120 7 A Primer in Stochastic Integration
To develop the natural degree of generality also for processes with jumps
we have to extend the theory of stochastic integration with respect to a local
martingaleSto the case, whereSis not necessarily locallyL^2 -bounded (but
only locallyL^1 -bounded). One has to replace the (easy)L^2 -theory by some
more refined functional analysis replacingL^2 (P)byH^1 (P)definedin(7.1).
Similarly the (easy) maximal inequality forL^2 (P)-bounded martingales has to
be replaced by the more subtle Burkholder-Davis-Gundy maximal inequality
pertaining to the norm ofH^1 (P). We don’t elaborate on these issues here and
refer, e.g., to [P 90], [RW 00].
Rather we now extend the theory to the case of (cadlag, adapted,Rd-
valued) processesSwhich are not necessarily local martingales. In the case
whenSis locally of bounded variation, i.e.
|S|t=sup
0 ≤t 0 <...<tn≤t∑ni=1∣
∣Sti−Sti− 1∣
∣<∞ a.s., for eacht<∞,the integration theory is, in fact, rather simple as we now can indeed argue
pathwise by considering eachω∈Ω separately. For almost eachω∈Ωthe
cadlag function (St(ω)) 0 ≤t<∞, which is of bounded variation on compact sub-
sets ofR+, defines a sigma-finiteRd-valued Borel-measuredS(ω)onR+;it
is defined on the intervals ]a, b], for 0≤a<b<∞,by
dS(ω)(]a, b]) =Sb(ω)−Sa(ω).Hence, for each bounded measurableRd-valued processH, the stochastic
integral
(H·S)t(ω):=∫t0(Hu(ω),dSu(ω)) (7.11)is well-defined, for almost eachω∈Ωandeacht∈R+, as a classical Lebesgue-
Stieltjes integral on the real positive lineR+. One can still extend the stochas-
tic integral (7.11) to the case, where the processHis not necessarily bounded,
but only such that for almost everyω∈Ωandeacht>0, (Hu(ω)) 0 ≤u≤tis
dS(ω)-integrable. This is anL^1 -theory as opposed to theL^2 -theory encoun-
tered in the setting of Brownian motion above.
We have thus briefly recapitulated the achievements of stochastic integra-
tion theory which were developed starting from the pioneering work of K. Itˆo
[I 44] until the late seventies, notably by the Japanese school and the Stras-
bourg school of probability around Paul Andr ́e Meyer. The notion of stochas-
tic integral was pushed to increasingly more general classes: if the (cadlag,
adapted,Rd-valued) processScanbewrittenasS=M+A,whereMis a
local martingale andAis of locally bounded variation, then there is a good
integration theory forS. For every locally bounded, predictableRd-valued
processHthe stochastic integral