关于恒定股息界线的一种危险模型毕业论文翻译

word文档可自由复制编辑ABSTRACT摘要Thispaperconsidersariskmodelwithaconstantdividendbarrier.Itfirst这篇文章谈论的是关于恒定股息界线的一种危险模型。pointsoutinterestingconnectionsbetweensomepreviousresultsforthismodelandthoseforspectrallynegativeLévyprocesses.Anexpressionisthenobtainedforthejointdistributionofthesurplusimmediatelypriortoruinandthedeficitatruin,discountedfromthetimeofruin.Suchanexpressioninvolvesknownresultsonthejointdistributionatruinforaclassicalriskmodelwithoutbarrier.Alsodiscussedarethejointdistributionsrelatedtothetimeperiodswhendividendsarepaid.Inparticular,thispaperobtainstheLaplacetransformforthetotaldividendpaymentsuntilruin,andanotherexpressionfortheexpectedpresentvalueofthetotalamountofdividendpaymentsuntilruin.Theresultsdonotrequirethepositiveloadingcondition.1.INTRODUCTIONThispaperconsidersthefollowingriskmodelwithaconstantdividendbarrier.Let{Nt}beaPoissonprocesswithintensity.Let,i_1,2,...,bei.i.d.positiverandomvariableswithacommondensityfunctionp,commondistributionfunctionP,andfinitemean.Theprocessesandareindependent.Forthesurplusprocessisdefinedby(1.1)whereHereisaconstant.Insuchasurplusprocessrepresentstheinitialsurplus.Theclaimsarriveaccordingto.Therandomvariablerepresentsthesizeofthei-thclaim.Premiumfunctionc(x)representstherateatwhichthepremiumiscollectedwhenthecurrentsurplusisx.Theprocessstandsforthesurplusprocessforariskmodelinwhichtheinsurancecompanywouldpayadividendatrateconcethesurplusreacheslevelbandstopthepaymentwheneverthenextincomingclaimbringsthesurplusdowntobelowlevelb.Noticethatruinoccurswithprobability1insuchamodel.Sothepositivesafetyloadingconditionisnotrequired.WriteThenisthesurplusprocessinaclassicalriskmodelwithasinglepremiumrate.Notethatisjustbydefinition.Theevolutionof{Ubt}canbedescribedintuitivelyasfollows.Theprocess{Utb}behaveslike{Ut}whenitsvalueisbetween0andb.Whenever{Utb}reacheslevelb,itstopsgrowingandkeepsitsvalueatbforanexponentialtimewithmean__1untilthenextclaimbringsittounderb.SetTb:_inf{t_0:Utb_0}and_b(u):__{Tb___U0b_u}.TherandomvariableTbistheruintime,and_b(u)istheruinprobabilitygiventhattheinitialsurplusisu.Writedy,dy_0,fortheincrementony.Write{Y_dy}fortheevent{y_Y_y_dy}.Foru,_,x,y_0,letWb(u;x,y)dxdy:__[e__Tb1(Ub_dx,_Ub_dy)_Ub_u],0_x_b,_Tb_Tb0andWb(u;b,y)dy:__[e__Tb1(Ub_b,_Ub_dy)_Ub_u]._Tb_Tb0ThenwriteT,_(u),andW_(u;x,y)forT_,__(u),andW_(u;x,y)._Riskmodelswithadividendbarrierhavebeenstudiedbymanyauthors.RefertoLin,Willmot,andDrekic(2003)andGerberandShiu(2004a)forsurveysonthepreviouswork.Inparticular,thedistributionofruintimeforamodelwithaconstantdividendbarrierwasdiscussedinGerber(1979).InLin,Willmot,andDrekictheGerber-Shiudiscountedpenaltyfunctionwasfurtherconsideredfortheclassicalriskmodelswithaconstantdividendbarrier.Moreprecisely,for