The Mathematics of Arbitrage

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 Lemma 6.6.1, which asserts that (Hn,∆S) converges a.s. tog 0.
Summing up, we deduce from the(NA)assumption and the fact that we
choose the maximising sequence (Hn)∞n=1 in the predictable rangeRthat
(Hn)∞n=1converges inRd. Denoting byĤthe limit, we deduce from Fatou’s
lemma thatĤis the optimiser for (6.5).
Now define the measureQonF 1 by
dQ
dP

=cU′

((

H,̂ ∆S

))

,

where the normalising constantc>0 is chosen such thatE

[

dQ

dP

]

=1(note

that by (6.6) we haveE

[∣∣

∣U′

((

H,̂∆S

))∣∣

∣

]

<∞).

To show thatQis indeed a martingale measure we have to show that

EQ[(H,∆S)] = 0, forH∈Rd

or, equivalently,
EQ[(H,∆S)]≤0, forH∈Rd.
To do so, we use a variational argument:

EQ[(H,∆S)]

=cEP

[

(H,∆S)U′

((

H,̂∆S

))]

=clim

α↘ 0

(

E

[

U

((

Ĥ+αH,∆S

))]

−E

[

U

((

H,̂∆S

))]

α

)

≤ 0

wherewehaveusedthefacthatU((
Ĥ+αH,∆S))−U((H,̂∆S))
α is a pointwise in-
creasing function ofα(by the concavity ofU) so that the monotone conver-
gence theorem applies.
We thus have found the desired martingale measureQunder the simpli-
fying assumption thatF 0 is trivial.

We now extend this argument to the case of an arbitraryσ-algebraF 0.

Proposition 6.6.2.Suppose that theRd-valued processS=(S 0 ,S 1 )based on
and adapted to(Ω,(Ft)^1 t=0,P)satisfies (NA). LetU(x)=−e−xand suppose
that
E[|U((H,∆S))|]<∞ (6.10)

for eachH∈L∞(Ω,F 0 ,P;Rd).DenotebyP theF 0 -measurable predictable
projection associated to∆S=S 1 −S 0. Then the following statements hold
true.