The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1347

Exactly as in Chap. 9 we introduce the cone

C 1 ,w={f|there is aw-admissible integrandHsuch thatf≤(H·M)∞}.

Theorem 15.4.10.Let Mbe aRd-valued local martingale andw ≥ 1 an
integrable function.
Given a sequence(Hn)n≥ 1 ofM-integrableRd-valued predictable processes
such that
(Hn·M)t≥−w, for alln, t ,

there are convex combinations

Kn∈conv{Hn,Hn+1,...},

and there is a super-martingale(Vt)t∈R+,V 0 =0, such that

lims↘t

s∈Q+

lim

n→∞

(Kn·M)s=Vt fort∈R+,a.s.,

and anM-integrable predictable processH^0 such that

((H^0 ·M)t−Vt)t∈R+ is increasing.

In addition,H^0 ·Mis a local martingale and a super-martingale.

Before proving Theorem 15.D we shall deduce a corollary which is similar
in spirit to Theorem 9.4.2, and which we will need in Sect. 15.5 below. For
a semi-martingaleSwe denote byMe(S) the set of all probability measures
QonFequivalent toP, such thatSis a local martingale underQ.

Corollary 15.4.11.LetSbe a semi-martingale taking values inRdsuch that
Me(S)=∅andw≥ 1 a weight function such that there is someQ∈Me(S)
withEQ[w]<∞.
Then the convex coneC 1 ,w is closed inL^0 (Ω,F,P)with respect to the
topology of convergence in measure.

Proof of Corollary 15.4.11.As the assertion of the corollary is invariant under
equivalent changes of measure we may assume that the original measurePis
an element ofMe(S) for whichEP[w]<∞, i.e., we are in the situation of
Theorem 15.B above. As in the proof of Theorem 15.B we also may assume
thatSis inH^1 (P) and therefore aP-uniformly integrable martingale.
Let
fn=(Hn·S)∞−hn

be a sequence inC 1 ,w,where(Hn)n≥ 1 is a sequence ofw-admissible integrands
andhn≥0. Assuming that (fn)n≥ 1 tends to a random variablef 0 in measure
we have to show thatf 0 ∈C 1 ,w.
It will be convenient to replace the time index set [0,∞[by[0,∞]byclosing
SandHn·Sat infinity, which clearly may be done as the martingale (St)t∈R+