__0thediscountedpenaltyfunctioncanbewrittenasm(u):__[e__Tbw(Ub,_Ub)_Ub_u],b,wTb_Tb0wherewisapositivefunctionoftwovariables.Bysolvingthecorrespondingintegro-differentialequations,expressionsofmb,w(u)wereobtainedforcertainchoicesoffunctionw.ClassicalriskmodelswithalineardividendbarrierwerediscussedrecentlyinAlbrecher,Hartinger,andTichy(2005).WorkonrenewalriskmodelswithaconstantdividendbarriercanbefoundinLiandGarrido(2004)andAlbrecher,Claramunt,andMa´rmol(2005).WorkonMarkovmodulatedriskmodelswithaconstantdividendbarriercanbefoundinFrostig(2005).Asimilarmodel(riskmodelwithatwo-steppremiumrate)wasstudiedinZhou(2004).Inthatmodelthesurplusprocessisalsodefinedbyequation(1.1),butwithapremiumfunctionc(x):c1,forx_b,__c2,forx_b,whereci___,i_1,2.ThejointdistributionatruinwasobtainedinZhou(2004)forsuchamodel.Althoughtheabove-mentionedmodelwithatwo-steppremiumratedoesnotexactlyincludetheriskmodelwithadividendbarrierconsideredinthispaper,nevertheless,theapproachtherecanbeimplementedunderthecurrentsetting.Onekeyinanalyzingsuchmodelsistoknowwhen{Ut}firstattainslevelbbeforeruin.For0_v_bwriteT(v):_inf{t_0:t_T,Ut_v}andTb(v):_inf{t_0:t_Tb,Ub_v}twiththeconventionthatinf/0:__.Tb(b)isthenthetimewhendividendisfirstpaidbeforeruin.NoticethatbothT(v)andTb(v)aredefectiverandomvariables.For0_u_vsetB(v_u):__[e__T(v)_U_u]._0ONACLASSICALRISKMODELWITHACONSTANTDIVIDENDBARRIER97ThenB_(v_u)___u]ifv_b.AlsonotethatB__[e__Tb(v)U0b(v_u)isthesameasB(0,v_u)inGerberandShiu(1998).Whenruinoccurs,distinguishaccordingtowhethertherunningmaximumofthesurplusprocessuptotheruintimeissmallerthanvornot.Since_[e__Tb1(Tb(v)__)_Ub_u]__[e__T1(T(v)__)_U_u],00bytheMarkovpropertywehavefor0_u_v_b,_[e__Tb_Ub_u]__[e__T_U_u]_B(v_u)_[e__T_U_v]_B(v_u)_[e__Tb_Ub_v].(1.2)00_0_0Wecanrearrangeitintoamoreappealingform:_[e__Tb_Ub_u]__[e__T_U_u]B(v_u)_00,0_u_v_b.(1.3)__[e__Tb_Ub_v]__[e__T_U_v]00Noticethattheright-handsideofequation(1.3)dependsonlyonbthroughv_b.Withasimilarargumentequation(1.3)canbegeneralizedasmb1,w(u)_mb2,w(u)B(v_u)_,0_u_v_b_b.(1.4)_m(v)_m(v)21b1,wb2,wAnexpressionfortheLaplacetransformofT(v)hasbeenobtainedbefore.Inexpression(6.25)ofGerberandShiu(1998)itwasshownthat,undertheconditionofpositivesafetyloading,e_(_)u__(u)B(v_u)__,0_u_v,(1.5)_e_(_)v__(v)_where_(v):__[e__T__(_)UT1(T__)_U_v]_0isthegeneralizedruinprobability,_(_)istheuniquenonnegativesolutiontoLundberg’sfundamentalequationc___(ˆp(_)_1)__,(1.6)andpˆdenotestheLaplacetransformofp.Equation(1.5)wasalsoobtainedinexpression(2.4)ofZhou(2004)forariskmodelperturbedbyaBrownianmotion.Underthissettingequation(1.6)becomes 