6.6 The DMW Theorem forT= 1 via Utility Maximisation 97defines the desired equivalent martingale measureQ,wherec>0 is a suitable
normalising constant. This is roughly the strategy of proof and we now have
to work through the details.
We need an auxiliary result (compare [S 92] and Sect. 9.8 below).
Lemma 6.6.1.LetU(x)=−e−xand let(fn)∞n=1be a sequence of real-valued
measurable functions defined on(Ω,F,P)such that
a:= lim
n→∞
sup
gn∈conv{fn,fn+1,...}E[U(gn)]>−∞. (6.8)Then there is a unique elementg 0 ∈L^0 (Ω,F,P;]−∞,∞])such that, for
each sequencegn∈conv{fn,fn+1,...}withlimn→∞E[U(gn)] =a, we have
that(gn)∞n=1converges tog 0 in probability.
Proof.Letgn∈conv{fn,fn+1,...}be such that limn→∞E[U(gn)] =a.Fix
n, mandα>0andlet
An,m,α={
|gn−gm|>αand min(gn,gm)<α−^1}
.
From the uniform concavity ofUon ]−∞,α−^1 +α] we conclude that there
exists aβ=β(α)>0 such that, forn, m∈Nandω∈An,m,α,wehave
U
(
gn(ω)+gm(ω)
2)
≥
U(gn(ω)) +U(gm(ω))
2+β.Forω∈An,m,αwe only apply the concavity ofUto obtain
U
(
gn(ω)+gm(ω)
2)
≥
U(gn(ω)) +U(gm(ω))
2.
Hence, forg=gn+ 2 gmwe have
E[U(g)]>
E[U(gn)] +E[U(gm)]
2+βP[An,m,α].If (gn)∞n=1is a maximising sequence for (6.8) we get therefore for eachα> 0
lim
n,m→∞
P[An,m,α]=0.This is tantamount to saying that (gn)∞n=1 is a Cauchy sequence in
L^0 (Ω,F,P;]−∞,∞]), i.e., with respect to convergence in probability on the
half-closed line ]−∞,∞]. Lettingg 0 = limn→∞gnwe have found our desired
limiting function. We observe that we cannot exclude the possibility, thatg 0
assumes the value +∞on a set of strictly positive probability.
As to the uniqueness ofg 0 ,letgn′be any other sequence in conv{fn,fn− 1 ,...}
such that limn→∞E[U(gn)] = a. Then by the same argument as above,
(g′n)∞n=1is also Cauchy inL^0 (Ω,F,P;]−∞,+∞]). By consideringg′′n:=gnfor