investment consumption and endowment

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1、MATH685Z—MathematicalModelsinFinancialEconomicsTopic5–Investment,consumptionandendowmentKeyquestionHowtochoosethebeststrategyfortransformingwealthinvestedattimet=0intowealthatt=1,withpossibleendowmentandaportionofwealthbeingconsumedattimet=0?Measureofperformance–expectedutilitycriterion

2、Letu(W,ω)representtheutilityofamountW,PbeaprobabilitymeasureonΩ,withP(ω)>0forallω∈Ω.u:R×Ω→RisafunctionsuchthatW→u(W,ω)isdifferentiable,concaveandstrictlyincreasingforeachω∈Ω,ωisthestateandEu(V1)=P(ω)u(V1(ω),ω).ω∈ΩInmostapplications,itsufficesforutodependonWonly.1Assumptions•Uncertainty–li

3、stingofallbasiceventsorstatesthatcouldoccurandtheirprobabilities;samplespaceisΩ={ω1,···,ωK};eachstateω∈ΩoccurswithapositiveprobabilityP(ω).•Securities–contractsforafuturedeliveryofcash,contingentontheprevailingstate•Endowments–cashthatthetradersreceivefromsourcesotherthantrading.•Therei

4、safinitenumberNofendogenoussecurities(allsecuri-tiesarecreatedbytraders,noentitiesoutsidethemodel).2Payoutsofthesecurities⎛⎞d1(ω1)···dN(ω1)⎜⎟D=⎝······⎠d1(ωK)···dN(ωK)dnisarandomvariabledefinedonthesamplespaceΩ,1≤n≤N.•ThereareafinitenumberIoftraders.Attime0,tradersknowonlythesetofpossiblest

5、atesΩ,andattimeTtheyknowtheprevailingstateω∈Ω.•Alltradersarepricetakers.Theydeterminetheirdemandsandsuppliesofsecuritieswithoutpayingattentiontotheimpactthattheiractionsontheultimatemarketpricesofsecurities.3•EndowmentprocessAttime0,traderireceivesanendowmentei(0).AttimeT,hereceivesthee

6、ndowmentei(T,ω)contingentontheprevailingstateω.ei={ei(0),ei(T)}istheendowmentprocessoftraderi.•ConsumptionprocessTheuncertainterminalendowmentsandpayoutsofsecuritiesintroduceuncertaintyintotheconsumptionattimeT.Attime0,givensecuritypricesP1,···,PN,eachtraderifacesconstraintsonconsumpti

7、onimposedbyherendowmentprocessei={ei(0),ei(T)}.4BudgetsetForanendowmentprocesseiandsecuritypricesP=(P1,···,PN)thebudgetsetB(ei,P)oftraderiisthesubsetoftheconsumptionsetXsuchthatc∈B(ei,P)ifftherearenumbersθ1,···,θNsuchthatNi(0)−θc(0)=enPnn=1Nc(T)=ei(T)+θndn.n=1θ=(θ1···θN)iscall