Surplus Analysis of Sparre Andersen Insurance Risk Processes

228 Pages·2017·2.89 MB·English

by  Gordon E. Willmot

Springer Actuarial
Gordon E. Willmot
Jae-Kyung Woo
Surplus Analysis
of Sparre Andersen
Insurance Risk
Processes
Springer Actuarial
Editors-in-chief
Hansjoerg Albrecher, Université de Lausanne, Switzerland
Ulrich Orbanz, University of Salzburg, Austria
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Daniel Bauer, University of Alabama, USA
Stéphane Loisel, ISFA, Université Lyon 1, France
Alexander J. McNeil, University of York, UK
Antoon Pelsser, Maastricht University, The Netherlands
Ermanno Pitacco, Università degli Studi di Trieste, Italy
Hailiang Yang, The University of Hong Kong, Hong Kong
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Gordon E. Willmot Jae-Kyung Woo
(cid:129)
Surplus Analysis of Sparre
Andersen Insurance Risk
Processes
123
Gordon E.Willmot Jae-Kyung Woo
Department ofStatistics andActuarial Schoolof Risk andActuarial Studies
Science University of NewSouthWales
University of Waterloo Australia
Canada
SpringerActuarial
ISBN978-3-319-71361-8 ISBN978-3-319-71362-5 (eBook)
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Preface
Thismonographisasummaryofourviewofthecurrentstateoftheartwithrespect
totheanalysisofsurplusandruin-theoreticanalysisfortheclassofSparreAndersen
(renewal)insuranceriskmodels.Thetimingseemsappropriatetous,astheanalysis
of this class of models seems to have reached a plateau. To be more precise, our
approach to the analysis uses defective renewal equation methodology in conjunc-
tionwithideaspromulgatedintheseminalpaperofGerberandShiu(1998),andit
appears(atleasttous)tobedifficulttoextenditsscopeofapplicationinasignificant
manneratthisjuncture.Wehavechosentofocusonanin-depthanalysisofstandard
models as opposed to an exhaustive treatment of models proposed which involve
variationoftheassumptionsinherentinthebasicmodels.Ourhopeisthatthewide
variety of ideas and techniques presented here may be adapted to other models as
appropriate. In many respects, the material presented here may be viewed as a
complement to the monograph by Kyprianou (2013) which focusses primarily on
Lévy-based models. The class of Sparre Andersen models differs from the Lévy
processmodels,withonenotableexception,namelytheclassicalcompoundPoisson
risk model. This latter model is discussed in detail in Chap. 3, after a preliminary
Chap. 2 dealing with various technical ideas which are needed in later chapters.
Structuralanddensityconceptswhicharecommontotheso-calleddependentSparre
AndersenmodelsarepresentedinChap.4.Statisticaldependencybetweentheclaim
amounts and the interclaim time preceding the claim is allowed because the basic
underlyingrandomwalkstructure isunaffectedbysuchdependency,andtheusual
independent Sparre Andersen model (without such dependency) is typically
recoverable easily in any event as a special case. A variety of approaches are
employedinamoredetailedanalysisinChap.5wheneithertheinterclaimtimeor
claim size distributions involve Erlang-type components. Chapter 6 discusses the
somewhat more technically challengingdistribution of the time of ruin in the clas-
sicalPoissonriskmodelandincludesadiscussionofthefinitetimeruin,whichwas
thesubjectoftheearliermonographofSeal(1978).Chapter7discussestherelated
delayed and discrete models, and Chap. 8 considers various quantities associated
with the time of ruin. Technical analysis in the monograph repeatedly involves
solutions of defective renewal equations, and as such, a discussion of bounds on
v
vi Preface
thesesolutionsisthenpresented.Itisworthmentioningthatavarietyofnewresults
are included in the monograph, in addition to new derivations of results obtained
previouslyintheliterature.Thesenoteshavebeenusedofteningraduatecoursesat
theUniversityofWaterlooandhaveundergonenumerousrevisionsinrecentyears
inordertostreamlinethetechnicaltreatmentofthesubjectmatter.Wewishtothank
various individuals for their helpful comments and suggestions on the manuscript,
which have undoubtedly improved its quality and presentation. These include Eric
Cheung, Mirabelle Huynh, David Landriault, Jeff Wong, and Ran Xu. Also, the
authorswishtogivespecialthankstoMs.JoanHattonforherexperttypingoflarge
portionsofthemanuscript.Finally,wewishtothankourfamilies(Deborah,Rachel,
Lauren,andKristenforGW,andEricandBoraforJW)fortheirexplicitandimplicit
supportof this project.
Waterloo, Canada Gordon E. Willmot
Sydney, Australia Jae-Kyung Woo
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Technical Preparation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Lagrange Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Dickson–Hipp Operators and Equilibrium Distributions . . . . . . . . 12
2.3 Defective Renewal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Mixed Erlang Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Coxian Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Gerber–Shiu Analysis in the Classical Poisson Risk Model. . . . . . . . 45
