The Mathematics of Arbitrage

6.3 The Selection Principle 89 HX= { f:Ω→Rd ∣ ∣fisF 0 -measurable andPf=f} (6.4) By construction we have forg∈EXandf∈HXthat (f, ...

90 6 The Dalang-Morton-Willinger Theorem inL^0 (Ω,F,P;K), i.e., a sequence ofK-valued functions depending onω∈Ω in a measurable ...

6.3 The Selection Principle 91 If the setIk−^1 (ω) is defined, find the smallest 1≤jk≤Nksuch that (fn(ω))n∈Ik− (^1) (ω)lies infi ...

92 6 The Dalang-Morton-Willinger Theorem As regards (ii) define ∆(ω) = lim sup n→∞ d(fn(ω),f 0 (ω)) so thatC={∆> 0 }. Modify ...

6.4 The Closedness of the ConeC 93 As regards (i) note that the assumption of the a.s. boundedness of ((Hn,∆S))∞n=1implies that ...

94 6 The Dalang-Morton-Willinger Theorem (ii)IfSsatisfies (NA) then the cone C=K−L^0 +(Ω,F 1 ,P) is closed inL^0 (Ω,F 1 ,P)too. ...

6.5 The DMW Theorem forT=1 95 Theorem 6.5.1.LetS=(S 0 ,S 1 )be an(F 0 ,F 1 )-adaptedRd-valued process satisfying the (NA) condit ...

96 6 The Dalang-Morton-Willinger Theorem 6.6 A Utility-based Proof of the DMW Theorem forT=1 forT=1 We give another proof of the ...

6.6 The DMW Theorem forT= 1 via Utility Maximisation 97 defines the desired equivalent martingale measureQ,wherec>0 is a suit ...

98 6 The Dalang-Morton-Willinger Theorem oddnandg′′n:=gn′ for evenn, we may also conclude that (gn′)∞n=1converges in probability ...

6.6 The DMW Theorem forT= 1 via Utility Maximisation 99 ‖Hnk−Hmk‖Rd≥α, so that E [( Hnk−Hmk α ,∆S ) − ∧ 1 ] ≥αγ, contradicting L ...

100 6 The Dalang-Morton-Willinger Theorem (i) There is a unique optimiserĤ∈L^0 (Ω,F 0 ,P;Rd)for the maximisation problem E[U((H ...

6.6 The DMW Theorem forT= 1 via Utility Maximisation 101 for which we then haveH ̃=P(H ̃),‖H ̃‖Rd= 1 a.s. onBcand by Lebesgue’s ...

102 6 The Dalang-Morton-Willinger Theorem Then on one hand side the strict positivity ofγimplies that E [( Hnk−Hmk α ,∆S ) − ∧ 1 ...

6.8 The Closedness ofKin the CaseT≥ 1 103 ∫ A S 0 f 1 dQ 1 = ∫ A S 1 f 1 dQ 1. Let us finally defineQonFTby the rule Q[A]= ∫ A f ...

104 6 The Dalang-Morton-Willinger Theorem Proof.The proof is by induction onT.ForT= 1 the statement is Stricker’s lemma, Theorem ...

6.9 The Closedness ofCin the CaseT≥1 under(NA)Condition 105 the functions H 1 τn |H 1 τn|^1 Aare still inE^1 sinceE^1 is closed ...

106 6 The Dalang-Morton-Willinger Theorem proved as follows. Let (H 1 ,∆S 1 )+f 2 = 0 whereH 1 is in canonical form and f 2 ∈K 2 ...

6.10 The DMW TheoremforT≥1 via Closedness ofC 107 Since (Hn·S)−T is a.s. bounded we have that ((ψ 1 ,∆S 1 )+f)≥0andby the(NA)con ...

108 6 The Dalang-Morton-Willinger Theorem AsCis closed inL^0 (Ω,FT,P), the setC 1 is closed inL^1 (Ω,FT,P 1 ). Obviously C 1 is ...