The Mathematics of Arbitrage

102 6 The Dalang-Morton-Willinger Theorem

Then on one hand side the strict positivity ofγimplies that

E

[(

Hnk−Hmk

α

,∆S

)

−

∧ 1

]

, k∈N,

remains bounded away from zero, whence ((Hnk−Hmk,∆S))∞k=1does not
converge to zero in measure. Combining this fact with theL^0 -boundedness
(6.12) of ((Hn,∆S))∞n=1we deduce that ((Hnk,∆S))∞k=1cannot converge in
L^0 (Ω,F 1 ,P;]−∞,+∞]), a contradiction to Lemma 6.6.1. Hence (Hn)∞n=1
converges a.s. to someĤ∈L^0 (Ω,F 0 ,P;Rd) for which we haveĤ=P(Ĥ)

and (H,̂∆S)=g 0. Assertions (i) and (ii) now follow by the same arguments
as discussed before.
As regards (iii) letH∈L∞(Ω,F 0 ,P;Rd) and estimate again

EQ̂[(H,∆S)|F 0 ]

=cEP

[

(H,∆S)U′

(

(H,̂∆S)

)

|F 0

]

=cEP

[

lim

α→ 0

U((Ĥ+αH,∆S))−U((H,̂ ∆S))

α

∣

∣

∣

∣

∣

F 0

]

≤ 0 ,

where again we have used in the last inequality the monotone convergence
theorem, the concavity ofUand (6.10).

6.7 Proof of the Dalang-Morton-Willinger Theorem forT≥

forT≥1 by Induction onT

Proof of Theorem 6.1.1.We proceed by induction onT.ForT= 1 Theorem
6.5.1 applies. So suppose Theorem 6.1.1 holds true forT−1.
We now consider the process (St)Tt=1 adapted to the filtration (Ft)Tt=1.
Because of the inductive hypothesis we suppose that there is a probability
measureQ^1 , defined onFT,equivalenttoP,andsothat

(i) dQ

1
dP is bounded,
(ii) S 1 ,...,STare inL^1 (Ω,FT,Q^1 ),
(iii) (St)Tt=1is aQ^1 -martingale, i.e., for allt≥1,A∈Ftwe have
∫

A

StdQ^1 =

∫

A

St+1dQ^1.

The one-step result in the DMW Theorem (Theorem 6.5.1 or Proposition
6.6.2) applied to the process (St)^1 t=0, the probability space (Ω,F 1 ,Q^1 )andthe
filtration (Ft)^1 t=0, gives us a bounded functionf 1 so that:f 1 isF 1 -measurable,
f 1 >0,EQ 1 [f 1 ]=1,EQ 1 [|S 1 |f 1 ]<∞,EQ 1 [|S 0 |f 1 ]<∞and for allA∈F 0
we have