The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1351

M)∗Un)n≥ 1 , showing that ((Ln·M)Un)n≥ 1 is a uniformly bounded sequence in
H^1 (P).
If we chooseC>0 sufficiently big we can makeP[Un<j] small, uniformly
inn; but the sequence of stopping times (Un)n≥ 1 still depends onnand we
have to replace it by just one stopping timeTjwhich works for all (Lnk)k≥ 1
for some subsequence (nk)k≥ 1 ; to do so, let us be a little more formal.
Assume thatT 0 =0,T 1 ,...,Tj− 1 have been defined as well as a subse-
quence, still denoted by (Ln)n≥ 1 , such that the claim is verified for 1,...,j−1;
we shall constructTj. Applying Lemma 15.4.13 toT=jwe may find a sub-
sequence (nk)k≥ 1 such that, for eachk,

P

[(

(Lnk+1·M)−(Lnk·M)

)∗

j≥^2

−k

]

< 2 −(k+j+2).

Now find a numberCj∈R+large enough such that

P

[

(Ln^1 ·M)∗j≥Cj

]

< 2 −(j+1)

and define the stopping timeTjby

Tj=inf

{

t

∣

∣

∣sup

k

|(Lnk·M)t|≥Cj+1

}

∧j

so thatTj≤jand
P[Tj=j]≥ 1 − 2 −j.
Clearly|(Lnk·M)t|≤Cj+1 fort<Tj, whence ((Lnk·M)(Tj)−)k≥ 1 is
uniformly bounded.
We have thatTj≤Unkfor eachk,whereUnkis the stopping time defined
above (withC=Cj+ 1). Hence we deduce from theH^1 (P)-boundedness of
((Lnk·M)Unk)k≥ 1 theH^1 (P)-boundedness of (Lnk·M)Tj. This completes the
inductive step and finishes the proof of Lemma 15.4.14.

Proof of Theorem 15.D.Given a sequence (Hn)n≥ 1 ofw-admissible integrands
choose the sequencesKn∈conv{Hn,Hn+1,...}andLnofw-admissible in-
tegrands and the super-martingalesV andW as in Lemma 15.4.12. Also fix
an increasing sequence (Tj)j≥ 1 of stopping times as in Lemma 15.4.14.
We shall argue locally on the stochastic intervals ]]Tj− 1 ,Tj]]. F i xj∈Nand
let

Ln,j=Ln (^1) ]]Tj− 1 ,Tj]].
By Lemma 15.4.14 there is a constantCj >0 such that (Ln,j)n≥ 1 is
a sequence of (w+Cj)-admissible integrands and such that (Ln,j·M)n≥ 1 is
a sequence of martingales bounded inH^1 (P) and such that the jumps of each
Ln,j·Mare bounded downward byw− 2 Cj. Hence — by passing to convex
combinations, if necessary — we may apply Theorem 15.B to splitLn,jinto
two disjointly supported integrandsLn,j=rLn,j+sLn,jand we may find an
integrandH^0 ,jsupported by ]]Tj− 1 ,Tj]] s u c h t h a t