242 12 Absolutely Continuous Local Martingale Measures
sign functionφis a predictable process equal to +1 or−1. The predictable
integrandg=φfstill satisfiesg·M = 0 but the componentg′·Anow
results in an arbitrage profit. This contradiction shows that the criterion of
Sect. 12.2 is fulfilled and hence the existence of the processhis proved. If we
writed〈M,M〉asσdλfor some control measureλand an operator-valued
predictable processσ, then we may, by the results of Sect. 12.2, suppose that
htis in the range of the operatorσt.
The following theorem is the basic theorem in dealing with the(NA)prop-
erty in the case of continuous price processes.
Theorem 12.3.6.If the continuous semi-martingaleSwith Doob-Meyer de-
compositionS=M+Asatisfies the (NA) property for general admissible
integrands, then we havedS=dM+d〈M,M〉h, where the predictable process
hsatisfies:
(i) T=inf
{
t∣
∣
∣
∫t
0 h′
ud〈M,M〉uhu=∞}
> 0 a.s..(ii)The[0,∞]-valued increasing process
∫t
0 h′
ud〈M,M〉uhuis continuous; in
particular it does not jump to∞.Proof.The existence of the processhfollows from the preceding theorem. The
stopping timeTis well-defined. The first claim on the stopping timeTfollows
from the second, so we limit the proof to the second statement. We will prove
that the set
F={T<∞}∩
{∫
T+εTh′td〈M, M〉tht=∞, ∀ε> 0}
has zero measure. ClearlyFis, by right continuity of the filtration, an element
of theσ-algebraFT. As the process〈M, M〉tis continuous, assertion (ii) will
follow from the fact thatP[F] = 0. Suppose now to the contrary thatFhas
strictly positive measure. We then look at the process (^1) F(S−ST), adapted
to the filtration (FT+t)t≥ 0 and we replace the probabilityPbyPF.Withthis
notation the theorem is reduced to the caseT=0.Thiscaseistreatedinthe
following theorem. It is clear that this will complete the proof.
Immediate Arbitrage Theorem 12.3.7.Suppose thed-dimensional con-
tinuous semi-martingaleShas a Doob-Meyer decomposition given by
dSt=dMt+d〈M, M〉tht
wherehis ad-dimensional predictable process. Suppose that a.s.
∫ε
0
h′td〈M, M〉tht=∞, ∀ε> 0. (12.1)
Then for allε> 0 , there is anS-integrable strategyHsuch thatH=H (^1) [[ 0,ε]],
H·S≥ 0 andP[(H·S)t>0] = 1,foreacht> 0 .Inotherwords,Sadmits
immediate arbitrage at timeT=0.