7.3 Strategies, Semi-martingales and Stochastic Integration 123topology induced by (7.13) (or, equivalently, by (7.14)). The limit process is
denoted byH·S.
The reader should note the subtle, but for our later applications crucial
difference to the sentence following (7.12) above: we shall see in Example 7.3.4
below that it may happen that, for a special semi-martingaleS=M+A,a
predictable processHisS-integrable in the sense introduced above, while the
stochastic integralH·Mdoes not exist as an integral in the sense of a local
martingale. The next theorem clarifies the situation. It will play a crucial role
in later sections.
Theorem 7.3.3.IfSis a special semi-martingale with canonical decomposi-
tionS=M+Aand ifHisS-integrable, thenH·Sis special if and only
if
(i) the processH·Mis defined as an integral of local martingales and
(ii)the processH·Ais defined as a Lebesgue-Stieltjes integral
∫
HudAu.In this case the canonical decomposition ofH·Sis given byH·S=H·M+H·A.
Before giving the proof we present an enlightening example due to M.
Emery. It will be of central importance for Chap. 14 below. ́
Example 7.3.4 ([E 80], see also Example 14.2.2 below).Let (Ω,F,P)bea
probability space on which the following objects are defined: an exponen-
tially distributed random variableTwith parameter 2 (the 2 is only to keep
in line with the notation in Example 14.2.2), i.e.,P[T>α]=e−^2 α,anda
Bernoulli variableB, i.e.,P[B=1]=P[B=−1] =^12 , which is independent
ofT.
The processM=(Mt)t≥ 0 is defined as follows
Mt={
0 , fort<T,
B, fort≥T.Denoting by (Ft)t≥ 0 the natural filtration generated by the processM,
it is straightforward to check (and intuitively rather obvious) thatM is a
martingale with respect to (Ft)t≥ 0. The process jumps at timeTwhere it has
a 50 : 50 chance to either jump up to 1 or down to−1.
Define the processHbyHu=^1 u,foru>0. This (deterministic) process
isM-integrable: indeed, the processes
(
H (^1) {|H|≤n}·S
)∞
n=1converge in the
semi-martingale topology to the processX=H·Mwhich is given by
Xt={
0 , fort<T,
B
T, fort≥T.(7.15)
Morally speaking, one is tempted to believe thatXshould still be a mar-
tingale: the processXhas the same chance of 50 : 50 to jump upwards or
downwards byT^1 at timeT. But, mathematically speaking,Xfails to be a
martingale as we encounter integrability problems: fort>0wehave