The Mathematics of Arbitrage

278 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory

Theorem 13.7.5.Suppose thatSis continuous and satisfies the (NFLVR)
property. Suppose thatf∈Kmaxand suppose also that for eachμ∈Mbawe
haveEμ[f]=0,thenf∈G∞.

Proof.LetHbe a maximal acceptable strategy such that (H·S)∞=f.For
eachn≥1 putTn=inf{t||(H·S)t|>n}which is the first time the process
H·Sexits the interval [−n,+n]. Clearlyfn=(H·S)Tndefines a sequence
inG∞and we will show thatfntends tofin the topology ofG. Because
−Eμ[f∧n] tends decreasingly to 0 forntending to infinity we infer from
Dini’s theorem and Theorem 13.5.2 that inf{Eμ[f∧n]|μ∈Mba}tends
to zero. It follows that sup{EQ[f−f∧n]|Q∈Me}tends to zero asn
tends to infinity. Because (f−fn)+=(f−n)+=(f−f∧n)weseethat
also sup{EQ[(f−fn)+]|Q∈Me}tends to zero asntends to infinity. By
Theorem 13.3.15 this means thatfntends toffor the norm onG.

Remark 13.7.6.The continuity assumption was only needed to obtain bounded
contingent claims and could be replaced by the assumption that the jumps of
H·Swere bounded.

Example 13.7.7 (Addendum).In the following corollary we use the same no-
tation as in Sect. 13.4, Example 13.4.4 and Theorem 13.4.7. Recall that the
contingent claimf=

∑∞

n=1fnsatisfiesEQ[f] = 0 for allQ∈M.

Corollary 13.7.8.The functionf=

∑∞

n=1fnis inK

maxbut its integral with

respect toμ∈Mbais not always zero.

Proof.Indeed if it were, thenfwould be inG∞.