7.3 Strategies, Semi-martingales and Stochastic Integration 121(H·S)t:= (H·M)t+(H·A)t,t∈R+, (7.12)is well-defined. One can even pass to not necessarily locally bounded pre-
dictable processes, provided the two terms on the right hand side make sense.
At this stage, around 1980, the pushing for greater and greater generality
came to an end. Through the work of Bichteler [B 81] and Dellacherie [DM 80]
it became clear that the class ofsemi-martingalesdefined in Definition 7.3.2
below is the largest class of processes for which the integration theory can
be generalized from simple integrands to more general ones by continuous
extension. The Bichteler-Dellacherie theorem (see, e.g., [P 90]) tells us that the
semi-martingalesSare precisely those processes which may be decomposed
asS=M+A,whereMis a local martingale andAis of locally bounded
variation. We shall briefly recall this theorem.
The spaceSof bounded simple strategies is equipped with the topology
of uniform convergence, which is given by the norm
‖H‖∞=sup{
‖Ht‖L∞(Ω,Ft,P)∣
∣t∈R+}
.
Definition 7.3.2.(i) Sis a strict semi-martingale if the operator
I:S→L^0 (Ω,F∞,P) and I(H)=(H·S)∞is continuous for the topologies defined respectively by‖.‖∞and by the
convergence in probability,
(ii)Sis a semi-martingale if it is locally a strict semi-martingale.
It is an easy exercise to show thatSis a semi-martingale if‖Hn‖∞→ 0
implies (Hn·S)t→0inL^0 for allt≥0. It is easy to check that, for a
processSof the formS=M+AwhereMis a local martingale andAis a
cadlag process of finite variation i.e. for alltwe have
∫t
0 |dAu|<∞a.s., this
continuity property is satisfied. (Here and in the sequel we follow the usual
terminology to call a process of bounded variation if it is locally of bounded
variation). The Bichteler-Dellacherie theorem asserts that also the converse is
true: a semi-martingaleSin the sense of Definition 7.3.2 can be decomposed
asS=M+Ain the above way.
We say thatSis aspecialsemi-martingale ifS=M+AwhereMis
a local martingale,Ais of finite variation andpredictable. In this case the
decomposition ofSas a sum of a local martingale and a predictable process
of finite variation is unique. We refer to it as the canonical decomposition
(see [DM 80] and [P 90]). One can show that a semi-martingaleSis special if
and only ifSis locally integrable, i.e. if there is a sequence of stopping times
Tn↗∞such thatE
[
sup 0 ≤t≤Tn|St|]
<∞.
We emphasize that being a semi-martingale does not depend onPbut
only on the null sets. In other words, ifSis a semi-martingale underPand
Q∼P,thenSis also a semi-martingale underQ. However, ifSis special for
PandQ∼Pis another probability equivalent toP,thenSdoes not need