM Review 40, 189-201.
[25] S. Asmussen (1999) On the ruin problem for some adapted premium rules. Probabilistic Analysis of Rare Events (V.K. Kalashnikov & A.M. Andronov, eds.), 3-15. Riga Aviation University. [26] S. Asmussen (2000) Matrix-analytic models and their analysis. Scand. J. Statist. 27, 193-226. [27] S. Asmussen & K. Binswanger (1997) Simulation of ruin probabilities for subexponential claims. Astin Bulletin 27, 297-318. [28] S. Asmussen, K. Binswanger & B. Hojgaard (2000) Rare events simulation for heavy-tailed distributions. Bernoulli 6, 303-322.
[29] S. Asmussen & M. Bladt (1996) Renewal theory and queueing algorithms for matrix-exponential distributions. Matrix-Analytic Methods in Stochastic Models (A.S. Alfa & S. Chakravarty, eds.), 313-341. Marcel Dekker, New York. [30] S. Asmussen & M. Bladt (1996) Phase-type distributions and risk processes with premiums dependent on the current reserve. Scand. Act. J. 1996, 19-36. [31] S. Asmussen, L. Floe Henriksen & C. Kliippelberg (1994) Large claims approximations for risk processes in a Markovian environment. Stoch. Proc. Appl. 54, 29-43.
[32] S. Asmussen, A. Frey, T. Rolski & V. Schmidt (1995) Does Markov-modulation increase the risk? ASTIN Bull. 25, 49-66. [33] S. Asmussen & B. Hojgaard (1996) Finite horizon ruin probabilities for Markovmodulated risk processes with heavy tails. Th. Random Processes 2, 96-107. [34] S. Asmussen & B. Hojgaard (1999) Approximations for finite horizon ruin prob-
abilities in the renewal model. Scand. Act. J. 1999 , 106-119.
BIBLIOGRAPHY
365
[35] S. Asmussen, B. Hojgaard & M. Taksar (2000) Optimal risk control and dividend distribution policies . Example of excess-off-loss reinsurance for an insurance corporation. Finance and Stochastics 4, 299-324. [36] S. Asmussen & C. Kliippelberg (1996) Large deviations results for subexponential tails, with applications to insurance risk Stoch. Proc. Appl. 64, 103-125. [37] S. Asmussen & G. Koole (1993). Marked point processes as limits of Markovian
arrival streams. J. Appl. Probab. 30, 365-372. [38] S. Asmussen , O. Nerman & M. Olsson (1996). Fitting phase-type distributions via the EM algorithm. Scand. J. Statist. 23, 419-441. [39] S. Asmussen & H.M. Nielsen (1995) Ruin probabilities via local adjustment coefficients . J. Appl. Prob 32 , 736-755. [40] S. Asmussen & C.A. O'Cinneide (2000/01) On the tail of the waiting time in a Markov-modulated M/G/1 queue. Opns. Res. (to appear). [41] S. Asmussen & C.A. O'Cinneide (2000/2001) Matrix-exponential distributions [Distributions with a rational Laplace transform] Encyclopedia of Statistical Sciences, Supplementary Volume (Kotz, Johnson, Read eds.). Wiley. [42] S. Asmussen & D. Perry (1992) On cycle maxima, first passage problems and extreme value theory for queues. Stochastic Models 8, 421-458.
[43] S. Asmussen & T. Rolski (1991) Computational methods in risk theory: a matrix-algorithmic approach. Insurance: Mathematics and Economics 10, 259274. [44] S. Asmussen & T. Rolski (1994) Risk theory in a periodic environment: Lundberg's inequality and the Cramer-Lundberg approximation. Math. Oper. Res. 410-433. [45] S. Asmussen & R.Y. Rubinstein (1995) Steady-state rare events simulation in queueing models and its complexity properties. Advances in Queueing: Models, Methods 8 Problems (J. Dshalalow ed.), 429-466. CRC Press, Boca Raton, Florida. [46] S. Asmussen & R.Y. Rubinstein (1999) Sensitivity analysis of insurance risk models. Management Science 45, 1125-1141.
[47] S. Asmussen, H. Schmidli & V. Schmidt (1999) Tail approximations for nonstandard risk and queueing processes with subexponential tails. Adv. Appl. Probab. 31, 422-447. [48] S. Asmussen & V. Schmidt (1993) The ascending ladder height distribution for a class of dependent random walks. Statistica Neerlandica 47, 1-9. [49] S. Asmussen & V. Schmidt (1995) Ladder height distributions with marks. Stoch. Proc. Appl. 58, 105-119. [50] S. Asmussen & S. Schock Petersen (1989) Ruin probabilities expressed in terms of storage processes. Adv. Appl. Probab. 20, 913-916.
[51] S. Asmussen & K. Sigman (1996) Monotone stochastic recursions and their duals. Probab. Th. Eng. Inf. Sc. 10, 1-20. [52] S. Asmussen & M. Taksar (1997) Controlled diffusion models for optimal dividend payout. Insurance: Mathematics and Economics 20, 1-15.
366
BIBLIOGRAPHY
[53] S. Asmussen & J.L. Teugels ( 1996) Convergence rates for M /G/1 queues and ruin problems with heavy tails . J. Appl. Probab. 33, 1181-1190. [54] K.B . Athreya & P. Ney ( 1972) Branching Processes . Springer , Berlin.
[55] B. von Bahr (1974) Ruin probabilities expressed in terms of ladder height distributions. Scand. Act. J. 1974, 190-204. [56] B. von Bahr (1975) Asymptotic ruin probabilities when exponential moments do not exist. Scand. Act. J. 1975, 6-10. [57] C.T.H. Baker (1977) The Numerical Solution of Integral Equations. Clarendon Press, Oxford. [58] O. Barndorff-Nielsen (1978) Information and Exponential Families in Statistical Theory. Wiley, New York. [59] O. Barndorff-Nielsen & H. Schmidli (1995) Saddlepoint approximations for the probability of ruin in finite time. Scand. Act. J. 1995, 169-186. [60] J. Beekman (1969) A ruin function approximation. Trans. Soc. Actuaries 21, 41-48, 275-279. [61] J. Beekman (1974) Two Stochastic Processes. Halsted Press, New York. [62] J.A. Beekman (1985) A series for infinite time ruin probabilities. Insurance: Mathematics and Economics 4, 129-134. [63] A. Berman & R.J. Plemmons (1994) Nonnegative Matrices in the Mathematical Sciences. SIAM. [64] P. Billingsley (1968) Convergence of Probability Measures. Wiley, New York. [65] N.H. Bingham (1975) Fluctuation theory in continuous time. Adv. Appl. Probab. 7, 705-766. [66] N.H. Bingham, C.M. Goldie & J.L. Teugels (1987) Regular Variation. Cambridge University Press, Cambridge.
[67] T. Bjork & J. Grandell (1985) An insensitivity property of the ruin probability. Scand. Act. J. 1985 , 148-156. [68] T. Bjork & J. Grandell (1988) Exponential inequalities for ruin probabilities in the Cox case. Scand. Act. J. 1988 , 77-111.
