The Mathematics of Arbitrage

14.5 Duality Results and Maximal Elements 307

Proof.LetZsbe a cadlag version of the density processZs=EQ 0

[

dQ

dQ 0 |Fs

]

.

Now we put

Zs^1 =1 fors<t

Zs^1 =1 fors≥tandω/∈A

Zs^1 =

Zs

Zt

fors≥tandω∈A.

Clearly the probability measureQ 1 defined bydQ 1 = Z∞^1 dQ 0 is in the
setMeσand satisfies the required properties. Indeed on the setAwe have
EQ 1 [w|Ft]=EQ[w|Ft]andEQ 1 [w]≤EQ 0 [w]+k<∞.

In Chap. 7 we recalled ́Emery’s example showing that a stochastic integral
with respect to a martingale is not always a local martingale. In [AS 94] Ansel
and Stricker gave necessary and sufficient conditions under which a stochastic
integral with respect to a local martingale remains a local martingale (see
Theorem 7.3.7). We rephrase part of their result in our context of sigma-
martingales.

Theorem 14.5.5.LetHbeS-integrable andw-admissible (wherew≥ 1 is
any random variable), thenH·Sisalocalmartingale(andhencealsoasuper-
martingale) for eachQ∈MeσsatisfyingEQ[w]<∞.

Proof.Simply writeH·Sas (Hφ−^1 )·(φ·S), where the strictly positive
predictable real-valued processφis such thatφ·Sis aH^1 (Q)-martingale.
Then apply the Ansel-Stricker result.

Remark 14.5.6.The statement of the preceding theorem becomes false if we
replace the conditionQ∈MeσbyQ∈Mes, introduced as in Proposition
14.4.5 above, as the set of equivalent measures, under whichH·Sis a super-
martingale for each admissibleH. To see this, take the processSdefined as
St =0fort≤1andSt=S 1 , a non-degenerate one-dimensional normal
variable, fort≥1. The filtration is simply the filtration generated byS.As
there are no admissible integrands, every equivalent probability measureQ
is inMes. But it is clearly false thatSbecomes aQ-super-martingale (i.e.
EQ[S 1 ]≤0) as soon asEQ[|S 1 |]<∞.

Definition 14.5.7.A random variablew:Ω→R+such thatw≥ 1 is called
a feasible weight function for the processS,if

(1)there is a strictly positive bounded predictable processφsuch that the maxi-
mal function of theRd-valued stochastic integralφ·Ssatisfies(φ·S)∗≤w.
(2)there is an elementQ∈Meσsuch thatEQ[w]<∞.

Remark 14.5.8.For feasible weight functionsw, it might happen that for some
elementsQ∈Meσwe have thatEQ[w]=∞, see the Example 14.5.23 below.