The Mathematics of Arbitrage

248 12 Absolutely Continuous Local Martingale Measures

stopping timeμ=∞onGcand equal to inf{t|Lt≤^12 Lσ}on the setG.The
outcome

(^1) G=

(

1

Lμ

−

1

Lσ

)

Lσ (^1) G
is the result of a 1-admissible strategy and clearly produces arbitrage. We may
therefore suppose thatP[G∩F]>0 and hence we also have

∫

GUσ>0. Again
this suffices to show thatUσ>0ontheset{Lσ> 0 }and again implies that
T≤ν. The proof of the lemma is complete now.

Proof of the Main Theorem 12.4.2.We now calculate the decomposition of
the continuous semi-martingaleSunderPF.IfS =M+Ais the Doob-
Meyer decomposition ofSunderPthen, underPF we writeS=M ̃+A ̃
whereA ̃t=At+

∫d〈M,U〉s
Us , see [L 77]. This integral exists for the measurePF
since onF the processUis bounded away from 0. A more explicit formula
forA ̃can be found if we use the structure of〈M, U〉. We thereto use the
Kunita-Watanabe decomposition of theL^2 -martingaleUwith respect to the
martingaleM. This is done in the following way (see [J 79]). The space of
allL^2 -martingales of the formα·Mis a stable space and in fact we have
‖(α·M)∞‖ 2 =E[

∫

α′d〈M, M〉α]. The orthogonal projection ofU∞on this
space is given by (β·M)∞for some predictable processβ, where of course

E

[∫

β′d〈M, M〉β

]

<∞.

In this notation we may write:

d〈M, U〉=d〈M, M〉β.

It follows that also

∫

β′d〈M, M〉β<∞a.s. for the measurePF and the

measuredA ̃can be written as

dA ̃=d〈M, M〉

(

ht+

βt

Ut

)

=d〈M, M〉kt.

Here we have putk=h+Uβto simplify notation.
To prove the(NFLVR)property forSunderPF we use the criterion of
Theorem 12.1.3 above.
Step1: the set of 1-admissible integrands forPF is bounded inL^0 (F).
From the properties ofβandhwe deduce that, for the measure,PF,the
integral ∫
∞

0

k′td〈M, M〉kt<∞ PF-a.s..

ThePF-local martingaleL ̃is now defined as

̃Lt=exp

(

−

∫t

0

k′udM ̃u−

1

2

∫t

0

ku′d〈M, M〉uku

)

.