[69] P. Bloomfield & D.R. Cox (1972) A low traffic approximation for queues. J. Appl. Probab. 9, 832-840. [70] A. Bobbio & M. Telek (1994) A bencmark for PH estimation algorithms: results for acyclic PH. Stochastic Models 10, 661-667. [71] P. Boogaert & V. Crijns (1987) Upper bounds on ruin probabilities in case of negative loadings and positive interest rates. Insurance: Mathematics and Economics 6, 221-232. [72] P. Boogaert & A. de Waegenaere (1990) Simulation of ruin probabilities. Insurance: Mathematics and Economics 9, 95-99. [73] A.A. Borovkov (1976) Asymptotic Methods in Queueing Theory. SpringerVerlag.
BIBLIOGRAPHY
367
[74] O. Boxma & J.W. Cohen (1998) The M/G/1 queue with heavy-tailed service time distribution. IEEE J. Sel. Area Commun. 16, 749-763. [75] O. Boxma & J.W. Cohen (1999) Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed service time distributions . Queueing Systems 33, 177-204. [76] N.L. Bowers , Jr., H.U. Gerber, J.C. Hickman, D.A. Jones & C.J. Nesbitt (1986)
Actuarial Mathematics . The Society of Actuaries , Itasca, Illinois. [77] P. Bratley, B.L. Fox & L. Schrage (1987) A Guide to Simulation. Springer, New York. [78] L. Breiman ( 1968) Probability. Addison-Wesley, Reading. [79] P.J. Brockwell , S.I. Resnick & R.L. Tweedie (1982) Storage processes with general release rule and additive inputs . Adv. Appl. Probab. 14, 392-433. [80] F. Broeck , M. Goovaerts & F. De Vylder (1986) Ordering of risks and ruin probabilities. Insurance : Mathematics and Economics 5, 35-39. [81] J.A. Bucklew ( 1990) Large Deviation Techniques in Decision, Simulation and Estimation. Wiley, New York. [82] H. Biihlmann ( 1970) Mathematical Methods in Risk Theory. Springer-Verlag, Berlin. [83] D.Y . Burman & D .R. Smith ( 1983) Asymptotic analysis of a queueing system with bursty traffic. Bell. Syst. Tech. J. 62, 1433-1453. [84] D.Y. Burman & D.R. Smith ( 1986) An asymptotic analysis of a queueing system with Markov-modulated arrivals . Oper. Res. 34, 105-119. [85] W. Bux & U. Herzog (1977) The phase concept : approximations of measured data and performance analysis. Computer Performance (K.M. Chandy & M. Reiser eds.), 23-38. North-Holland, Amsterdam.
[86] K.L. Chung (1974) A Course in Probability Theory (2nd ed.). Academic Press, New York San Francisco London. [87] E. Qinlar (1972) Markov additive processes. II. Z. Wahrscheinlichkeitsth. verve. Geb. 24, 93-121. [88] J.W. Cohen (1982) The Single Server Queue (2nd ed.) North-Holland, Amsterdam. [89] M. Cottrell, J.-C. Fort & G. Malgouyres (1983) Large deviations and rare events in the study of stochastic algorithms . IEEE Trans. Aut. Control AC-28, 907920. [90] D.R. Cox (1955) Use of complex probabilities in the theory of stochastic processes. Proc. Cambr. Philos. Soc. 51, 313-319. [91] H. Cramer (1930) On the Mathematical Theory of Risk. Skandia Jubilee Volume, Stockholm.
[92] H. Cramer (1955) Collective risk theory. The Jubilee volume of Forsakringsbolaget Skandia, Stockholm. [93] K. Croux & N. Veraverbeke (1990). Non-parametric estimators for the probability of ruin. Insurance : Mathematics and Economics 9, 127-130.
368
BIBLIOGRAPHY
[94] M. Csorgo & J. Steinebach ( 1991 ) On the estimation of the adjustment coefficient in risk theory via intermediate order statistics . Insurance: Mathematics and Economics 10, 37-50. [95] M. Csorgo & J.L. Teugels ( 1990) Empirical Laplace transform and approximation of compound distributions . J. Appl. Probab. 27, 88-101.
[96] D. Daley & T. Rolski (1984) A light traffic approximation for a single server queue. Math. Oper. Res. 9 , 624-628. [97] D. Daley & T. Rolski (1991) Light traffic approximations in queues. Math. Oper. Res. 16, 57-71. [98] A. Dassios & P. Embrechts (1989). Martingales and insurance risk. Stochastic Models 5, 181-217. [99] R. van Dawen (1986) Ein einfacher Beweis fur Ohlins Lemma. Blotter der deutschen Gesellschaft fur Versicherungsmathematik XVII, 435-436. [100] A. Davidson (1946) On the ruin problem of collective risk theory under the assumption of a variable safety loading (in Swedish). Forsiikringsmatematiska Studier Tillagnade Filip Lundberg, Stockholm. English version published in Skand. Aktuar. Tidskr. Suppl., 70-83 (1969).
[101] C.D. Daykin, T. Pentikainen & E. Pesonen (1994) Practical Risk Theory for Actuaries. Chapman & Hall. [102] P. Deheuvels & J. Steinebach (1990) On some alternative estimators of the adjustment coefficient in risk theory. Scand. Act. J. 1990 , 135-159.
[103] F. Delbaen & J. Haezendonck (1985) Inversed martingales in risk theory. Insurance: Mathematics and Economics 4, 201-206. [104] F. Delbaen & J. Haezendonck (1987) Classical risk theory in an economic environment. Insurance: Mathematics and Economics 6, 85-116. [105] A. Dembo & O. Zeitouni (1992) Large Deviations Techniques and Applications. Jones and Bartlett, Boston. [106] A.P. Dempster, N.M. Laird & D.B. Rubin (1977) Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. 22, 583-602. [107] L. Devroye (1986) Non-uniform Random Variate Generation. Springer, New York.
[108] F. De Vylder (1977) A new proof of a known result in risk theory. J. Comp. Appl. Math. 3, 227-229. [109] F. De Vylder (1978) A practical solution to the problem of ultimate ruin probability. Scand. Actuarial J., 114-119. [110] F. De Vylder (1996) Advanced Risk Theory. Editions de 1'Universite de Bruxelles. [111] F. De Vylder & M. Goovaerts (1984) Bounds for classical ruin probabilities. Insurance: Mathematics and Economics 3, 121-131. [112] F. De Vylder & M. Goovaerts (1988) Recursive calculation of finite-time ruin probabilities. Insurance: Mathematics and Economics 7, 1-7.
BIBLIOGRAPHY
369
[113] D.C.M. Dickson (1992) On the distribution of the surplus prior to ruin . Insurance : Mathematics and Economics 11, 191-207.
[114] D.C.M. Dickson (1994) An upper bound for the probability of ultimate ruin. Scand. Act. J. 1994 , 131-138. [115] D.C.M. Dickson (1995) A review of Panjer's recursion formula and its applications. British Actuarial J. 1, 107-124.
[116] D.C.M. Dickson & J.R. Gray (1984) Exact solutions for ruin probability in the presence of an upper absorbing barrier. Scand. Act. J. 1984 , 174-186. [117] D.C.M. Dickson & J.R. Gray (1984) Approximations to the ruin probability in the presence of an upper absorbing barrier. Scand. Act. J. 1984 , 105-115. [118] D.C.M. Dickson & C. Hipp (1998) Ruin probabilities for Erlang (2) risk processes. Insurance : Mathematics and Economics 22, 251-262.
