13.7 The Closure ofG∞and Related Problems 277implies that for alln, necessarily,Eμ[f∧n]≤0. The sequenceEμ[f∧n]is
increasing and bounded above, so it converges and sincef∧ntends tofin
μ-measure the limn→∞Eμ[f∧n] is necessarily the integral offwith respect
toμ. It follows that alsoEμ[f]≤0.
In the same style we can prove thatf∈Kmaxis the limit of a sequence
obtained by stopping. Iffis of the formf=(H·S)∞for someS-integrable
admissible processH,letforn≥1:
Tn=inf{t|(H·S)t>n}.Proposition 13.7.2.Suppose that S is a locally bounded semi-martingale
that satisfies the (NFLVR) property. Iff is 1 -admissible and maximal and
ifμ∈Mba,thenfTntends tofinμ-measure.
Proof.Simply remark that for eachQ∈Me,wehaveQ[Tn<∞]≤n^1.
Theorem 13.7.3.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Iffis in the closureG∞ofG,thenEμ[f]=0
for eachμ∈Mba.
Proof.Take (fn)n≥ 1 a sequence of bounded contingent claims inGthat tends
toffor the topology ofG. This means that sup{‖f−fn‖L (^1) (Q)|Q∈Me}
tends to zero. In particular the sequence (fn)n≥ 1 is a Cauchy sequence inG
and hence for allμ∈Mbawe have thatEμ[|fn−fm|] tends to zero asn, m
tend to infinity. Since, as easily seen, the sequence (fn)n≥ 1 tends tofinμ-
measure, we obtain thatfisμ-integrable andEμ[f] = limn→∞Eμ[fn]=0.
Proposition 13.7.4.Suppose that S is a locally bounded semi-martingale
that satisfies the (NFLVR) property. Supposef∈Kmaxandf=(H·S)∞for
a maximal strategyH.Ifforeachμ∈Mbathe functionfsatisfiesEμ[f]=0,
then for each stopping timeT and each μ∈Mba, the functionfT isμ-
integrable and satisfiesEμ[fT]=0.
Proof.We already showed thatfT is inGand hence isμ-integrable for all
μ∈Mbaand thatEμ[fT]≤0 for allμinMba.
Let us prove the opposite inequality. The sequenceEμ[f∧n] of continuous
functions onMbatends increasingly to 0. As follows from Dini’s theorem, we
have that for eachδ>0 there is a numbernsuch thatEQ[f∧n]>−δfor
allQ∈Me.ButforeachQ∈Mewe have thatEQ[f|FT]=fTand hence
thatEQ[f∧n|FT]≤fT∧n.ThisimpliesthatforallQ∈Meand for alln
large enough, we haveEQ[fT∧n]>−δ. We therefore obtain thatEμ[fT]≥0.
Since the converse inequality was already shown we obtainEμ[fT]=0.
The converse of Theorem 13.7.3 is less trivial and we need the extra as-
sumption thatSis continuous.