352 15 A Compactness Principle
lim
n→∞‖(rLn,j−H^0 ,j)·M‖H (^1) (P)=0
(Zj)t= limq↘t
q∈Q+
lim
n→∞
(sLn,j·M)q
whereZjis a well-defined adapted cadlag increasing process.
Finally we paste things together by definingH^0 =
∑∞
j≥ 1 H
0 ,jandZ=
∑∞
j≥ 1 Zj. By Lemma 15.4.12 we have that
Wt= lims↘t
s∈Q+
(Ln·M)sis a well-defined super-martingale. As
Z=(H^0 ·M)−Wis an increasing process and as (H^0 ·M) is a local martingale and a super-
martingale by [AS 94] we deduce from the maximality ofW thatH^0 ·M
is in fact equal toW. Hence (H^0 ·M)−V is increasing and the proof of
Theorem 15.D is finished.
15.5 Application
In this section we apply the above theorems to give a proof of theOptional De-
composition Theoremdue to N. El Karoui, M.-C. Quenez [EQ 95], D. Kramkov
[K 96a], F ̈ollmer-Kabanov [FK 98], Kramkov [K 96b] and F ̈ollmer-Kramkov
[FK 97]. We refer the reader to these papers for the precise statements and
for the different techniques used in the proofs.
We generalise the usual setting in finance in the following way. The process
Swill denote anRd-valued semi-martingale. In finance theory, usually the idea
is to look for measuresQsuch that underQthe processSbecomes a local
martingale. In the case of processes with jumps this is too restrictive and
theideaistolookformeasuresQsuch thatSbecomes asigma-martingale.
A processSis called a Q-sigma-martingale if there is a strictly positive,
predictable processφsuch that the stochastic integralφ·Sexists and is a
Q-martingale. We remark that it is clear that we may require the process
φ·Sto be anH^1 -martingale and that we also may require the processφto
be bounded (compare Chap. 14). As easily seen, local martingales are sigma-
martingales. In the local martingale case the predictable processφcan be
chosen to be decreasing and this characterises the local martingales among the
sigma-martingales. The concept of sigma-martingale is therefore more general
than the concept of local martingale. The setMe(S) denotes the set of all
equivalent probability measuresQonFsuch thatSis aQ-sigma-martingale.
It is an easy exercise to show that the setMe(S) is a convex set. We suppose
that this set is non-empty and we will refer to elements ofMe(S)asequivalent