[119] D.C.M. Dickson & C. Hipp (1999) Ruin problems for phase-type(2) risk processes. Scand. Act. J. 2000 , 147-167. [120] D.C.M. Dickson & H.R. Waters (1996 ) Reinsurance and ruin . Insurance: Mathematics and Economics 19, 61-80. [121] D.C.M. Dickson & H.R. Waters (1999) Ruin probabilities wih compounding. Insurance: Mathematics and Economics 25, 49-62. [122] B. Djehiche (1993) A large deviation estimate for ruin probabilities. Scand. Act. J. 1993 , 42-59.
[123] E. van Doorn & G.J.K. Regterschot (1988) Conditional PASTA. Oper. Res. Letters 7, 229-232. [124] N.G. Duffield & N. O'Connell (1995) Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Philos. Soc. 118, 363-374. [125] F. Dufresne & H.U. Gerber (1988) The surpluses immediately before and at ruin, and the amount of claim causing ruin. Insurance : Mathematics and Economics 7, 193-199.
[126] F. Dufresne & H.U. Gerber (1991) Risk theory for the compound Poisson process that is perturbed by a diffusion. Insurance : Mathematics and Economics 10, 5159. [127] F. Dufresne, H.U. Gerber & E.S.W. Shiu (1991) Risk theory with the Gamma process. Astin Bulletin 21, 177-192. [128] E.B. Dynkin (1965) Markov Processes I. Springer , Berlin Gottingen Heidelberg.
[129] D .C. Emanuel , J.M. Harrison & A.J. Taylor ( 1975) A diffusion approximation for the ruin probability with compounding assets. Scand. Act. J. 1975 , 37-45. [130] P. Embrechts ( 1988 ). Ruin estimates for large claims . Insurance: Mathematics and Economics 7, 269-274. [131] P. Embrechts , J. Grandell & H. Schmidli (1993) Finite-time Lundberg inequalities in the Cox case . Scand. Act. J. 1993 , 17-41.
370
BIBLIOGRAPHY
[132] P. Embrechts, R. Griibel & S.M. Pitts (1993) Some applications of the fast Fourier transform in insurance mathematics . Statistica Neerlandica 47, 59-75. [133] P. Embrechts & T. Mikosch (1991). A bootstrap procedure for estimating the adjustment coefficient. Insurance: Mathematics and Economics 10, 181-190. [134] P. Embrechts, C. Kliippelberg & T. Mikosch (1997) Modelling Extremal Events for Finance and Insurance. Springer, Heidelberg.
[135] P. Embrechts & H. Schmidli (1994) Ruin estimation for a general insurance risk model. Adv. Appl. Probab. 26, 404-422. [136] P. Embrechts & N. Veraverbeke (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims . Insurance : Mathematics and Economics 1, 55-72. [137] P. Embrechts & J.A. Villasenor (1988) Ruin estimates for large claims. Insurance: Mathematics and Economics 7, 269-274. [138] P. Embrechts, J.L. Jensen, M. Maejima & J.L. Teugels (1985). Approximations for compound Poisson and Polya processes. Adv. Appl. Probab. 17, 623-637. [139] A.K. Erlang (1909) Sandsynlighedsregning og telefonsamtaler . Nyt Tidsskrift for Matematik B20, 33-39. Reprinted 1948 as 'The theory of probabilities and telephone conversations' in The Life and work of A.K. Erlang 2, 131-137. Trans. Danish Academy Tech. Sci. [140] J.D. Esary, F. Proschan & D.W. Walkup (1967) Association of random variables, with applications. Ann. Math. Statist. 38, 1466-1474.
[141] F. Esscher (1932) On the probability function in the collective theory of risk. Skand. Akt. Tidsskr. 175-195.
[142] W. Feller (1966) An Introduction to Probability Theory and its Applications I (3nd ed.). Wiley, New York. [143] W. Feller (1971) An Introduction to Probability Theory and its Applications II (2nd ed.). Wiley, New York. [144] P.J. Fitzsimmons (1987). On the excursions of Markov processes in classical duality. Probab. Th. Rel. Fields 75, 159-178. [145] P. Franken, D. Konig, U. Arndt & V. Schmidt (1982) Queues and Point Processes. Wiley. [146] E.W. Frees (1986) Nonparametric estimation of the probability of ruin. Astin Bulletin 16, 81-90. [147] M. Frenz & V. Schmidt (1992) An insensitivity property of ladder height distributions. J. Appl. Probab. 29, 616-624. [148] C.D. Fuh (1997) Corrected ruin probabilities for ruin probabilities in Markov random walks. Adv. Appl. Probab. 29, 695-712. [149] C.D. Fuh & T. Lai (1998) Wald's equations, first passage times and moments of ladder variables in Markov random walks. J. Appl. Probab. 35, 566-580. [150] H. Furrer (1998) Risk processes perturbed by a-stable Levy motion. Scand. Act. J. 1998 , 59-74.
BIBLIOGRAPHY
371
[151] H. Furrer (1996) A note on the convergence of the infinite-time ruin probabilities when the weak approximation is a-stable Levy motion. Unpublished, contained in the authors PhD thesis, ETH Zurich. [152] H. Furrer, Z. Michna & A. Weron (1997) Stable Levy motion approximation in collective risk theory. Insurance: Mathematics and Economics 20, 97-114.
[153] H. Furrer & H. Schmidli (1994) Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance: Mathematics and Economics 15, 23-36.
[154] H.U. Gerber (1970) An extension of the renewal equation and its application in the collective theory of risk. Skand. Aktuarietidskrift 1970 , 205-210. [155] H.U. Gerber (1971) Der Einfluss von Zins auf die Ruinwahrscheinlichkeit. Mitt. Ver. Schweiz. Vers. Math.71, 63-70.
[156] H.U. Gerber (1973) Martingales in risk theory. Mitt. Ver. Schweiz. Vers. Math. 73, 205-216. [157] H.U. Gerber (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation Monographs, University of Pennsylvania.
[158] H.U. Gerber (1981) On the probability of ruin in the presence of a linear dividend barrier. Scand. Act. J. 1981 , 105-115. [159] H.U. Gerber (1986). Life Insurance Mathematics. Springer. [160] H.U. Gerber (1988) Mathematical fun with ruin theory. Insurance : Mathematics and Economics 7, 15-23.
[161] P. Glasserman (1991) Gradient Estimation via Perturbation Analysis. Kluwer, Boston, Dordrecht, London. [162] P. Glasserman & S.-G. Kou (1995) Limits of first passage times to rare sets in regenerative processes. Ann. Appl. Probab. 5, 424-445. [163] P.W. Glynn & W. Whitt (1994) Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. In Studies in Applied Probability (J. Galambos & J. Gani, eds.). J. Appl. Probab. 31A, 131-156.
[164] B.V. Gnedenko & I.N. Kovalenko (1989) Introduction to Queueing Theory (2nd ed.). Birkhauser , Basel. [165] M.J. Goovaerts, F. de Vylder & J. Haezendonck (1984) Insurance Premiums. North-Holland, Amsterdam. [166] M.J. Goovaerts, R. Kaas, A.E. van Heerwarden & T. Bauwelinckx (1990) Effective Actuarial Methods. North-Holland, Amsterdam. [167] C.M. Goldie & R. Griibel (1996) Perpetuities with thin tails. Adv. Appl. Probab. 28, 463-480. [168] J. Grandell (1977) A class of approximations of ruin probabilities. Scand. Act. J., Suppl., 1977, 37-52. [169] J. Grandell (1978) A remark on 'A class of approximations of ruin probabilities'. Scand. Act. J. 1978 , 77-78.
