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Stochastic Processes and their Applications 108(2003) estimates for the ruin probability in$nite horizon in a discrete-time model withheavy-tailed insurance and$nancial risksQihe Tanga;,Gurami TsitsiashvilibaDepartment of Quantitative Economics,University of Amsterdam,Roetersstraat 11,1018 WB Amsterdam,The NetherlandsbInstitute of Applied Mathematics,Far Eastern Scienti%c Center,Russian Academy of Sciences,690068 Vladivostok,RussiaReceived 17 October 2002;received in revised form 15 May 2003;accepted 7 July 2003AbstractThis paper investigates the probability of ruin within$nite horizon for a discrete time riskmodel,in which the reserve of an insurance business is currently invested in a risky asset.Under assumption that the risks are heavy tailed,some precise estimates for the$nite time ruinprobability are derived,which con$rm a folklore that the ruin probability is mainly determinedby whichever of insurance risk and$nancial risk is heavier than the other.In addition,somediscussions on the heavy tails of the sum and product of independent random variables areinvolved,most of which have their own merits.c?2003 Elsevier B.V.All rights reserved.Keywords:Asymptotics;Dominated variation;Matuszewska indices;Moment index;Ruin probability;Subexponentiality1.Introduction1.1.Background of the present studyRecently,a vast amount of papers has been published on the issue of ruin of aninsurer who is exposed to a stochastic economic environment.Such the environmenthas two kinds of risk,which were called by Norberg(1999)as insurance risk and$nancial risk,respectively.The$rst kind of risk is the traditional liability risk relatedCorresponding author.Tel.:+31-20-5254107;fax:+31-20-5254349.E-mail addresses:q.tanguva.nl(Q.Tang),guramiam.dvo.ru(G.Tsitsiashvili).0304-4149/$-see front matter c?2003 Elsevier B.V.All rights reserved.doi:10.1016/j.spa.2003.07.001300Q.Tang,G.Tsitsiashvili/Stochastic Processes and their Applications 108(2003)299325to the insurance portfolio,and the second is the asset risk related to the investmentportfolio.The aim of this paper is to derive precise estimates for the probability of ruin within$nite time for a discrete time risk model as the initial capital tends to in$nity,withemphasis on heavy-tailed insurance risk and$nancial risk.The stochastic economicenvironment is considered in the following way.First we denote by a random variable(r.v.)Xnthe net payout of the insurer at year n,and by a positive r.v.Ynthe discountfactor(from year n to year n1)related to the return on the investment,n=1;2;:.Then the discounted value of the total risk amount accumulated till the end of year ncan be modelled by a discrete time stochastic processWn=n?i=1Xii?j=1Yj;n=1;2;:(1.1)One sees that model(1.1)is only slightly diFerent from the one proposed by Nyrhinen(1999),as commented by him on p.320.Let the initial capital of the insurer bex0.We denote by (x)=P(Wnx:for some 16n),respectively,(x;T)=P(Wnx:for some 16n6T),the probabilities of the ultimate ruin and of the ruinwithin$nite horizon T.Nyrhinen(1999,2001)investigated the asymptotic behavior of the ruin probabilities(x)and (x;T).Under a general assumption that both sequences Xn:n=1;2;:and Yn:n=1;2;:are independent,Nyrhinen(1999)employed large deviationstechniques in the discrete time model(1.1)and determined a rough(or crude)estimatefor the ruin probability (x)in the formlimx(logx)1log(x)=w;(1.2)where w is a positive parameter which can explicitly be expressed by the distributionsof Yn:n=1;2;:.What is really interesting is that,for the particular case where bothXn:n=1;2;:and Yn:n=1;2;:are sequences of independent and identicallydistributed(i.i.d.)r.v.s,the asymptotic relation(1.2),combining with a result byGoldie(1991),implies a stronger formula for the ruin probability (x)thatlimxxw(x)=C:(1.3)We call that relation(1.3)gives the ultimate ruin probability (x)a precise(or re$ned)estimate.Here,the words rough and precise are adopted from the study on largedeviations;see,for instance,Mikosch and Nagaev(1998,p.83).Unfortunately,theconstant C in relation(1.3)is so involved and ambiguous that it is even not easyto infer directly from the representation given by Goldie(1991)whether or not itis positive.Lately,Nyrhinen(2001)further improved the results to a more generalstochastic case by adding another sequence Ln:n=1;2;:to the above-mentionedstochastic model such that(Xn;Yn;Ln),n=1;2;:,constitute a sequence of i.i.d.randomvectors.The advantage of the modelling in Nyrhinen(2001)is that with the help ofthe sequence Ln:n=1;2;:it is possible to treat continuous time models.Kalashnikov and Norberg(2002)investigated the probability of ultimate ruin in thebivariate LL evy driven risk process.Applying the result in Goldie(1991),they showedQ.Tang,G.Tsitsiashvili/Stochastic Processes and their Applications 108(2003)299325301once again that the ultimate ruin probability decreases at a power rate as given in(1.3)as the reserve increases and is invested