The Mathematics of Arbitrage

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
theorem,
γ=E[(H,∆S)−∧ 1 |F 0 ].

LettingA={γ=0}we have ( (^1) AH, ̃∆S)−= 0 a.s., whence the(NA)
assumption implies that ( (^1) AH, ̃∆S)=0almostsurely.As (^1) AH ̃ satisfies
(^1) AH ̃=P( (^1) AH ̃)wemusthave (^1) AH ̃=0a.s.,sothatA =B. Hence we
have shown thatγis a.s. strictly positive onBc.
Now let (Hn)∞n=1∈L^0 (Ω,F 0 ,P;Rd) be a maximising sequence for (6.11).
By passing to (P(Hn))∞n=1we may assume thatHnis in canonical form, i.e.,
Hn=P(Hn). We infer from Lemma 6.6.1 thatfn=(Hn,∆S)convergesin
measure to someg 0 ∈L^0 (Ω,F 1 ,P;]−∞,∞]).
We now show that (Hn)∞n=1remains bounded inL^0 (Ω,F 0 ,P;Rd), i.e., for
ε>0thereisM>0 such that
P[‖Hn‖Rd≥M]<ε,forn≥ 1. (6.12)
Indeed, otherwise there isα>0 and an increasing sequence (nk)∞k=1such
that
P

[

‖Hnk‖Rd≥k

]

≥α. (6.13)
On the other hand, the boundedness from below of (E[U((Hn,∆S))])∞n=1
implies in particular the boundedness of (E[(Hn,∆S)−])∞n=1, say by a constant
M>0. Hence

H ̃nk=Hnk

k

(^1) {‖Hnk‖≥k}
is in canonical form, satisfies

E

[(

H ̃nk,∆S

)

−

]

≤

M

k

and‖H ̃nk‖Rd≥1onAk={‖Hnk‖Rd≥k}. It follows that

E[γ (^1) Ak]≤E

[(

H ̃nk,∆S

)

−

∧ 1

]

≤

M

k

,

which in view of (6.13) leads to a contradiction to the strict positivity ofγ.
TheL^0 -boundedness of (Hn)∞n=1given by (6.12) implies in particular that
g 0 is a.s. finite.
We now show that (Hn)∞n=1is a Cauchy-sequence inL^0 (Ω,F,P;Rd), i.e.,
with respect to convergence in measure. Indeed, suppose to the contrary that
there isα>0 and sequences (nk)∞k=1,(mk)∞k=1tending to infinity such that

P[‖Hnk−Hmk‖Rd>α]>α.