372
BIBLIOGRAPHY
[170] J. Grandell (1979) Empirical bounds for ruin probabilities. Stoch. Proc. Appl. 8,243-255. [171] J. Grandell (1990) Aspects of Risk Theory. Springer-Verlag. [172] J. Grandell ( 1992) Finite time ruin probabilities and martingales . Informatica 2, 3-32. [173] J. Grandell (1997) Mixed Poisson Processes. Chapman & Hall. [174] J. Grandell (1999) Simple approximations of ruin functions. Preprint, KTH. [175] J. Grandell & C.-O. Segerdahl (1971) A comparison of some approximations of ruin probabilities. Skand. Aktuar Tidskr. 54, 143-158. [176] B. Grigelionis ( 1992) On Lundberg inequalities in a Markovian environment. Proc . Winter School on Stochastic Analysis and Appl .. Akademie-Verlag, Berlin. [177] B. Grigelionis ( 1993) Two-sided Lundberg inequalities in a Markovian environment. Liet. Matem. Rink. 33, 30-41. [178] B. Grigelionis (1996) Lundberg-type stochastic processes. Probability Theory and Mathematical Statistics (I.A. Ibragimov, ed.), 167-176. [179] R. Grnbel (1991) G/G/1 via FFT. Statistical Algorithm 265. Applied Statistics 40, 355-365.
[180] R. Griibel & R. Hermesmeier (1999) Computation of compound distributions I: aliasing errors and exponential tilting. Astin Bulletin 29, 197-214. [181] D.V. Gusak & V.S. Korolyuk (1969) On the joint distribution of a process with stationary increments and its maximum . Th. Probab. Appl. 14, 400-409. [182] A. Gut (1988 ) Stopped Random Walks . Theory and Applications. SpringerVerlag, New York. [183] M. Gyllenberg & D. Silvestrov (2000) Cram€r-Lundberg approximations for nonlinearly perturbed risk processes . Insurance: Mathematics and Economics 26, 75-90.
[184] H . Hadwiger (1940) Uber die Wahrscheinlichkeit des Ruins bei einer grossen Zahl von Geschiiften . Archiv v. mathematische Wirtschaft - and Sozialforschung 6,131-135. [185] J. M. Harrison (1977) Ruin problems with compounding assets . Stoch. Proc. Appl. 5 , 67-79.
[186] J. M. Harrison & A.J. Lemoine (1977) Limit theorems for periodic queues. J. Appl. Probab. 14, 566-576. [187] J. M. Harrison & S.I. Resnick (1976 ) The stationary distribution and first exit probabilities of a storage process with general release rule . Math. Opns . Res. 1, 347-358.
[188] J. M. Harrison & S.I. Resnick (1977) The recurrence classification of risk and storage processes . Math. Opns . Res. 3 , 57-66. [189] A. E. van Heerwarden (1991) Ordering of Risks: Theory and Actuarial Applications. Tinbergen Institute Research Series 20, Amsterdam.
BIBLIOGRAPHY
373
[190] P. Heidelberger (1995) Fast simulation of rare events in queueing and reliability models. ACM TOMACS 6, 43-85. [191] W.-R. Heilmann (1987) Grundbegriffe der Risikotheorie. Verlag Versicherungswirtschaft e.V., Karlsruhe. [192] U. Herkenrath (1986) On the estimation of the adjustment coefficient in risk theory by means of stochastic approximation procedures. Insurance: Mathematics and Economics 5, 305-313.
[193] U. Hermann (1965) Bin Approximationssatz fiir Verteilungen stationarer zufalliger Punktfolgen. Math. Nachrichten 30, 377-381. [194] 0. Hesselager (1990) Some results on optimal reinsurance in terms of the adjustment coefficient. Scand. Act. J. 1990 , 80-95. [195] B.M. Hill (1975) A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163-1174. [196] C. Hipp (1989a) Efficient estimators for ruin probabilities. Proc. Foruth Prague Symp. on Asymptotic Statistics (P. Mandl & M. Huskova, eds.), 259-268. [197] C. Hipp (1989b) Estimators and bootstrap confidence intervals for ruin probabilities. Astin Bulletin 19, 57-70. [198] C. Hipp & R. Michel (1990) Risikotheorie: Stochastische Modelle and Statistische Methoden. Versicherungswirtschaft, Karlsruhe.
[199] R.V. Hogg & S.A. Klugman (1984) Loss Distributions. Wiley, New York. [200] M.L. Hogan (1986) Comment on corrected diffusion approximations in certain random walk problems. J. Appl. Probab. 23, 89-96. [201] L. Horvath & E. Willekens (1986) Estimates for the probability of ruin starting with a large initial reserve. Insurance: Mathematics and Economics 5, 285-293. [202] 0. Haggstrom, S. Asmussen & 0. Nerman (1992) EMPHT - a program for fitting phase-type distributions. Studies in Statistical Quality Control and Reliability 1992 :4. Mathematical Statistics, Chalmers University of Technology and the University of Goteborg. [203] T. Hoglund (1974) Central limit theorems and statistical inference for Markov chains. Z. Wahrscheinlichkeitsth. verve. Geb. 29, 123-151. [204] T. Hoglund (1990) An asymptotic expression for the probability of ruin within finite time. Ann. Probab. 18, 378-389.
[205] T. Hoglund (1991). The ruin problem for finite Markov chains. Ann. Prob. 19, 1298-1310. [206] B. Hojgaard & M. Taksar (2000) Optimal proportional reinsurance policies for diffusion models. Scand. Act. J. 1998 , 166-180.
[207] D.L. Iglehart (1969) Diffusion approximations in collective risk theory. J. Appl. Probab. 6, 285-292.
[208] V.B. Iversen & L. Staalhagen (1999) Waiting time distributions in M/D/1 queueing systems. Electronic Letters 35, No. 25.
374
BIBLIOGRAPHY
[209] D. Jagerman (1985) Certain Volterra integral equations arising in queueing. Stochastic Models 1, 239-256. [210] D. Jagerman (1991) Analytical and numerical solution of Volterra integral equations with applications to queues. Manuscript, NEC Research Institute, Princeton, N.J. [211] J. Janssen (1980) Some transient results on the M/SM/1 special semi-Markov model in risk and queueing theories. Astin Bull. 11, 41-51. [212] J. Janssen & J.M. Reinhard (1985) Probabilities de ruine pour une classe de modeles de risque semi-Markoviens. Astin Bull. 15, 123-133.
[213] P.R. Jelenkovic & A.A. Lazar (1998) Subexponential asymptotics of a Markovmodulated random walk with a queueing application. J. Appl. Probab. 35, 325247. [214] A. Jensen (1953) A Distribution Model Applicable to Economics. Munksgaard, Copenhagen. [215] J.L. Jensen (1995) Saddle Point Approximations. Clarendon Press, Oxford. [216] M. Johnson & M. Taaffe (1989/90) Matching moments to phase distributions. Stochastic Models 5, 711-743; ibid. 6, 259-281; ibid. 6, 283-306.
[217] R. Kaas & M.J. Goovaerts (1986) General bound on ruin probabilities. Insurance: Mathematics and Economics 5, 165-167. [218] V. Kalashnikov (1996) Two-sided bounds of ruin probabilities. Scand. Act. J. 1996, 1-18. [219] V. Kalashnikov (1997) Geometric Sums: Bounds for Rare Event with Applications. Kluwer.