in a risky asset.They concluded that riskyinvestments may impair the insurers solvency just as severely as do large claims.We mention that there are enormous papers which are devoted to the ultimate ruinof the continuous and discrete time risk models with risky assets since the pioneeringwork by Harrison(1977).We do not plan,it is also impossible for us,to cite here acomplete list of references.In this connection we refer to the survey paper by Paulsen(1998).We address in the present paper the asymptotic behavior of the$nite time ruinprobability of the risk model(1.1).Compared with the study on the probability ofultimate ruin,the research on the probability of ruin in$nite time in the stochasticeconomic environment is quite scarce.Of course the ruin in$nite time for the casewithout risky investment has been extensively investigated in the past.In this latteraspect we refer to BaltrM unas(1999)and Malinovskii(2000),among others.Both refer-ences aimed at precise estimates for the$nite time ruin probability in the renewal riskmodel,where BaltrM unas(1999)handled the$nite time ruin probability (x;n)for each$xed n=1;2;:in the discrete time version under the assumption that the claimsizeis heavy tailed,and Malinovskii(2000)considered the case where the safety loadingcoeNcient depends on the initial capital x and tends to 0 as x ,and derived someprecise estimates for the$nite time ruin probability (x;T)uniformly for T 0 underthe assumption that the claimsize is light tailed,i.e.satis$es the CramL er conditions.The most related reference on the$nite time ruin corresponding to our case is stillNyrhinen(2001),which derived an asymptotic result for the ruin probability in$nitetime in the rough form thatlimx(logx)1log(x;t logx)=R(t)(1.4)for every large t,where R(t)is an appropriate positive constant,mainly determined bythe distribution of the$nancial risk Y1.All the cited references above except BaltrM unas(1999)did not pay special attention to the case of heavy-tailed risks in their models.In the present paper,we will derive some precise estimates for the ruin probability(x;n),where the$nite horizon n=1;2;:is$xed when we let the initial capitalx tend to in$nity.In doing so,we assume that the insurance risk X1and/or$nancialrisk Y1are heavy tailed.Such the assumption is reasonable in view of the facts that,as remarked by Embrechts et al.(1997),the ruin is mainly due to one large claim,and that,corresponding to our model,the ruin is mainly due to one large insuranceor$nancial risk.Researchers in mathematical$nance usually have special intereston$nite horizon models.They often fail,however,to$nd convenient numerical andanalytical tools in their investigation.The advantage of our consideration is that we$rst derive a recurrence expression for the$nite time ruin probability,which givesrise to the possibility of quantitative investigation and the convenience in calculationon the$nite time ruin probability.Our method originates from the paper by Clineand Samorodnitsky(1994)and some related references,allowing us to derive preciseestimates for the$nite time ruin probability step by step.This diFers from those appliedin the papers cited above.Our results con$rm the folklore that the ruin probability ismainly determined by whichever of insurance risk and$nancial risk is heavier than302Q.Tang,G.Tsitsiashvili/Stochastic Processes and their Applications 108(2003)299325the other.To a certain extent our work also shows that,for the case of heavy-tailedrisks,the$nite time ruin probability decreases approximately at a power rate as theinitial capital tends to in$nity.For the case of light-tailed risks,however,the continuinginvestigation in our next paper Tang and Tsitsiashvili(2003)will show that the$nitetime ruin probability may decrease at an exponential rate,which diFers from those inthe literature.1.2.The outline of the paperSection 2 describes the framework of the present investigation and de$nes the$nitetime ruin probability (x;n)with emphasis on the insurance risk X and$nancial riskY.Speci$cally,an expression for (x;n)is derived,based on a backward recurrenceformula.This result plays a fundamental role in the present work.Section 3 listssome preliminaries about heavy-tailed distributions and related important distributionclasses.Special attention is paid to the tail equivalency of the sum and product oftwo independent random variables.Some discussions on the moment and Matuszewskaindices are also given,most of which are of interests on their own right.The mainresults with their proofs are presented in the last three sections.In Section 4,we give arough look at (x;n)via the moment and Matuszewska indices of X and Y,illustratingthat (x;n)decreases approximately at a power rate as the initial capital x tends toin$nity provided that X or Y has a dominatedly varying tail.Section 5 presents someprecise estimates for (x;n)under the assumption that the insurance risk X is heavytailed and dominates the$nancial risk Y in the sense that P(Y x)=o(P(X x).The