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)isdi?erentiable,concaveandstrictlyincreasingforeachω∈Ω,ωisthestateandEu(V1)=P(ω)u(V1(ω),ω).ω∈ΩInmostapplications,itsu?cesforutodependonWonly.1Assumptions?Uncertainty–li

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

4、sa?nitenumberNofendogenoussecurities(allsecuri-tiesarecreatedbytraders,noentitiesoutsidethemodel).2Payoutsofthesecurities??d1(ω1)···dN(ω1)??D=?······?d1(ωK)···dN(ωK)dnisarandomvariablede?nedonthesamplespaceΩ,1≤n≤N.?Therearea?nitenumberIoftraders.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)i?therearenumbersθ1,···,θNsuchthatNi(0)?θc(0)=enPnn=1Nc(T)=ei(T)+θndn.n=1θ=(θ1···θN)iscall