[220] V. Kalashnikov (1999) Bounds for ruin probabilities in the presence of large claims and their comparison. N. Amer. Act. J. 3, 116-129. [221] E.P.C. Kao (1988) Computing the phase-type renewal and related functions. Technometrics 30, 87-93. [222] S. Karlin & H.M. Taylor (1981) A Second Course in Stochastic Processes. Academic Press, New York. [223] J. Keilson (1966) A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37, 866-870. [224] J. Keilson & D.M.G. Wishart (1964) A central limit theorem for processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 60, 547-567. [225] J. Keilson & D.M.G. Wishart (1964) Boundary problems for additive processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 61, 173-190. [226] J. Keilson & D.M.G. Wishart (1964). Addenda to for processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 63, 187-193. [227] J.H.B. Kemperman (1961) The Passage Problem for a Markov Chain. University of Chicago Press, Chicago. [228] J. Kennedy (1994) Understanding the Wiener-Hopf factorization for the simple random walk. J. Appl. Probab. 31, 561-563.
BIBLIOGRAPHY
375
[229] J.F.C. Kingman (1961) A convexity property of positive matrices . Quart. J. Math. Oxford 12, 283-284. [230] J.F.C. Kingman (1962) On queues in heavy traffic. Quart. J. Roy. Statist. Soc. B24, 383-392. [231] J.F.C. Kingman (1964) A martingale inequality in the theory of queues. Proc. Camb. Philos. Soc. 60 , 359-361. [232] C. Kliippelberg (1988) Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132-141.
[233] C. Klnppelberg (1989) Estimation of ruin probabilities by means of hazard rates. Insurance: Mathematics and Economics 8, 279-285. [234] C. Klnppelberg (1993) Asymptotic ordering of risks and ruin probabilities. Insurance : Mathematics and Economics 12, 259-264.
(235] C. Klnppelberg & T. Mikosch (1995) Explosive Poisson shot noise with applications to risk retention. Bernoulli 1 , 125-147. [236] C. Klnppelberg & T. Mikosch (1995) Modelling delay in claim settlement. Scand. Act. J. 1995 , 154-168. [237] C. Kliippelberg & U. Stadtmiiller (1998) Ruin probabilities in the presence of heavy-tails and interest rates . Scand. Act. J. 1998 , 49-58. [238] V.S. Korolyuk, I.P. Penev & A.F. Turbin (1973) An asymptotic expansion for the absorption time of a Markov chain distribution. Cybernetica 4, 133-135 (in Russian). [239] H. Kunita (1976) Absolute continuity of Markov processes. Seminaire de Probabilties X . Lecture Notes in Mathematics 511, 44-77. Springer-Verlag. [240] U. Kuchler & M. Sorensen (1997) Exponential Families of Stochastic Processes. Springer-Verlag.
[241] G. Latouche & V. Ramaswami (1999) Introduction to Matrix-Analytic Methods in Stochastic Modelling. SIAM. [242] A.J. Lemoine (1981) On queues with periodic Poisson input. J. Appl. Probab. 18, 889-900. [243] A.J. Lemoine (1989) Waiting time and workload in queues with periodic Poisson input. J. Appl. Probab. 26, 390-397. [244] T. Lehtonen & H. Nyrhinen (1992a) Simulating level-crossing probabilities by importance sampling. Adv. Appl. Probab. 24, 858-874. [245] T. Lehtonen & H. Nyrhinen (1992b) On asymptotically efficient simulation of ruin probabilities in a Markovian environment. Scand. Actuarial J., 60-75. [246] D. Lindley (1952) The theory of queues with a single server. Proc. Cambr. Philos. Soc. 48, 277-289. [247] L. Lipsky (1992) Queueing Theory - a Linear Algebraic Approach. Macmillan, New York. [248] D. Lucantoni (1991) New results on the single server queue with a batch Markovian arrival process. Stochastic Models 7, 1-46.
376
BIBLIOGRAPHY
[249] D. Lucantoni , K.S. Meier-Hellstern & M .F. Neuts ( 1990) A single server queue with server vacations and a class of non-renewal arrival processes . Adv. Appl. Probab. 22, 676-705. [250] F. Lundberg (1903) I Approximerad Framstdllning av Sannolikhetsfunktionen. II Aterforsdkring av Kollektivrisker . Almqvist & Wiksell, Uppsala. [251] F. Lundberg (1926) Forsdkringsteknisk Riskutjdmning. F. Englunds Boktryckeri AB, Stockholm.
[252] A.M. Makowski ( 1994) On an elementary characterization of the increasing convex order , with an application . J. Appl. Probab . 31, 834-841. [253] V. Mammitzsch ( 1986 ) A note on the adjustment coefficient in ruin theory. Insurance : Mathematics and Economics 5, 147-149. [254] V .K. Malinovskii ( 1994) Corrected normal approximation for the probability of ruin within finite time. Scand. Act. J. 1994, 161-174.
[255] V.K. Malinovskii ( 1996) Approximation and upper bounds on probabilities of large deviations of ruin within finite time. Scand. Act. J. 1996 , 124-147. [256] A. Martin-L6f (1983) Entropy estimates for ruin probabilities . Probability and Mathematical Statistics (A. Gut & J. Hoist eds.), 29-39.
[257] A. Martin-L3f (1986) Entropy, a useful concept in risk theory. Scand. Act. J. 1986 , 223-235. [258] K .S. Meier ( 1984). A fitting algorithm for Markov-modulated Poisson processes having two arrival rates . Europ. J. Opns. Res. 29 , 370-377.
[259] Z . Michna ( 1998) Self-similar processes in collective risk theory. J. Appl. Math. Stoch. Anal. 11 , 429-448. [260] H.D. Miller ( 1961 ) A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1261 (0?)-1270.
[261] H.D. Miller (1962 ) A matrix factorization problem in the theory of random variables defined on a finite Markov chain . Proc. Cambridge Philos. Soc. 58, 268-285. [262] H .D. Miller ( 1962 ) Absorption probabilities for sums of random variables defined on a finite Markov chain . Proc. Cambridge Philos. Soc. 58 , 286-298.
[263] M . Miyazawa & V. Schmidt ( 1993) On ladder height distributions of general risk processes . Ann. Appl. Probab. 3, 763-776. , [264] G.V. Moustakides ( 1999) Extension of Wald 's first lemma to Markov processes. J. Appl. Probab. 36, 48-59.
[265] S.V. Nagaev (1957) Some limit theorems for stationary Markov chains. Th. Probab. Appl. 2, 378-406. [266] P. Ney & E. Nummelin (1987) Markov additive processes I. Eigenvalue properties and limit theorems . Ann. Probab. 15, 561-592.
BIBLIOGRAPHY
377
[267] M .F. Neuts (1977) A versatile Markovian point process. J. Appl. Probab. 16, 764-779. [268] M .F. Neuts ( 1978) Renewal processes of phase type. Naval Research Logistics Quarterly 25, 445-454. [269] M .F. Neuts ( 1981 ) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore . London. [270] M .F. Neuts ( 1989) Structured Stochastic Matrices of the M/G/1 Type and their Applications. Marcel Dekker, New York.