2_2/2_c___(pˆ(_)_1)__.ItsproofisanapplicationoftheMarkovproperty.Thepositivesafetyloadingconditionis,indeed,notrequiredthere.Theotherkeyinourapproachistomakeuseofknownresultsonclassicalriskmodelswithoutbarriers.SincethecelebratedworkofGerberandShiu(1997),thejointdistributionsoftheruintime,thesurplusimmediatelybeforeruin,andthedeficitatruinhavebeenstudiedintensively,oftenundertheconditionofpositivesafetyloading.WorkalongthislinealsocanbefoundinChiuandYin(2003),Wu,Wang,andWei(2003),andZhangandWang(2003).TofindexpressionsforWb(u;x,y)andWb(u;b,y)withnorestrictiononsafetyloading,alittle__fluctuationtheoryforLe´vyprocesseswithonlynegativejumpswillbeintroduced.SuchprocessesarealsocalledspectrallynegativeLe´vyprocesses,orprocesseswithstationaryandindependentincrementsandwithskip-freeupwardssamplepaths.Theruinproblemforriskmodelscorrespondstotheso-calledexitprobleminthetheoryforLe´vyprocesses.TwoexcellentreferencesareChapterVIIinBertoin(1996)andBertoin(1997).AlsoseePistorius(2004)foranintroductiononLe´vyprocesses.InthefollowingsomerelatedresultsfromBertoin(1997)willbepresentedunderthesimplersettingofthispaper.98NORTHAMERICANACTUARIALJOURNAL,VOLUME9,NUMBER4AspectrallynegativeLe´vyprocess{Xt}withinitialvalueX0_0isconsideredinBertoin(1997).ItsLaplacetransformisgivenby_[e_Xt]_et(_),__0,wheretheLaplaceexponenthastheform0(_)_m__ 2_2/2__x_(e_1__x1(x__1))_(dx),__and_(dx)isameasureon(__,0)satisfying_0(1_x2)_(dx)__.When{Xt}isreplacedby__{Ut_U0},thecorrespondingLaplaceexponentbecomes(_)_c___(ˆp(_)_1).WriteW_,__0,fortheso-calledscalefunctionfor{Ut_U0}.Byexpression(7)inBertoin(1997)itisdeterminedbyitsLaplacetransform_1_e__xW(x)dx_,___(_).(1.7)_0c___(ˆp(_)_1)__NoticethatW_(0)_1/candthedenominatoroftheright-handsideofequation(1.7)isrelatedtoLundberg’sfundamentalequation.Itisknownfromexpression(6)inBertointhatfor0_u_v,W(u)B(v_u)__.(1.8)_W(v)_ByTheorem1andCorollary2inBertoin(1997),wefurtherhave,forx_(0,b)andy_0,_[e__T*1(U_dx,_U_dy)_U_u]T*_T*0W(u)W(b_x)_____1(u_x)W(u_x)__p(x_y)dxdy,(1.9)W(b)__whereT*:_inf{t_0:Ut_(0;b)}.Moreover,byTheorem7.1ofBertoin(1996),wehave_[e___(x)_U_u]_e_(x_u)_(_),x_u,(1.10)0where_(x):_inf{t_0:Ut_x}.Lettingb→_inequation(1.9),itfollowsfromequations(1.8)and(1.10)thatW(u)W(b_x)W(u;x,y)_lim____1(u_x)W(u_x)__p(x_y)_W(b)_b→___lim(W(u)_[e__T(b)_U_b_x]_1(u_x)W(u_x))_p(x_y)_0_b→__(W(u)e_x_(_)_1(u_x)W(u_x))_p(x_y).(1.11)__Let__0inequations(1.11).ThenW(u;x,y)_(W(u)e_x_(0)_1(u_x)W(u_x))_p(x_y).(1.12)000Noticethatunderthepositivesafetyloadingcondition_(0)_0.ThescalefunctionW_iscloselyrelatedtotheruinprobability_.Itispointedoutinexpression(9)ofBertoin(1997)that_W(x)___kW*(k_1)(x),_0k_0whereW0*nstandsforthen-foldconvolutionofW0.Underthepositiveloadingcondition,acomparisonoftheLaplacetransformsforW0andfor1__revealsthatONACLASSICALRISKMODELWITHACONSTANTDIVIDENDBARRIER991__(x)W0(x)_.