1TheCompoundPoissonprocessDefinitionofcompoundPoissonprocessNatureofthecompoundPoissonprocessApplicationofcompoundPoissonprocess2Definitions:

{N(t),t≥0}isaPoissonprocessofintensityλ,{Yk,k=1,2,...}isasequenceofindependentandidenticallydistributedrandomvariables,andwiththe{N(t),t≥0}independence,if

Definitions{X(t),t≥0}isacompoundPoissonprocess3N(t)isthetimeperiod(0,t]withinacertainnumberofcustomerstothestore,{N(t),t≥0}isaPoissonprocessifYkisthek-thcustomerinthestoreofmoneyspent,then{Yk,k=1,2,...}isasequenceofindependentandidenticallydistributedrandomvariables,andwiththe{N(t),t≥0}independence.

HutchisonX(t)forthestoresin(0,t]periodofturnover,theIsacompoundPoissonprocess.CompoundPoissonProcess4NatureofthecompoundPoissonproces{X(t),t≥0}isacompoundPoissonprocess,then(1){X(t),t≥0}areindependentgrowthprocess;(2)X(t)isthecharacteristicfunctiongY(u)isthecharacteristicfunctionoftherandomvariableY1,λisthearrivalrateofevents.(3)If<∞,thenE[X(t)]=λtE[Y1],D[X(t)]=λtE[YI2]5Proof:Let0≤t0<t1<...<tm,then{Yk,k=1,2,...}isanindependentandidenticallydistributedrandomvariables,soX(t)withindependentincrementsproperty.6(2)7⑶natureoftheconditionalexpectationE[X(t)]=E{E[X(t)|N(t)]}knowledgeandassumptions:Then:Similarly8CompoundPoissonprocessconsistsofarandomvariable{Yn}andconstitute,whenYn≡1,X(t)=N(t),X(t)istheusualPoissonprocess.WhenaskedtodefineacompoundPoissonprocess,analyzespecificissues,wemustfirstdetermineaPoissonprocesswithasequenceofrandomvariables,andthentoverifytheindependenceoftherandomvariablesequenceandthesequenceofrandomvariableswithaPoissonprocess.Onlyaftertheseconditionsaremet,theissuebeforeprocessingorcomputing.9Example:

LetimmigrantstosettleinaplaceofhouseholdsisaPoissonprocess,knowntoanaverageoftwotosettleweek.Lethouseholdpopulationisarandomvariable,andtheprobabilityofahouseholdisa4/6,3istheprobabilityof1/3,thereisaprobabilityof2/3,thereisaprobabilitythatahuman1/6.Andinformthepopulationofallhouseholdsareindependent.Seeking[0,t]weekstosettleinthenumberofimmigrantstothemathematicalexpectationandvariance.10Solution:LetYiisthei-thpopulationofhouseholds,{Yi}areindependent,thetotalnumberofimmigrants

X(t)=

IsacompoundPoissonprocess.Accordingtothemeaningofproblems,λ=2.

E[Y1]=4.1/6+3·1/3+2·1/3+1·1/6=5/2;

E[Y12]=42.1/6+32·1/3+22·1/3+12·1/6=43/6;

So,

E[X(t)]=λtE[Y1]=2t·5/2=5t;

D[X(t)]=λtE[Y12]=2t43/6=43t/3.11Example:

Considertheinsurancecompanywaspreparedtopaymoneytothetotalamountofinsurancereserves.

Assumingtheinsurancepolicyholderatthetime0<τ1<τ2<...<τn

Death;insuranceamountrequestedforthefamiliesofYn.

YnareindependentandaresubordinatetoU[1500-2000]evenlydistributed.

AssumingX(t)representsthenumberof[0~t]perioddeaths.

X(t)isahomogeneousPoissonprocessλ=3.Insurancecompaniesprepare

TheinsuranceamountZt=FindcompoundPoissonprocessE[Z(t)],D[Z(t)]12Solution:toknowthemeaningofthequestions,λ=3Seenbytheprobabilitytheory,mathematicalexpectationevenlydistributedSoBecauseBythe13So

14Example:

SetupanumberofairportpassengerarrivalPoissonprocess,thenumberofaircraftperhourarrivesforfive,passengertotalA,B,Carethreetypes,thenumberofpassengerstheycancarrywere180people,145people,80people.Andthesameprobabilityofthesethreeplanesappeared.Seekingtoreachtheairportwithinthreehoursthemaximumnumberofpassengersmathematicalexpectationofhowmany?Howmuchvariance?15Solution:LetthenumberofpassengeraircraftYirepresentsthefirstnnumberofpassengers,X(t)representativesarriveattheairport,then16λ=5,Sowithinthreehoursofpassengersmathematicalexpectationandvarianceasfollows:SummaryEquivalenttoacompoundPoissonprocesscorrespondingtothetimeinterval[0,t]auxiliaryrandomvariableYnsuperimposedtoaPoissonprocess{X(t),t≥0}togo.SocompoundPoissonprocesscanbecalledoverlayprocess.Thus,accordingtothecompoundPoissonprocesscanoftenfindsomesimilarstoreturnover,insurancecompanyreservesandthelike,