other estimates are given in Section 6 corresponding to the inverse case,i.e.thatY is heavy tailed and dominates X.Regretfully,the study on the inverse case inSection 6 is not so complete as that in Section 5.Simple numerical results are added inSection 7.1.3.Notational conventionsThroughout,for a given r.v.X concentrated on(;)with a distribution function(d.f.)F,we denote its right tail byOF(x)=1F(x)=P(X x),and denote its positivepart by X+=max0;X.For two d.f.s F1and F2concentrated on(;),we writeby F1F2(x)=?F1(xt)F2(dt),x,the convolution of F1and F2,andwrite by F21=F1F1the convolution of F1with itself.All limiting relationships,unlessotherwise stated,are for x .Let a(x)0 and b(x)0 be two in$nitesimals,satisfyingl6liminfxa(x)b(x)6 limsupxa(x)b(x)6l+:We write a(x)=O(b(x)if l+,a(x)=o(b(x)if l+=0,and a(x)?b(x)if bothl+and l0;we write a(x).b(x)if l+=1,a(x)&b(x)if l=1,anda(x)b(x)if both.We say that a(x)and b(x)are weakly equivalent if a(x)?b(x),and that a(x)and b(x)are(strongly)equivalent if a(x)b(x).Q.Tang,G.Tsitsiashvili/Stochastic Processes and their Applications 108(2003)2993253032.Framework model2.1.Ruin probabilities,insurance risk and%nancial riskThe basic assumptions of this paper are as follows,as applied by Nyrhinen(1999,2001):P1.The successive net incomes An,n=1;2;:,constitute a sequence of i.i.d.r.v.s withcommon d.f.concentrated on(;),where the net income Anis understoodas the total incoming premium minus the total claim amount within year n;P2.The reserve is currently invested into a risky asset which may earn negative in-terest rnat year n,and rn,n=1;2;:,also constitute a sequence of i.i.d.r.v.s,with common d.f.concentrated on(1;);P3.The two sequences An:n=1;2;:and rn:n=1;2;:are mutually independent.To save notation,we may say that the An,n=1;2;:,are independent replicates ofa generic r.v.A.We will be using this device throughout,letting the symbols speakfor themselves.In the literature,the r.v.Bn=1+rnis often called as the inPationcoeNcient from year n 1 to year n and the r.v.Yn=B1nthe discount factor fromyear n to year n 1,n=1;2;:.In the terminology of Norberg(1999),we call ther.v.s X=A and Y as the insurance risk and$nancial risk,respectively.Clearly,P(0Y)=1.Let the initial capital of the insurance company be x0.We tacitly assume that theincome Anis made or calculated at the end of year n,n=1;2;:.Hence,the surplusof the company accumulated till the end of year n can be characterized by Snwhichsatis$es the recurrence equation below:S0=x;Sn=BnSn1+An;n=1;2;:;(2.1)where Bn=1+rn,n=1;2;:.Clearly,if we assume that the income Anis madeor calculated at the beginning of year n,then this recurrence equation should berewritten asS0=x;Sn=Bn(Sn1+An);n=1;2;:(2.2)Related discussions can be found in Cai(2002),where the author considered twononstandard risk models where the interest rates rn,n=1;2;:,follow a dependentautoregressive structure,and established some Lundberg bounds for the ultimate ruinprobability under some CramL er conditions.In this paper,we shall primarily investigatemodel(2.1),and sometimes simply list some parallel results related to model(2.2)intoremarks accordingly.The model we handle in the sequel,unless otherwise stated,willautomatically be related to(2.1).By the recurrence equation(2.1),we immediatelyobtainS0=x;Sn=xn?j=1Bj+n?i=1Ain?j=i+1Bj;n=1;2;:;(2.3)where?nj=n+1=1 by convention.304Q.Tang,G.Tsitsiashvili/Stochastic Processes and their Applications 108(2003)299325The Snin expression(2.3)is immediately recognized as the value of a perpetuityat the end of year n,n=0;1;:;see Embrechts et al.(1997,Chapter 8.4)for asimple review,where we$nd that the limit behavior(as n )of the processSn:n=0;1;:in(2.3)has been extensively investigated.Kalashnikov and Norberg(2002,p.214)pointed out that the process Sn:n=0;1;:in(2.3)coincides withthe bivariate LL evy driven risk process when embedded at the occurrence times of thesuccessive claims in their model.We de$ne,as usual,the time of ruin in the considered risk model with initial capitalx0 by?(x)=infn=1;2;:Sn0|S0=x:Hence,the probabilities of ruin within$nite time,(x;n),and of ultimate ruin,(x),can be de$ned by(x;n)=P(?(x)6n);respectively,(x)=(x;)=P(?(x):It is obvious that the function (x;n)is nonincreasing in x0;)and nondecreasingin n=1;2;:.This paper also gives some asymptotic results on the time of ruin.Clearly,the probability that the ruin occurs exactly at year n,which is naturally de$nedby?(x;n)=P(?(x)=n),satis$es?(x;n)=(x;n)(x;n 1);n=1;2;:(2.4)Remark 2.1.We have de$ned the ruin probabilities by these formulae mainly to bemore compatible with related earlier studies in this$eld.Unfortunately,as remarkedby our referee,these de$nitions are rather arguable,although this tradition has becomeembedded in the recent literature.A more relevant calculation might be P(?y(x)6n)or P(?y(x)for x0 and n=1;2;:,where?y(x)is a stopping time,de$ned by?y(x)=infn=1;2;:Sny|S0=xfor any regulatory or trigger boundary y0.This stopping time?y(x)may be inter-preted as the$rst time at which th
