The Mathematics of Arbitrage

122 7 A Primer in Stochastic Integration

to be special forQ. For more details on how to define stochastic integrals
for bounded strategies and for general predictable strategies we refer to the
literature (e.g. [DM 80], [P 90], [J 79], [B 81]).

IfSis a specialRd-valued semi-martingale and ifKis a one-dimensional,
predictable and bounded process, then theRd-valued stochastic integralK·S
which is defined coordinatewise is still special. Indeed, suppose thatS=M+A
whereMis a local martingale andAis a predictable process of finite variation.
Therefore there is a sequence of stopping timesTntending to∞, such that,
for eachn∈N,MTn is inH^1 (P)andATn is of integrable variation and
predictable. The Burkholder-Davis-Gundy ([DM 80], [J 79]) inequalities show
thatK·MTnis still inH^1 and ordinary integration theory shows thatK·ATn
is still of integrable variation. As a result we see thatK·Sis a special semi-
martingale. Moreover the canonical decomposition ofK·SisK·M+K·A.

On the space of one-dimensional semi-martingales we put a vector space
topology, the so-calledsemi-martingale topology, induced by the quasi-norm
or distance function [E 79]

D[S]=

∑∞

n=1

2 −nsup{E[|(K·S)n|∧1]||K|≤ 1 }, (7.13)

where the processesKare assumed to be real-valued and predictable.
In this topology we have, for a sequence (Sk)∞k=1of semi-martingales, that

Sk→0 if and only if, for eacht,wehavethat(K·Sk)t
P
→0 uniformly inK,
|K|≤1,Kreal-valued, predictable. An equivalent metric also inducing the
semi-martingale topology is

D∗[S]=

∑

2 −nsup{E[(K·S)∗n∧1]||K|≤ 1 }. (7.14)

As usualY∗denotes the maximal function defined asYt∗=sup 0 ≤u≤t|Yu|.For
cadlag processesY, the processY∗is again cadlag.
We can also define a stronger distance function,D∞, inducing the semi-
martingale topology onT=[0,∞] as opposed to the time index setT=R+.
This distance is defined as

D∞[S]=sup{E[(K·S)∗∞∧1]||K|≤ 1 }.

For integration theory this topology is typically too strong but in Chap. 9 this
notion will turn out to be useful.

We now extend the class of integrands for a given semi-martingaleSfrom
locally boundedpredictableRd-valued processesHto processesH,whichare
not necessarily locally bounded. We say that a predictableRd-valued pro-

cessHisS-integrable if (H (^1) {|H|≤n}·S)∞n=1forms a Cauchy sequence in the
space of one-dimensional semi-martingales with respect to the semi-martingale