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,∆S))]→max,H∈L^0 (Ω,F 0 ,P;Rd), (6.11)which is in canonical form (i.e.,Ĥ=P(Ĥ)).
(ii) Every maximising sequenceHn∈L^0 (Ω,F 0 ,P;Rd)for (6.11), verifying
Hn=P(Hn), converges toĤin measure.
(iii)The equation
dQ̂
dP:=
exp(
−
(
H,̂∆S
))
E
[
exp(
−
(
H,̂∆S
))]
defines a measureQ̂onF 1 such thatSis a martingale underQ̂.Before starting the proof of the proposition we remark that condition
(6.10) is a “without loss of generality” assumption: the argument (6.6) above
also applies to the present setting to yield a measureP′ ∼Psuch that
EP′[|U((H,∆S))|]<∞for eachH∈L∞(Ω,F 0 ,P;Rd).
Proof.We shall mimic the above argument in an “F 0 -parameterised” way.
The crucial step is the extension of (6.9) to the present setting. LetPdenote
theF 0 -measurable predictable range projection associated to ∆S.LetB=
{E[‖∆S‖Rd∧ 1 |F 0 ]=0}so thatBis the biggest set (modulo null-sets) in
F 0 on which ∆S=0.NotethatPvanishes onB.
The caseB= Ω again is trivial as we then have ∆S= 0 a.s. so thatĤ=0
andd
Q̂
dP=1.
Excluding this case we define
H 1 ={
H∈H∆S^1
∣
∣‖H‖Rd≥ (^1) Bca.s.}.
Define theF 0 -measurable non-negative function
γ=essinf{E[(H,∆S)−∧ 1 |F 0 ]|H∈H 1 }.
We claim that the functionγis a.s. strictly positive onBc=Ω\B. Indeed
forH 1 ,H 2 ∈H 1 andA∈F 0 , the functionH=H 11 A+H (^21) Ω\Ais inH 1 too;
hence we may — similarly as in the proof of Lemma 6.2.1 — find a sequence
(Hn)∞n=1∈H 1 such that
γ= lim
n→∞
E[(Hn,∆S)−∧ 1 |F 0 ], a.s..
We may supp ose thatHn=PHnand, by multiplying with‖Hn‖−Rd^1 ,that
‖Hn‖Rd= 1 almost surely onBc. We may apply Proposition 6.3.4 above to
find anF 0 -measurably parameterised subsequence (Hτk)∞k=1converging a.s. to
H∈L^0 (Ω,F,P;Rd) such that (H, ̃∆S)−∧1 = lim infn→∞(Hn,∆S)−∧1 a.s.,