(1.13)c___Equation(1.8)hadalreadybeenpointedoutinGerber(1979).ThefollowingidentityisfoundinSection10.1there:h(u)B(v_u)_,(1.14)_h(v)wherehsolvestheintegro-differentialequationxch(x)_(___)h(x)___h(x_y)dP(y)_0.(1.15)0TheLaplacetransformforhcanbeeasilydeterminedfromequation(1.15)asˆhch(0)ˆh(_)_.c___(ˆp(_)_1)__Thenbysettingh(0)_1/cwereachthath_W_.Thefunctionh(x),whichwascalledv(x),wasalsoanalyzedinSection4ofLin,Willmot,andDrekic(2003)andShiu(1998).Wecanreconcileequation(1.5)withequation(1.8)bydirectlyshowingthatW_(v)isproportionaltoe_(_)v___(v).Tothisend,makeuseofexpression(2.16)inGerberandShiu(1998)forpenaltyfunction_(x,y):_e__(_)yandthefactthat_(_)solvesequation(1.6).Itthenfollowsthate_(_)x___(x)solvesequation(1.15).Consequently,undertheconditionofpositivesafetyloading,W(0)e_(_)v__(v)W(v)__(e_(_)v__(v))__.(1.16)_1__(0)_c(1__(0))__Moreover,fromequations(1.11)and(1.16)wecanrecoverexpressions(6.34)and(6.35)inGerberandShiu.Byequation(2.6)inLin,Willmot,andDrekic(2003)weseethatmb1,w(x)_mb2,w(x)solvesequation(1.15).Sowecanalsoreconcileequations(1.4)and(1.14).Theoptimaldividendproblemshavebeenaddressedbyseveralauthors.SeeGerberandShiu(1998)foranearlierworkandreferencestherein,andDicksonandWaters(2004)forrecentworkontheclassicalriskmodel.AlsoseeGerberandShiu(2004a)forworkonaBrownianriskmodel.Inparticular,_[D_(u,b)],theexpectedpresentvalueofthedividendpaymentsuntilruinwhentheinterestrateis_,isstudiedinGerberandShiu(1998)(explicitdefinitionof_[D_(u,b)]willbegiveninSection3).Moreprecisely,theyprovedthat__[D(u,b)]__1.(1.17)__uu_bApplyingequation(1.5),whichrequirestheconditionofpositivesafetyloading,andequation(1.17)totheidentity_[D(u,b)]_B(b_u)E[D(b,b)]___yieldse_(_)u__(u)__[D(u,b)]_,(1.18)__(_)e_(_)b__(b)_wheredenotesthederivativeof___._Combingequations(1.16)and(1.18)yields100NORTHAMERICANACTUARIALJOURNAL,VOLUME9,NUMBER4W(u)h(u)__[D(u,b)]__(1.19)_W(b)h(b)_underthepositivesafetyloadingcondition.Identity(1.19)wasalreadyobtainedinexpression(8.9)ofGerber(1972)foraperturbedriskmodel.Alsoseeexpression(10.1.13)ofGerber(1979).Inthispaperadifferentapproachisadopted.Eitherbystudyingthejointdistributiononthedurationsofdividendpaymentsorbyconditioningonthefirstclaim,wecouldexpress_[D_(b,b)]intermsofpandB_(b_v).Inthiswaywecouldeventuallyexpress_[D_(u,b)]intermsofpandW_.Again,thepositivesafetyloadingconditionisnotrequired.Therestofthispaperisarrangedasfollows.InSection2,first,anexpressionforWb(u;x,y)isfound._Sinceb__m(u)_bb_dx_dyw(x,y)W(u;x,y)_b,w__dyw(b,y)W_(u;b,y),000anexpressionofmb,w(u)followsreadilyforanyw.InthiswaywegeneralizetheresultsinLin,Willmot,andDrekic(2003)toallthepossiblechoicesoffunctionw.TheproofisanadaptationofthatinZhou(2004),whichinvolvesintensiveapplicationsoftheMarkovproperty.TheLaplacetransformisalsofoundinSection2forthelasttimewhenthedividendispaidbeforeruin.InSection3theLaplacetransformisobtainedforthetotalamountofdividendsbeforeruin.Anotherformulaisderivedfortheexpectedpresentvalueofthetotalamountofdividendpaymentsuntilruin.ExplicitresultsarefoundforamodelwithexponentialclaimsinSection4.OurapproachisdifferentfromthatusedinLin,Willmot,andDrekic(2003),GerberandShiu(1998),andsomeotherrelatedworkthatreliesonsolvingcertainintegro-differentialequations.OneadvantageisthatitiscloselyconnectedtothepreviousworkonGerber-ShiufunctionsfortheclassicalriskmodelsandthegeneraltheoryforLe´vyprocesses.Thisapproachdoesnotrequirethepositivesafetyloadingcondition.Italsogoesaroundsometechnicalitiessuchasthedifferentiabilitywhendealingwithdifferentialequations.ItshouldbepointedoutthatthecontinuityassumptiononthedistributionforXiinequation(1.1)isnotnecessary.InBertoin’soriginalworkthereisnoassumptiononthedistributionforXi.Butthispaperwillnotgetintothedetailsinthisrespect.Ofcourse,thisapproachhasitsownshortcoming,mainlybecauseequation(1.7)cannotalwaysbeinvertedanalytically.Inmanycaseswehavetocountonnumericalinversions.2.JOINTDISTRIBUTIONATRUINTofindanexpressionforWbwefirstneedtofindanexpressionfor__T¯W(u;v,x,y)dxdy:__[e1(T(v)__,U_dx,_U_dy)_U_u],_T_T0where0_u_v,0_x_v,y_0.Notethat(u;v,x,y)dxdygivesthediscountedjointdistributionatruinwhenthesurplusprocess¯W_{Ut}nevercanreachlevelvbeforeruin.ApplyingthestrongMarkovpropertyatT(v),wehave__T¯W(u;v,x,y)dxdy__[e1(T(v)__,U_dx,_U_dy)_U_u]_T_T0__[e__T1(U_dx,_U_dy)_U_u]T_T0__[e__T1(T(v)__,U_dx,_U_dy)_U_u]T_T0_W(u;x,y)dxdy__[e__T(v)_U_u]_[e__T1(U_dx,_U_dy)_U_v]_0T_T0_W(u;x,y)dxdy_B(v_u)W(v;x,y)dxdy.___Therefore,ONACLASSICALRISKMODELWITHACONSTANTDIVIDENDBARRIER101¯W(u;v,x,y)_W(u;x,y)_B(v_u)W(v;x,y).(2.1)____Thenextpropositionisoneofourmainresults,inwhichWbcanbeexpressedintermsofpandW_.Proposition2.1Given0_u_b,0_x_b,andy_0,wehave_p(b_y)Wb(b;b,y)_,(2.2)_______bp(z)B(b_b_z)dz0___bp(z)W¯(b_z;b,x,y)dzWb(b;x,y)_0_,(2.3)_______bp(z)B(b_b_z)dz0_andWb(u;x,y)_W¯(u;b,x,y)_B(b_u)Wb(b;x,y).(2.4)____PROOFStartingfromlevelbthesurplusprocess{Utb}spendsanexponentialtimeatlevelbuntilthefirstclaimarrives.Fortheevent{UbTb__b,_UTbb_dy}tooccur,eitherthatclaimcausesruin,orelse{Utb}firstjumpstobetween0andb,thencomesbacktolevelbbeforeruinandstartsalloveragain.ApplyingtheMarkovpropertyatthejumpingtime,wehave_bWb(b;b,y)__p(b_y)_Wb(b;b,y)_p(z)B(b_b_z)dz_.______0Thenequation(2.2)followsbysolvingtheaboveequation.Similarly,startingfromb,fortheevent{UbTb__dx,_UTbb_dy}tooccur,thefirstclaimcannotcauseruin.Consequently{Utb}firstjumpstosomewherebetween0andb.Thenruinoccurseitherbefore{Utb}comesbacktolevelb,orelse{Utb}comesbacktolevelbandstartsalloveragain.Itfollowsthat_bbWb(b;x,y)_¯b__p(z)W(b_z;b,x,y)dz_W(b;x,y)_p(z)B(b_b_z)dz_._______00Thusequation(2.3)alsofollows.Startingfromu,ruinoccurseitherbefore{Utb}everreacheslevelb,orelse{Utb}reachesbbeforeruin.Identity(2.4)alsofollowsfromthestrongMarkovproperty._WecanalsofindanexpressionfortheLaplacetransformofTb.Becauseofequation(1.2)wehaveonlytofind__b]._[e__TbUb0Proposition2.2Wehave_(1_P(b))___bp(z)(_[e__T_U_b_z]_B(b_b_z)_[e__T_U_b])dz_[e__Tb_Ub_b]_00_0.0______bp(z)B(b_b_z)dz0_(2.5)PROOFIntegratingequation(2.1)yieldsb__dx_W¯(u;v,x,y)dy__[e__T_U_u]_B(v_u)_[e__T_U_v]._0_000Thenintegratingequations(2.2)and(2.3),andsummingthemup,weobtainequation(2.5)._102NORTHAMERICANACTUARIALJOURNAL,VOLUME9,NUMBER4FormulasofW0(u;x,y)foraclassicalriskmodelwitheitherpositive,negative,orzerosafetyloadingalsocanbefoundinSchmidli(1999).Byletting_→0_inequation(1.8),wehaveW0(u)_{T(v)___U0_u}_.W0(v)Moreover,¯W0(u;v,x,y)_W0(u;x,y)__{T(v)___U0_u}W0(v;x,y)W0(u)_W0(u;x,y)_W0(v;x,y).W0(v)Wecanthenobtainthejointdistributionof(UbTb_,UbTb)byletting_→0_inProposition2.1.Proposition2.3Given0_u_b,0_x_b,andy_0,wehaveb_p(b_y)W0(b)W0(b;b,y)_b,_W0(b)___0p(z)W0(b_z)dz__bp(z)[W(b)W(b_z;x,y)_W(b_z)W(b;x,y)]dzWb(b;x,y)_00000,0_W(b)___bp(z)W(b_z)dz000andbW0(u)W0(u)bW0(u;x,y)_W0(u;x,y)_W0(b;x,y)_W(b;x,y).W0(b)W0(b)Thelastresultinthissectionconcernsthelasttimewhenadividendispaidbeforeruin.SetT(b):_sup{t_0:t_T,Ub_b}twiththeconventionthatsup/0_0.Proposition2.4For0_u_b,wehave_(1_P(b))B(b_u)_[e__T(b)_Ub_u]__.0______bp(z)B(b_b_z)dz0_PROOFStartingfromb,thenextclaimarrivesafteranexponentialtime.Conditioningonthesizeofthatclaim,wehavethat_b_[e__T(b)_Ub_b]__1_P(b)__[e__T(b)_Ub_b]_p(z)B(b_b_z)dz_.00____0Moreover,T(b)_T(b)ifandonlyifT(b)__.AnapplicationofthestrongMarkovpropertyattimeT(b)gives_[e__T(b)_Ub_u]_B(b_u)_[e__T(b)_Ub_b].0_0Theassertionofthispropositionthusfollows._3.PRESENTVALUEOFDIVIDENDSUNTILRUINWriteRi(resp.,Li)forthetimewhen{Utb}reaches(resp.,leaves)levelbfrombelow(resp.,above)forthei-thtimebeforeruin.Then0_R1_L1_R2_L2_...,andLi_RirepresentsthedurationofONACLASSICALRISKMODELWITHACONSTANTDIVIDENDBARRIER103thei-thperiodofdividendpayment.TheLaplacetransformofR1isgivenbyeitherequation(1.5)or(1.8).LetM:_sup{i:Ri_T(u)}bethetotalnumberofdividendpaymentperiodsbeforeruin.Let__0betheforceofinterestforvaluation.GivenUb0_u,thepresentvalueofthetotalamountofdividendpaymentsuntilruinisdefinedbyMLicMD(u,b):_1__ce__tdt_1_(e__Ri_e__Li).(3.1)_{M_1}{M_1}i_1Ri_i_1Writep0:__{R1___U0_u}andp1:__{R2___R1__}.ThenW0(u)p0__{T(b)___U0_u}_.W0(b)Wewillderiveanexpressionforp1.Startingfromlevelb,afterthefirstdownwardjump,theoverallprobabilityofruinbefore{Utb}everclimbsbacktobagainisbbW(b_z)1_P(b)_0_p(z)_{T(b)___U_b_z}dz_1_P(b)__p(z)_1__dz.000W0(b)Sobp_1_1_P(b)___p(z)_{T(b)___U_b_z}dz_100bW(b_z)0__p(z)dz.0W0(b)Wewanttounderstandthejointdistributionfor{Ri}and{Li}.OurnextresultisaconsequenceoftheMarkovpropertyappliedto{Utb}.Proposition3.1TherandomvariableMfollowsadistributionsuchthat_{M_0}_1_p0and_{M_k}_p0p1k_1(1_p1)fork_1.GivenM_k,{Li_Ri,i_1,...,k}arei.i.d.exponentialrandomvariableswithmean__1;{Ri_1_Li,i_1,...,k_1}arei.i.d.randomvariableswithacommonconditionalLaplacetransform1b_[e__(R2_L1)_M_k]__p(y)B(b_y,b)dy,__0;_p10andR1hasaconditionalLaplacetransformB_(b_u)/p0.Moreover,thetwosequencesandR1areindependent.REMARK3.2ItfollowsfromProposition3.1thatthetotaldividendtimeMD(u,b)_(L_R)_0iici_1isa(defective)geometricsummationofi.i.d.exponentialrandomvariables.AsomewhatsimilarworkondurationofthetimeinredfortheclassicalriskmodelcanbefoundinDosReis(1993)andDicksonandDosReis(1996).WecanfurthercomputetheLaplacetransformoftheundiscounteddividendsuntilruin.104NORTHAMERICANACTUARIALJOURNAL,VOLUME9,NUMBER4Proposition3.3Given__0,wehave__D0(u,b)p0c_(1_p1)_[e]_1_p0_.(3.2)__c_(1_p1)PROOFProposition3.1gives_ck__[e__D0(u,b)]_1_p__p(1_p)pk_1__.0011c___k_1Thenequation(3.2)followsreadily._REMARK3.4NotethatD0(b,b)followsanexponentialdistribution,andconditionalonD0(u,b)_0,D0(u,b)alsofollowsanexponentialdistribution.REMARK3.5ThedistributionforD0(u,b)wasconsideredinSection5ofGerberandShiu(2004a)fortheBrownianriskmodel.SincetheBrownianriskmodelarisesasatime-spacescalinglimitofthePoissonriskmodel(seeGerberandShiu2004b),expression(5.2)inGerberandShiu(2004a)canbeobtainedfromequation(3.2)bytakinganappropriatelimit.Observethat_[D(u,b)]_B(b_u)_[D(b,b)].___Therefore,weneedtofindonlyanexpressionfor_[D_(b,b)].Proposition3.1essentiallygivesanexplicitdescriptiononthedistributionofD_(b,b).ItallowsustocomputedirectlytheexpectedvalueofD_(u,b)similartotheproofforProposition3.3.HansGerberpointsoutadifferentproof,whichthispaperwillborrow.Proposition3.6TheexpectedpresentvalueofdividendsuntilruiniscB(b_b)__[D(b,b)]_.(3.3)_______bp(y)B(b_b_y)dy0_PROOFConsiderthesurplusprocess{Utb},startingatb,uptotimet.Conditioningonthearrivaltimeandthesizeofthefirstclaim,wehavecD(b,b)_e__t_(1_e__t)_e__tD(b,b)____tcb___s__s__s__e (1_e)_e_p(y)D(b_y,b)dyds.(3.4)_0_0Differentiatingequation(3.4)withrespecttot,wehavec0___e__t_(1_e__t)_e__tD(b,b)__e__t(ce__t__e__tD(b,b))___cb__e__t (1_e__t)_e__t_p(y)D(b_y,b)dy.(3.5)__0Lettingt_0inequation(3.5)yieldsONACLASSICALRISKMODELWITHACONSTANTDIVIDENDBARRIER105b0___D(b,b)_c__D(b,b)___p(y)D(b_y,b)dy___0b___D(b,b)_c__D(b,b)__D(b,b)_p(y)B(b_b_y)dy.(3.6)____0Finally,equation(3.3)isobtainedbysolvingequation(3.6)forD_(b,b)._REMARK3.7Combiningequations(1.8)and(3.3),wehavecW(u)__[D(u,b)]_.(3.7)_(___)W(b)___bp(y)W(b_y)dy_0_REMARK3.8SinceW__hsatisfiesequation(1.15),thenequation(3.7)canberewrittenasW(u)__[D(u,b)]_,_W(b)_andwehaverecoveredequation(1.19).Thispaperassumesthatdividendsarepaidaccordingtoabarrierstrategydescribedearlier.Byanoptimaldividendbarrierismeantavalueb*suchthatb*isthevalueofb_0thatminimizesthedenominatorofequation(3.7),b(___)W(b)___p(y)W(b_y)dy.