The Mathematics of Arbitrage

326 15 A Compactness Principle

15.2 Notations and Preliminaries

We fix a filtered probability space (Ω,F,(Ft)t∈R+,P), where the filtration
(Ft)t∈R+satisfies theusual conditionsof completeness and right continuity.
We also assume thatFequalsF∞. In principle, the letterMwill be reserved
for a cadlagRd-valued local martingale. We assume thatM 0 =0toavoid
irrelevant difficulties att=0.
We denote byO(resp.P)theσ-algebra of optional (resp. predictable)
subsets ofR+×Ω. For the notion of anM-integrableRd-valued predictable
processH=(Ht)t∈R+and the notion of the stochastic integral

(H·M)t=

∫t

0

HudMu

we refer to [P 90] and to [J 79]. Most of the time we shall assume that the
processH·Mis a local martingale (for the delicacy of this issue compare
[E 80] and [AS 94]) and, in fact, a uniformly integrable martingale.
For the definition of the bracket process [M, M] of the real-valued local
martingaleMas well as for theσ-finite, non-negative measured[M, M]on
theσ-algebraOof optional subsets of Ω×R+, we also refer to [P 90]. In the
cased>1 the bracket process [M, M] is defined as a matrix with components
[Mi,Mj]whereM=(M^1 ,...,Md). The process [M, M] takes values in the
cone of non-negative definite (d×d)-matrices. This is precisely the Kunita-
Watanabe inequality for the bracket process. One can select representations so
that for almost eachω∈Ω the measured[M, M] induces aσ-finite measure,
denoted byd[M, M]ω,ontheBorelsetsofR+(and with values in the cone
of non-negative definite (d×d)-matrices).
For anRd-valued local martingaleX,X 0 = 0, we define theH^1 -norm by

‖X‖H 1 =‖(trace([X, X]∞))

(^12)
‖L (^1) (Ω,F,P)

=E

[(∫

∞

0

d(trace([X, X]t))

)^12 ]

≤∞

where trace denotes the trace of a (d×d)-matrix and theL^1 -norm by

‖X‖L^1 =sup

T

E[|XT|]≤∞,

where|.|denotes a fixed norm onRd, where the sup is taken over all finite
stopping timesTand which, in the case of a uniformly integrable martingale
X,equals
‖X‖L 1 =E[|X∞|]<∞.