[271] M.F. Neuts (1992) Models based on the Markovian arrival process. IEICE Trans. Commun. E75-B , 1255-1265. [272] J . Neveu ( 1961 ) Une generalisation des processus a accroisances independantes. Sem. Math. Abh. Hamburg. [273] R. Norberg ( 1990) Risk theory and its statistics environment . Statistics 21, 273-299. [274] H . Nyrhinen ( 1998 ) Rough descriptions of ruin for a general class of surplus processes. Adv. Appl. Probab. 30, 1008-1026. [275] H. Nyrhinen (1999) Large deviations for the time of ruin. J. Appl. Probab. 36, 733-746.
[276] C.A. O'Cinneide (1990) Characterization of phase-type distributions. Stoch. Models 6, 1-57. [277] J. Ohlin (1969) On a class of measures for dispersion with application to optimal insurance. Astin Bulletin 5 , 249-266.
[278] E. Omey & E. Willekens (1986) Second order behaviour of the tail of a subordinated probability distributions. Stoch. Proc. Appl. 21, 339-353. [279] E. Omey & E. Willekens (1987) Second order behaviour of distributions subordinate to a distribution with finite mean . Stochastic Models 3 , 311-342.
[280] A.G. Pakes (1975) On the tail of waiting time distributions. J. Appl. Probab. 12, 555-564. [281] J. Paulsen (1993) Risk theory in a stochastic economic environment. Stoch. Proc. Appl. 46, 327-361. [282] J . Paulsen ( 1998) Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Proc. Appl. 75, 135-148. [283] J. Paulsen (1998) Ruin theory with compounding assets - a survey. Insurance: Mathematics and Economics 22, 3-16. [284] J. Paulsen & H.K. Gjessing (1997a) Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: Mathematics and Economics 20, 215-223.
[285] J. Paulsen & H.K. Gjessing (1997b) Present value distributions with applications to ruin theory and stochastic equations. Stoch. Proc. Appl. 71, 123-144.
378
BIBLIOGRAPHY
[286] J. Paulsen & H.K. Gjessing (1997c) Ruin theory with stochastic return on investments. Adv. Appl. Probab. 29, 965-985. [287] F. Pellerey (1995) On the preservation of some orderings of risks under convolution. Insurance: Mathematics and Economics 16, 23-30. [288] S. Schock Petersen (1989) Calculation of ruin probabilities when the premium depends on the current reserve. Scand. Act. J. 1989 , 147-159.
[289] C. Philipson (1968) A review of the collective theory of risk. Skand. Aktuar. Tidskr. 61, 45-68, 117-133. [290] E.J.G. Pitman (1980) Subexponential distribution functions. J. Austr. Math. Soc. 29A, 337-347. [291] S. Pitts (1994) Nonparametric estimation of compound distributions with applications in insurance. Ann. Inst. Stat. Math. 46, 537-555. [292] S.M. Pitts, R. Griibel & P. Embrechts (1996) Confidence bounds for the adjustment coefficient. Adv. Appl. Probab. 28, 820-827.
[293] N.U. Prabhu (1961). On the ruin problem of collective risk theory. Ann. Math. Statist. 32, 757-764. [294] N.U. Prabhu (1965) Queues and Inventories. Wiley. [295] N.U. Prabhu (1980) Stochastic Storage Processes. Queues, Insurance Risk, and Dams. Springer, New York, Heidelberg, Berlin.
[296] N.U. Prabhu & Zhu (1989) Markov-modulated queueing systems. Queueing Systems 5, 215-246. [297] R. Pyke (1959) The supremum and infimum of the Poisson process. Ann. Math. Statist. 30, 568-576.
[298] V. Ramaswami (1980) The N/G/1 queue and its detailed analysis. Adv. Appl. Probab. 12, 222-261. [299] C.M. Ramsay (1984) The asymptotic ruin problem when the healthy and sick periods form an alternating renewal process. Insurance: Mathematics and Economics 3, 139-143. [300] C.R. Rao (1965 ) Linear Statistical Inference and Its Applications. Wiley.
[301] G.J.K. Regterschot & J.H.A. de Smit (1986) The queue M/G/1 with Markovmodulated arrivals and services. Math. Oper. Res. 11, 465-483. [302] J.M. Reinhard (1984) On a class of semi-Markov risk models obtained as classical risk models in a markovian environment. Astin Bulletin XIV, 23-43. [303] S. Resnick & G. Samorodnitsky (1997) Performance degradation in a single server exponential queueing model with long range dependence. Opns. Res., 235-243. [304] B. Ripley (1987) Stochastic Simulation . Wiley, New York.
[305] C.L.G. Rogers (1994) Fluid models in queueing theory and Wiener-Hopf factorisation of Markov chains. Ann. Appl. Probab. 4, 390-413. [306] T. Rolski (1987) Approximation of periodic queues. J. Appl. Probab . 19, 691707.
BIBLIOGRAPHY
379
[307] T. Rolski, H. Schmidli, V. Schmidt & J. Teugels (1999) Stochastic Processes for Insurance and Finance. Wiley. [308] S.M. Ross (1974) Bounds on the delay distribution in GI/G/1 queues. J. Appi. Probab. 11, 417-421. [309] H.-J. Rossberg & G--Siegel (1974) Die Bedeutung von Kingmans Integralgleichungen bei der Approximation der stationiiren Wartezeitverteilung im Modell GI/C/1 mit and ohne Verzogerung beim Beginn einer Beschiiftigungsperiode. Math. Operationsforsch. Statist. 5, 687-699.
[310] R.Y. Rubinstein (1981) Simulation and the Monte Carlo Method. Wiley. [311] R.Y. Rubinstein & B. Melamed (1998) Modern Simulation and Modelling. Wiley. [312] R.Y. Rubinstein & A. Shapiro (1993) Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization via the Score Function Method. Wiley.
[313] M. Rudemo (1973) Point processes generated by transitions of a Markov chain. Adv. Appl. Probab. 5, 262-286. [314] T. Ryden (1994) Parameter estimation for Markov modulated Poisson processes. Stochastic Models 10, 795-829.
[315] T. Ryden (1996) An EM algorithm for estimation in Markov-modulated Poisson processes. Comp. Statist. Data Anal. 21, 431-447
[316] S. Schlegel ( 1998) Ruin probabilities in perturbed risk models. Insurance: Mathematics and Economics 22, 93-104. [317] L. Schmickler ( 1992 ) MEDA : mixed Erlang distributions as phase-type representations of empirical distribution functions . Stochastic Models 6, 131-156. [318] H. Schmidli ( 1994) Diffusion approximations for a risk process with the possibility of borrowing and interest. Stochastic Models 10 , 365-388. [319] H. Schmidli ( 1995 ) Cramer-Lundberg approximations for ruin probabilities of risk processes perturbed by a diffusion . Insurance: Mathematics and Economics 16, 135-149. [320] H. Schmidli ( 1996) Martingales and insurance risk. Lecture Notes of the 8th Summer School on Probability and Mathematical Statistics ( Varna), 155-188. Science Culture Technology Publishing , Singapore.
