The Mathematics of Arbitrage

354 15 A Compactness Principle

sense: for each semi-martingaleW ̃withW ̃−W increasing and such that ̃W
is aQ-local super-martingale, for eachQ∈Me(S), we haveW=W ̃.

Proof of the Optional Decomposition Theorem 15.5.1.For the given semi-
martingaleV we find a maximal semi-martingaleW as in the preceding
Lemma 15.5.4. We shall find anS-integrable predictable processHsuch that
we obtain a representation of the processW as the stochastic integral over
H, i.e.,
W=H·S

which will in particular prove the theorem.
FixQ 0 ∈Me(S) and apply Lemma 15.5.3 to theQ 0 -local super-martingale
W to find (Tj)j≥ 1 andwj∈L^1 (Ω,F,Q 0 ). Note that it suffices — similarly
as in [K 96a] — to prove Theorem 15.5.1 locally on the stochastic intervals
]]Tj− 1 ,Tj]]. Hence we may and do assume that|W|≤wfor someQ 0 -integrable
weight-functionw≥1. SinceSis a sigma-martingale for the measureQ 0 ,we
can by the discussion preceding the Theorem 15.5.1, and without loss of gener-
ality, assume thatSis anH^1 (Q 0 )-martingale. So we suppose that the weight
functionwalso satisfies|S|≤w,where|.|denotes any norm onRd.
Fix the real numbers 0≤u<vand consider the processuWvstarting at
uand stopped at timev, i.e.,

uWv

t =Wt∧v−Wt∧u,

which is aQ-local super-martingale, for eachQ∈Me(S),andsuchthat
|uWv|≤ 2 w.

Claim 15.5.5.There is an S-integrable 2 w-admissible predictable process
uHv, which we may choose to be supported by the interval]u, v], such that

(uHv·S)∞=(uHv·S)v≥f=uWvv=uWv∞.

Assuming this claim for a moment, we proceed similarly as D. Kramkov
([K 96a, Proof of Theorem 2.1]): fixn∈Nand denote byT(n)thesetoftime
indices

T(n)=

{

j

2 n

∣

∣

∣

∣^0 ≤j≤n^2

n

}

and denote byHnthe predictable process

Hn=

n∑ 2 n

j≥ 1

(j−1)2−nHj 2 −n,

where we obtain(j−1)2

−n

Hj^2

−n
as a 2w-admissible integrand as above with
u=(j−1)2−nandv=j 2 −n. ClearlyHnis a 2w-admissible integrand such
that the process indexed byT(n)

((Hn·S)j 2 −n−Wj 2 −n)j=0,...,n 2 n