108 6 The Dalang-Morton-Willinger Theorem
AsCis closed inL^0 (Ω,FT,P), the setC 1 is closed inL^1 (Ω,FT,P 1 ). Obviously
C 1 is a convex cone (sinceCis a convex cone). The(NA)condition implies
thatC 1 ∩L^1 +(Ω,FT,P 1 )={ 0 }. The Kreps-Yan Theorem 5.2.2 now gives the
existence of an equivalent probability measureQso thatddPQ 1 is bounded and
so thatEQ[f]≤0 for allf ∈C 1. Obviously allSt∈L^1 (Q)sinceddPQ 1 is
bounded. Since for each coordinatej=1,...,dand eachA∈Ftwe have
(^1) A(Stj+1−Stj)∈C 1 and− (^1) A(Stj+1−Stj)∈C 1 ,wemusthaveEQ[ (^1) AStj+1]=
EQ[ (^1) ASjt]. This shows thatSt=EQ[St+1|Ft].
Let us finally verify assertion (iii) of Theorem 6.1.1. SinceddQP=ddPQ 1 ddPP^1
we have thatddQPis bounded.
6.11 Interpretation of theL∞-Bound in the DMW Theorem.......
in the DMW Theorem
This section is based on [De 00, Chap. VII]. We will suppose that the process
(St)Tt=0adapted to (Ft)Tt=0satisfies(NA)and that it isintegrablewith respect
toP.Theset
Ma={
dQ
dP∈L∞
∣
∣
∣
∣Qa probability such thatSis aQ-martingale}
is non-empty by the DMW Theorem. The space
W 1 ={
(H·S)T|Hpredictable (H·S)T∈L^1 (Ω,FT,P)}
is closed inL^1 (Ω,FT,P)sinceW 1 =K∩L^1 (Ω,FT,P)andKisL^0 -closed.
The setMais the intersection ofW 1 ⊥with the set of probability measures.
For eachk≥1 we define a utility functionuk:L^1 →Ras follows. The setPk
is defined byPk=
{
dQ
dP∣
∣
∣Qa probability,ddQP≤k}
. With this set we define
the coherent monetary utility function
uk(Y) = inf{EQ[Y]|Q∈Pk}=min{EQ[Y]|Q∈Pk}.The utility functionuk is concave andL^1 -continuous since it is Lipschitz.
The setOk={Y|uk(Y)> 0 }is therefore open and convex inL^1 (Ω,FT,P).
Furthermore it contains the cone of strictly positive integrable random vari-
ables. The setPkcan be described asPk={Q|Qa probability,EQ[Y]≥
0 for allY∈Ok}. The interpretation of theL∞-bound in the DMW Theorem
is described in the subsequent result:
Theorem 6.11.1.Under the above assumptions we have, for eachk≥ 1 ,
Ma∩Pk=∅⇐⇒W 1 ∩Ok=∅.