156 9 Fundamental Theorem of Asset Pricing
to [P 90] for the details. The predictableσ-algebraPonR+×Ωistheσ-
algebra generated by the stochastic intervals [[0,T]] , w h e r e Truns through all
the stopping times. A predictable processHis a process that is measurable
for theσ-algebraP. For the theory of stochastic integration we refer to [P 90]
and to [CMS 80]. IfXis a real-valued stochastic process the variableX∗is
defined asX∗=supt≥ 0 |Xt(ω)|. This variable is measurable ifXis right or left
continuous. Sometimes we will useX∗twhich is defined as supt≥u≥ 0 |Xu(ω)|.
X∗is called the maximum function and it plays a central role in martingale
theory. IfXis a cadlag process, i.e. a right continuous process possessing left
limits for eacht>0, then ∆Xdenotes the process that describes the jumps
ofX. More precisely (∆X)t=Xt−Xt−and (∆X) 0 =X 0.
IfXis a semi-martingale thenX defines a continuous operator on the
space of bounded predictable processes of bounded support into the spaceL^0.
The space of semi-martingales can therefore be considered as a space of linear
operators. The semi-martingale topology is precisely induced by the topology
of linear operators. It is therefore metrisable by a translation invariant metric
given by the distance ofXto the zero semi-martingale:
D(X)=sup⎧
⎨
⎩
∑
n≥ 12 −nE[min (|(H·X)n|,1)]∣
∣
∣∣
∣
∣
H predictable,|H|≤ 1⎫
⎬
⎭
.
For this metric, the space of semi-martingales is complete, see [E 79].
A semi-martingaleXis called special if it can be decomposed asX=M+A
whereMis a local martingale andAis apredictableprocess of finite varia-
tion. In this case such a decomposition is unique and it is called the canonical
decomposition. It is well-known (see [CMS 80]) that a semi-martingale is spe-
cial if and only ifXis locally integrable, i.e. there is an increasing sequence
of stopping timesTn, tending to∞such thatXT∗nis integrable. The follow-
ing theorem on special semi-martingales will be used on several occasions, for
a proof we refer to [CMS 80].
Theorem 9.2.2.IfXis a special semi-martingale with canonical decompo-
sitionX=M+Aand ifHisX-integrable then the semi-martingaleH·X
is special if and only if
(1)HisM-integrable in the sense of stochastic integrals of local martingales
and
(2)HisA-integrable in the usual sense of Stieltjes-Lebesgue integrals.
In this case the canonical decomposition ofH·Xis given byH·X=H·M+
H·A.
The following theorem seems to be folklore. Essentially it may be deduced
from (the proof of) an inequality of Stein ([St 70]), see also [L 78, Y 78b].
For a survey of these results and related inequalities see [DS 95d]. For con-
venience of the reader we include the easy proof, suggested by Stricker, of
Theorem 9.2.3.