(3.8)__0Wefirstconsiderthecasethat__0.Withthepositivesafetyloadingcondition,itisevidentfromequation(1.13)that1limW0(b)__0.b→_c___Consequently,blim_W(b)____p(y)W(b_y)dy__0.00b→_0Sothereisnomeaningfuloptimaldividendbarrierinthissituation.Ontheotherhand,for__0,withthepositivesafetyloadingconditionornot,wealwayshavethat,forfixedu,T(b)increasesto_asb→_.Byequation(1.8)wethushavethatW_(b)increasesto_asb→_.Consequently,blim(___)W(b)____p(y)W(b_y)dy___.__b→_0Sotheoptimaldividendbarrieralwaysexistsfor__0.Tofindtheoptimaldividendbarrierwhen__0,wecantakederivativesonexpression(3.8)andsolvethefollowingequationforb:b(___)W(b)__p(b)W(0)___p(y)W(b_y)dy_0,(3.9)___0wheredenotesthederivativeofW_W.Wemightnothavetheclosed-formexpressionforWingeneral.__Thenwehavetosettleforanumericalsolutiontoequation(3.9).106NORTHAMERICANACTUARIALJOURNAL,VOLUME9,NUMBER44.ANEXAMPLEThissectionconsidersamodelinwhichtheclaimsizeXifollowsanexponentialdistributionwithmean__1.Thepositiveloadingconditionisnotassumed.Becausep(x)__e__xandpˆ(_)__/(___),equation(1.6)thenbecomesc_2_(c_____)_____0,whichhasapositivesolution____c___(c_____)2_4c___(_)_2candanegativesolution____c___(c_____)2_4c__¯_(_)_.2cInvertingtheLaplacetransform(1.7),wehave(___(_))e_(_)x_(___¯(_))e_¯(_)xW(x)_.__(c_____)2_4c__WethushaveanexpressionforW_(u;x,y)byequation(1.11).Itfollowsfromequation(1.8)(orExample2.6inZhou2004)that(___(_))e_(_)u_(___¯(_))e_¯(_)uB(v_u)_._(___(_))e_(_)v_(___¯(_))e_¯(_)vThenbyequation(2.1)wecanreachanexpressionfor(u;v,x,y)thatleadstoexplicit,butrather¯W_complicated,expressionsforWb(u;x,y)byProposition2.1._Letuscontinuetoworkoutanexpressionfor__u].Since_[e__TbUb0bb(___(_))e_(_)(b_z)_(___¯(_))e_¯(_)(b_z)_p(z)B(b_b_z)dz___z__edz__(_)b_¯(_)b00(___(_))e_(__¯_(_))e_e_(_)b__e_¯(_)b_(___(_))e_(_)b_(___¯(_))e_¯(_)bandbb¯_(_)_p(z)_[e__T_U_b_z]dz___z_¯(_)(b_z)__ee_1__dz000__e_¯(_)b_e__b,thenbyequation(2.5),_¯(_)_e_(_)b__e_¯(_)b_e__b_e_¯(_)b_e__b_e_¯(_)b1______(___(_))e_(_)b_(___¯(_))e_¯(_)b_[e__Tb_Ub_b]_0__(e_(_)b_e_¯(_)b)____(___(_))e_(_)b_(___¯(_))e_¯(_)b_(_(_)__¯(_))e(_(_)__¯(_))b_.(___(___)_(_))e_(_)b_(___(___)_¯(_))e_¯(_)bHence,__u]followsfromequation(1.2)readily.Suchanexpressionwasobtainedinex-_[e__TbUb0pression(6.3)ofLin,Willmot,andDrekic(2003).ONACLASSICALRISKMODELWITHACONSTANTDIVIDENDBARRIER107ByProposition2.4,wehave__e__(b_y)[(___(_))e_(_)b_(___¯(_))e_¯(_)b]_[e__T(b)_Ub_u]_.0[___(___)_(_)]e_(_)b_[___(___)_¯(_)]e_¯(_)bInaddition,_e(__c_)u/c_c_p0_(__c_)b/c_e_c_andc_(e(__c_)b/c_1)p1_(__c_)b/c._e_c_WecaneasilyfindtheLaplacetransformforD0(u,b)byProposition3.3.Moreover,byProposition3.6wehave(___(_))e_(_)u_(___¯(_))e_¯(_)