[321] H. Schmidli ( 1997a) Estimation of the Lundberg coefficient for a Markov modulated risk model . Scand. Act. J. 1997, 48-57. [322] H. Schmidli ( 1997b) An extension to the renewal theorem and an application in risk theory. Ann. Appl. Probab . 7, 121-133. [323] H. Schmidli ( 1999a) On the distribution of the surplus prior and at ruin. Astin Bulletin 29, 227-244. [324] H . Schmidli ( 1999b) Perturbed risk processes: a review . Th. Stoch. Proc. 5, 145-165. [325] H . Schmidli ( 2001 ) Optimal proportional reinsurance policies in a dynamic setting . Scand. Act. J. 2001 , 40-68. [326] H.L. Seal ( 1969) The Stochastic Theory of a Risk Business. Wiley.
380
BIBLIOGRAPHY
[327] H.L. Seal (1972) Numerical calculcation of the probability of ruin in the Poisson/Exponential case. Mitt. Verein Schweiz. Versich. Math. 72, 77-100. [328] H. L. Seal ( 1972). Risk teory and the single server queue . Mitt. Verein Schweiz. Versich. Math. 72, 171-178. [329] H.L. Seal (1974) The numerical calculation of U(w, t), the probability of nonruin in an interval (0, t). Scand. Act. J. 1974, 121-139. [330] H.L. Seal (1978) Survival Probabilities. Wiley, New York. [331] G.A.F. Seber (1984) Multivariate Observations. Wiley. [332] C : O. Segerdahl ( 1942) Uber einige Risikotheoretische Fagestellungen . Skand. Aktuar Tidsskr. 61, 43-83. [333] C: O. Segerdahl (1955) When does ruin occur in the collective theory of risk? Skand. Aktuar Tidsskr. 1955, 22-36. [334] C: O. Segerdahl (1959) A survey of results in the collective theory of risk. In Probability and statistics - the Harald Cramer volume (ed. Grenander), pp. 279299. Almqvist & Wiksell, Stockholm. [335] B . Sengupta ( 1989) Markov processes whose steady-state distribution is matrixgeometric with an application to the GI/PH/1 queue. Adv. Appl. Probab. 21, 159-180.
[336] B. Sengupta (1990) The semi-Markov queue: theory and applications. Stochastic Models 6, 383-413. [337] M. Shaked & J.G. Shantikumar (1993) Stochastic Orders and Their Applications. Academic Press. [338] E.S. Shtatland ( 1966) On the distribution of the maximum of a process with independent increments. Th. Probab. Appl. 11, 483-487. [339] A. Shwartz & A. Weiss (1995) Large Deviations for Performance Analysis. Chapman & Hall.
[340] E.S.W. Shin (1987) Convolution of uniform distributions and ruin probability. Scand. Act. J. 1987, 191-197. [341] E.S.W. Shin (1989) Ruin probability by operational calculus. Insurance: Mathematics and Economics 8, 243-249. [342] D. Siegmund (1975) The time until ruin in collective risk theory. Mitteil. Verein Schweiz Versich. Math. 75, 157-166.
[343] D. Siegmund (1976) Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4, 673-684. [344] D. Siegmund (1976) The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes . Ann. Probab . 4, 914-924. [345] D. Siegmund (1979) Corrected diffusion approximations in certain random walk problems. Adv. Appl. Probab. 11, 701-719. [346) D . Siegmund ( 1985) Sequential Analysis . Springer-Verlag.
[347] K. Sigman (1992) Light traffic for workload in queues. Queueing Systems 11, . 429-442. [348] K . Sigman ( 1994 ) Stationary Marked Point Processes: An Intuitive Approach. Chapman and Hall.
BIBLIOGRAPHY
381
[349] E. Slud & C. Hoesman ( 1989) Moderate- and large-deviations probabilities in actuarial risk theory. Adv. Appl. Probab. 21, 725-741. [350] W.L. Smith ( 1953) Distribution of queueing times. Proc. Cambridge Philos. Soc 49, 449-461. [351] D .A. Stanford & K.J. Stroinski ( 1994) Recursive method for computing finitetime ruin probabilities for phase-distributed claim sizes. Astin Bulletin 24, 235254.
[352] D. Stoyan (1983) Comparison Methods for Queues and Other Stochastic Models (D.J. Daley ed.). John Wiley & Sons, New York. [353] E. Straub (1988) Non-Life Insurance Mathematics. Springer, New York. [354] B. Sundt ( 1993) An Introduction to Non-Life Insurance Mathematics (3rd ed.). Verlag Versicherungswirtschaft e.V., Karlsruhe. [355] B . Sundt & W.S. Jewell ( 1981 ) Further results on recursive evaluation of compound distributions . Astin Bulletin 12, 27-39.
[356] B . Sundt & J.L. Teugels ( 1995 ) Ruin estimates under interest force . Insurance: Mathematics and Economics 16, 7-22. [357] B. Sundt & J.L. Teugels (1997) The adjustment coefficient in ruin estimates under interest force . Insurance: Mathematics and Economics 19, 85-94.
[358] R. Suri ( 1989) Perturbation analysis : the state of the art and research issues explained via the GI/G/1 queue. Proceeedings of the IEEE 77, 114-137.
[359] L. Takhcs (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley. [360] G.C. Taylor (1976 ) Use of differential and integral inequalities to bound ruin and queueing probabilities . Scand. Act. J. 1976, 197-208.
[361] G.C. Taylor (1978) Representation and explicit calculation of finite-time ruin probabilities. Scand. Act. J. 1978, 1-18. [362] G.C. Taylor (1979) Probability of ruin under inflationary conditions or under experience ratings . Astin Bulletin 16 , 149-162.
[363] G.C. Taylor (1979) Probability of ruin with variable premium rate. Scand. Act. J. 1980 , 57-76. [364] G.C. Taylor (1986) Claims reserving in Non-Life Insurance. North-Holland. [365] J.L. Teugels (1982) Estimation of ruin probabilities . Insurance: Mathematics and Economics 1, 163-175.
[366] O. Thorin (1974) On the asymptotic behaviour of the ruin probability when the epochs of claims form a renewal process. Scand. Act. J. 1974, 81-99. [367] O. Thorin (1977). Ruin probabilities prepared for numerical calculations. Scand. Actuarial J., 65-102.
[368] O. Thorin (1982) Probabilities of ruin. Scand. Act. J. 1982 , 65-102. [369] O. Thorin (1986) Ruin probabilities when the claim amounts are gamma distributed. Unpublished manuscript.
382
BIBLIOGRAPHY
[370] 0. Thorin & N. Wikstad (1977) Numerical evaluation of ruin probabilities for a finite period. Astin Bulletin VII, 137-153. [371] 0. Thorin & N. Wikstad (1977) Calculation of ruin probabilities when the claim distribution is lognormal. Astin Bulletin IX, 231-246. [372] H. Thorisson (2000) Coupling, Stationarity and Regeneration. Springer-Verlag. [373] S. Tacklind (1942) Sur le risque dans les jeux inequitables. Skand. Aktuar. Tidskr. 1942, 1-42.
[374] F. Vazquez-Abad (1999) RPA pathwise derivatives of ruin probabilities. Submitted. [375] N. Veraverbeke (1993) Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insurance : Mathematics and Economics 13, 5762.
[376] A. Wald (1947) Sequential Analysis. Wiley. [377] V. Wallace (1969) The solution of quasi birth and death processes arising from multiple access computer systems. Unpublished Ph. D. thesis, University of Michigan. [378] H.R. Waters (1983) Probability of ruin for a risk process with claims cost inflation. Scand. Act. J. 1983 , 148-164.
