12.3 The No-Arbitrage Property and Immediate Arbitrage 241We now give some more motivation why such a form of arbitrage is called
immediate arbitrage. In the preceding example, for each stopping timeT> 0
the processS−STadmits an equivalent martingale measureQ(T)givenby
the densityfT=exp(−^12
∫ 1
T
√^1
udWu−1
8∫ 1
T1
udu). We can check this by means
of the Girsanow-Maruyama formula or we can check it even more directly via
Itˆo’s rule. This statement shows that if one wants to make an arbitrage profit,
one has to be very quick since a profit has to be the result of an action taken
before timeT.
Let us also note that the processSalso satisfies the(NA)property for
simple integrands. As is well-known it suffices to consider integrands of the
formf (^1) ]]T 0 ,T 1 ]]wherefisFT 0 -measurable (see Chap. 9). Let us show that such
an integrand does not allow an arbitrage. Take two stopping timesT 0 ≤T 1.
We distinguish betweenP[T 0 >0] = 1 andT 0 = 0. (The 0-1-law forF 0
(Blumenthal’s theorem) shows that one of the two holds).
IfT 0 >0,P-a.s., then the result follows immediately from the existence
of the martingale measureQ(T 0 ) for the processS−ST^0.
IfT 0 =0,wehavetoprovethatST 1 ≥0(orST 1 ≤0) implies that
ST 1 =0a.s..
We concentrate on the first case and assume to the contrary thatST 1 ≥
0andP{ST 1 > 0 }>0. Note that it follows from the law of the iterated
logarithm that inf{t|St< 0 }= 0 almost surely, hence the stopping times
Tε=inf{t>ε|St<−ε}
tend to zero a.s. asεtends to zero. Letε>0 be small enough such that
{Tε<T 1 }has positive measure to arrive at a contradiction:
0 >EQ(Tε)[STε (^1) {Tε<T 1 }]=EQ(Tε)[ST 11 {Tε<T 1 }]≥ 0.
The following theorem, which is based on the material developed in
Sect. 12.2, is well-known and has been around for some time. At least in
dimensiond= 1 the result should be known for a long time. For dimension
d>1, the presentation below is, we guess, new.
Theorem 12.3.5.If thed-dimensional, locally bounded semi-martingaleS
satisfies the (NA) property for general admissible integrands, then the Doob-
Meyer decompositionS=M+AsatisfiesdA=d〈M,M〉h,wherehis a
d-dimensional predictable process and whered〈M,M〉denotes the operator-
valued measure defined by the(d×d)-matrix process(〈M,M〉)i,j≤d. The process
hmay be chosen to take its values in the infinitesimal rangeR(d〈M, M〉).
Proof.We apply the criterion of Sect. 12.2. Takefad-dimensional predictable
process such that the measured〈M,M〉fis zero and such that eitherfhas
norm 1 or norm 0. It is obvious that the stochastic integralf′·Mexists and
results in the zero process. If the processf′·Ais not zero then we replacefby
the sign function coming from the Jordan-Hahn decomposition off′·A.This