158 9 Fundamental Theorem of Asset Pricing
Remark 9.2.6.In Corollary 9.2.4 we cannot replace (∆X)∗by (∆X)T.The
following example illustrates this. We construct a bounded semi-martingale
X such that for eachε>0 there is a stopping timeT with|∆AT|=1
and|∆XT|≤ε. This clearly shows that there is no constantK such that
‖(∆A)T‖p≤K‖(∆X)T‖p. The construction is as follows: For 0≤t<1 put
Xt= 0. We now proceed by recursion. Forna natural number we suppose
the processXis already constructed fort<n. The filtrationFsis defined
asFs=σ(Xu;u≤s)andFs−=σ(Xu;u<s). Att=nwe put a jump
(∆X)nsuch that|(∆X)n|is uniformly distributed over the interval [0,2] and
is independent of the pastFn−of the process. This means that|(∆X)n|is
independent of the variables (∆X) 1 ,...,(∆X)n− 1 .If(X)n−≥0then(∆X)n
is uniformly distributed over the interval [− 2 ,0], otherwise if (X)n−<0then
(∆X)nis uniformly distributed over [0,2]. Forn≤t<n+ 1 we putXt=Xn.
The filtrationFsis clearly right continuous and if we augment it with the null
sets we obtain that the natural filtration ofXsatisfies the usual conditions.
Forε>0 we now defineT=inf{t||(∆X)t|≤ε}. ClearlyT<∞almost
surely and satisfies the desired properties.
IfAis a predictable process of finite variation withA 0 = 0, we can as-
sociate with it a (random) measure onR+.ThevariationofA, a process
denoted byV,isgivenby
Vt=sup{ n
∑k=1|Ask−Ask− 1 |∣
∣
∣
∣
∣
0=s 0 <s 1 < ... < sn=t}
.
The processV is predictable and it also defines a (random) measure on
R+. The processVdefines aσ-finite measureμVon the predictableσ-algebra
onR+×Ω. The definition ofμVis, forKa predictable subset ofR+×Ω:
μV(K)=E[∫∞
0( (^1) K)udVu
]
.
The measureμAis defined in a similar way, but its definition is restricted to
aσ-ring to avoid expressions like∞−∞. It is well-known, see [M 76, Chap. I]),
that the measureμVis precisely the variation measure ofμA. From the Hahn
decomposition theorem we deduce that there is a partition ofR+×Ω, in two
sets,B+andB−, both predictable, such that ( (^1) B+·A)and(− (^1) B−·A)are
increasing. MoreoverV=(( (^1) B+− (^1) B−)·A). For almost allωthe measuredA
onR+is absolutely continuous with respect todV and the Radon-Nikod ́ym
derivative is precisely (^1) F+− (^1) F− whereF± ={t|(t, ω)∈B±}. We will
refer to this decomposition as theHahn decompositionofA.Notethat
the difficulty in the definition of the pathwise decomposition of the measures
dA(ω) comes from the fact that the setsF+andF−have to be glued together
in order to form the predictable setsB+andB−. See [M 76, Chap. I] for the
details of this result which is due to Cath ́erine Dol ́eans-Dade.