[379] M. Van Wouve, F. De Vylder & M. Goovaerts (1983) The influence of reinsurance limits on infinite time ruin probabilities. In: Premium Calculation in Insurance (F. De Vylder, M. Goovaerts, J. Haezendonck eds.). Reidel, Dordrecht Boston Lancaster. [380] W. Whitt (1989) An interpolation approximation for the mean workload in a GI/G/1 queue. Oper. Res. 37, 936-952. [381] W. Willinger, M. Taqqu, R. Sherman & D. Wilson (1995) Self-similarity in highspeed traffic: analysis and modeling of ethernet traffic measurements. Statistical Science 10, 67-85. [382] G.E. Willmot (1994) Refinements and distributional generalizations of Lundberg's inequality. Insurance: Mathematics and Economics 15, 49-63. [383] G.E. Willmot & X. Lin (1994) Lundberg bounds on the tails of compound distributions. J. Appl. Probab. 31, 743-756.
[384] R.W. Wolff (1990) Stochastic Modeling and the Theory of Queues. Prentice-Hall.
Index differential equation 16, 245-248, 341, 361
adjustment coefficient 17, 70-79, 9396, 97, 170-173, 180-182, 201-214, 308,314-316,328330,359 aggregate claims 103-106, 189, 316323
diffusion 3, 15, 17, 205, 302-303 diffusion approximation 17, 117-127 corrected 121-127 duality 13-14, 30-32, 33-34, 141144,185-187,272,292-293
Bessel function 102, 201 Brownian motion 3 , 25-26, 40, 117128,200-201,269,299,301
Edgeworth expansion 113, 318-319 Erlang distribution 7, 86, 217, 226, 360
central limit theorem 60 , 94-96, 110113,281,293-294,318-320 change of measure 26-30, 34-36, 3839,44-47,67-79,98-99,100, 111-117,121-129,135,137141,160-167,178-184,203, 283,287-292,307-312 compound Poisson model 4, 11-12, 14-15, 24-25, 37, 39, 4851. 57-96, 97-129, 135, 227229,242,259-261,285-292, 323 Coxian distribution 147, 218 Cox process 4, 5, 12 Cramer-Lundberg approximation 1617, 71-79, 138-139, 162164,182,203,308 Cramer-Lundberg model: see compound Poisson model cumulative process 334
excursion 155-156, 271-274, 278 gamma distribution 6-7, 79, 91, 207 heavy-tailed distribution 6, 14, 17, 18-19,251-280 heavy traffic 76, 80 -81, 82-83 hyperexponential distribution 7, 7879,86,150,217,226,228229,249-250 integral equation 16 Lindley 143
renewal 64, 74-75, 89, 332-333 Volterra 192-194, 248 Wiener-Hopf 144 interest rate 190, 196-201 inverse Gaussian distribution 76, 9293, 119, 122, 301 Kronecker product- and sum 221, 239,249,346-349
dams: see storage process
383
384
INDEX
ladder heights 47-56, 61-62, 71, 100, 106-108,152-160,227-230, 261-264,275-278,336-339 Laplace transform 15, 65, 99, 108109,123,234,240-244,339 large deviations 129, 203-204, 213214, 306-316 Levy process 3, 15, 36-39, 57-58, 108 life insurance 5, 134, 175 light traffic 81-83 Lindley integral equation 143 process 33-34, 142 likelihood ratio : see change of measure lognormal distribution 9, 251, 257, 260 Lundberg conjugation 69-79 , 98-99, 112113,128-129,134-135,161164,178-182,287-291
equation 16, 25, 69-70, 75-76, 134-135,161,180,287,315 inequality 17-18, 25, 71-79, 113114, 138, 162, 203 Markov additive process 12, 39-47, 52-
53,139-141,148,160-161, 171, 178 -modulation 12, 132-133, 145187,234-240,269-271,304 process 28-30, 38, 39-47, 154, 271-274,348 terminating 215-216, 227-228, 245
martingale 24-26, 27-30, 35, 39, 42, 44,108,161,238,298-299, 304-305
matrix equation , non-linear 155, 157, 230, 234
matrix-exponential distribution 240244 matrix-exponentials 14, 16, 41, 4446,218-221,340-350 multiplicative functional 28-30, 35,
38, 44, 179 NP approximation 318-320
Palm distribution 52-53, 149, 267269 Panjer's recursion 320-323 Pareto distribution 9-10, 86 periodicity 12, 176-185, 269 Perron-Frobenius theory 41-42,349-
350 perturbation 172-173, 295; see also sensitivity analysis phase-type distribution 8, 14, 16, 133,146-148,174,201,215250,350-361 Poisson process Markov-modulated 12 periodic 12, 176-185 non-homogeneous 60 Pollaczeck-Khinchine formula 61-67, 80,259-261,285-287 queue 14 , 141-144, 185-187 GI/G/1 141-144 M/D/1 66-67
M/G/1 13, 32, 37, 96, 144, 229 M/M/1 101 Markov-modulated 185-187 periodic 187
random walk 33-36, 59, 133, 137139,261-264,288-290,297299,302,336-339
385
INDEX
rational Laplace transform 8, 222, 240; see also matrix-exponential distribution regenerative process 264 -268, 280, 292-294, 333-334
regular variation 10, 251, 253, 256258, 260 reinsurance 8, 326-330
renewal process 131, 146, 174, 223226, 331-336 equation 64, 74-75, 89, 332-333 model 12, 131-144, 229-234, 261264 reserve-dependent premiums 14, 189214,244-250,279-280 Rouche roots 158, 233-234, 238 saddlepoint method 115-117, 123, 307-308, 317-318 semi-Markov 147, 162, 335-336 sensitivity analysis 86-93, 172-173, 294-296 shot-noise process 314 simulation 19, 213, 281-296 stable process 15, 120 statistics x, 11, 12, 18-19, 96-93, 152,314,359-361 stochastic control x stochastic ordering 18, 83-86, 168172
storage process 13, 30-32, 191-192, 279-280 subexponential distribution 11, 251280 time change 4, 60, 87, 147, 177 time-reversion 14, 31, 49-50, 54-55, 107,154-157,186,273-274, 338
utility 324, 327
waiting time 141, 186-187 virtual: see workload
Weibull distribution 9, 251, 257, 260 Wiener-Hopf theory 144, 160, 233, 244,262-263,336-339 workload 13, 37, 141-144, 186-187
Advanced Series on Statistical Science & Applied Probability - Vol. 2
A I 11 JjVb
.
I 1!
Ruin Probabilities
l' i | i Yj
,T [Ail i The book is a comprehensive treatment of || I i I \ classical and modern ruin probability theory. Some i (||l I JL I J r of the topics are Lundberg's inequality, the ^W A l \ i l ' ''' Cramer-Lundberg approximation, exact solutions, P'i yfliother approximations (e.g. for heavy-tailed claim size distributions), y finite horizon ruin probabilities, extensions of the classical compound Poisson model to allow f o r reserve-dependent premiums, Markov-modulation or periodicity. Special features of the book are the emphasis on change of measure techniques, phase-type distributions as a computational vehicle and the connection to other applied probability areas like queueing theory.
"This book is a must for anybody working in applied probability. It is a comprehensive treatment of the known results on ruin probabilities..." Short Book Reviews
ISBN 981-02-2293-9 mi u inn i nun I I I I I I i in u
www. worldscientific.com 2779 he
9 "789810ll22293211