3.1 The Classical Poisson Risk Model. . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Time of Ruin and Related Quantities . . . . . . . . . . . . . . . . . . 46
3.3 Derivation of the Classical Poisson Gerber–Shiu Function . . . . . . 49
3.4 Analysis of the Classical Poisson Gerber–Shiu Function. . . . . . . . 52
4 Gerber–Shiu Analysis in the Dependent Sparre Andersen Model. . . 61
4.1 The Dependent Sparre Andersen Model. . . . . . . . . . . . . . . . . . . . 61
4.2 Conditioning on the Time and Amount of the First Claim . . . . . . 62
4.3 Density Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Conditioning on the First Drop in Surplus . . . . . . . . . . . . . . . . . . 66
4.5 The Distribution and Moments of the Deficit . . . . . . . . . . . . . . . . 72
5 Models Involving Erlang Components . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 A Dependent Coxian Interclaim Time Model . . . . . . . . . . . . . . . . 80
5.2 The Independent Exponential Claim Size Model . . . . . . . . . . . . . 90
5.3 A Dependent Coxian Claim Size Model . . . . . . . . . . . . . . . . . . . 104
5.4 A Dependent Mixed Erlang Claim Size Model. . . . . . . . . . . . . . . 120
6 The Time of Ruin in the Classical Poisson Risk Model. . . . . . . . . . . 127
6.1 Moments of the Time of Ruin. . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Finite Time Ruin and a Partial Integrodifferential Equation. . . . . . 133
6.3 Finite Time Ruin Probabilities for Mixed Erlang Claim
Amounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
vii
viii Contents
6.4 The Joint Distribution of the Time of Ruin and the Deficit . . . . . . 147
6.5 Further Remarks on the Density of the Time of Ruin . . . . . . . . . . 148
7 Related Risk Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.1 Delayed and Stationary Renewal Risk Models . . . . . . . . . . . . . . . 151
7.2 Discrete Renewal Risk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8 Other Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.1 Additional Variables in the Penalty Function . . . . . . . . . . . . . . . . 179
8.1.1 The Surplus Immediately After the Second Last Claim
and the Minimum Surplus Before Ruin. . . . . . . . . . . . . . . 180
8.1.2 The Maximum and the Minimum Surplus Levels
Before Ruin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.1.3 The Discounted Aggregate Claim Costs Until Ruin . . . . . . 189
8.1.4 The Number of Claims Until Ruin . . . . . . . . . . . . . . . . . . 195
8.2 Ordering Properties of Some Ruin-Related Quantities. . . . . . . . . . 201
8.3 Bounds on Solutions to Renewal Equation. . . . . . . . . . . . . . . . . . 204
References.... .... .... .... ..... .... .... .... .... .... ..... .... 217
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 223
Chapter 1
Introduction
Oneofthecentraltopicsininsuranceriskmanagementhashistoricallybeenthestudy
ofrisktheory,andthisremainsthecaseeventoday.Thedangerassociatedwiththe
amountofclaimstobeexperiencedonablockofinsurancebusinessisofinterestnot
onlyoverafixedperiodoftime,butthemannerinwhichtheclaimsdevelopovertime
isalsoveryimportant.Toaddressthisdevelopmentofclaimsexperience,stochastic
processmodelshavelongbeenproposedandstudied.TheearlyworkofLundberg,
Cramer,andothersinthefirstpartofthetwentiethcenturyproposedtheuseofthe
classicalcompoundPoissonmodel,anditisratherstrikingthattheiranalysisofsuch
afundamentalprobleminvolvedveryelegantandsophisticatedmathematics.Inthis
context,interestintheseearlieryearswasprimarilyontheprobabilitythattheinsurer
wouldbecomeinsolvent,referredtoastheruinprobability.
Theunderlyingstochasticprocesswasactuallyofinterestmuchmoregenerally
in applied probability in connection with the analysis of dams, storage processes,
andinqueueingtheory.Assuch,similardevelopmentsinthesefieldsparalleledthat
ofinsurancerisktheory.Thisconnectionwithotherdisciplinesfirstappearstohave
been noticed in an actuarial context by Seal, as is described with references in his
monographSeal(1978).Theconnectionwithqueueingisnowwellunderstoodand
isdescribedelegantlybyChap.6inAsmussenandAlbrecher(2010).
In the latter part of the twentieth century increased attention was being paid to
other functionals associated with the event of ruin. Of central importance in this
contextwasthedeficitortheseverityofruin.Thatis,notonlywastheprobabilityof
theruineventofinterest,butalsotheamountofthedeficitatthetimeofruin.This
quantity is clearly an important and natural extension in the context of insurance
risk management. The idea of incorporating other ruin related quantities into the
analysiswasunifiedandextendedbyGerberandShiu(1998),aswillsubsequently
bediscussedindetail.Itisnoteworthyatthispointinthediscussiontomentionthat
thislatterapproachofGerberandShiu(1998)involvestechnicaltoolswhichseem
to arise less naturally in a queueing framework, but are nevertheless of interest in
©SpringerInternationalPublishingAG2017 1
G.E.WillmotandJ.-K.Woo,SurplusAnalysisofSparreAndersenInsurance
RiskProcesses,SpringerActuarial,https://doi.org/10.1007/978-3-319-71362-5_1