The Mathematics of Arbitrage

236 12 Absolutely Continuous Local Martingale Measures

[DM 80]). One of these results says that there is a sequence of predictable
stopping times (Tn)n≥ 1 that exhausts all the jumps ofA.Fixnand let
(τk) 0 ≤k≤Nn be the finite ordered sequence of stopping times obtained from
the set{ 0 , 21 n,..., 2 nn,T 1 ,...,Tn}.

PutVn=

∑Nn− 1
k=0 |Aτk+1−Aτk|^1 [[τk+1,∞[[.
BecauseAis predictable, the variablesAτkareFτk−-measurable and hence
the processesVnare predictable. BecauseVntends pointwise toV,thispro-
cess is also predictable.
(ii) The second part is proved using a constructive proof of the Hahn-
Jordan decomposition theorem. It could be left as an exercise but we promised
to give details. LetV=var(A) as obtained in the first part. Being predictable
and cadlag, the process is locally bounded ([DM 80]) and hence there is an
increasing sequence (Tn)n≥ 1 of stopping times such thatTn↗∞andVTn≤n.
Define now

H=

{

φ

∣

∣

∣

∣

∣

φpredictable andE

[∫

R+

φ^2 dVu

]

<∞

}

.

With the obvious inner product〈φ, ψ〉 =E[

∫

φuψudVu], the spaceH
divided by the obvious subspace{φ|E[

∫

φ^2 dVu]=0}, is a Hilbert space. For
eachnwe define the linear functionalLnonHas

Ln(φ)=E

[∫

[0,Tn]

φudAu

]

.

Since
∣
∣
∣
∣
∣

∫

[0,Tn]

φudAu

∣

∣

∣

∣

∣

≤

∫

[0,Tn]

|φu|dVu≤

√

n

(∫

[0,Tn]

φ^2 udVu

)^12

,

the functionalLnis well-defined. Therefore there isψnsuch that

Ln(φ)=E

[∫

[0,Tn]

φuψundVu

]

.

Clearly the elementsψnandψn+1agree for functionsφsupported on [[0,Tn]].
Hence (with the convention thatT 0 =0)wehavethatψ=

∑

n≥ 1 ψ

n 1
]]Tn− 1 ,Tn]]
is predictable and satisfies for alln:

Ln(φ)=E

[∫

[[ 0,Tn]]

φψ dV

]

.

Let nowCt=At−

∫t

0 ψudVu. We will show thatC= 0. First we show thatCis

continuous. Letτbe a predictable stopping time. Defineφ=∆Cτ (^1) [[τ]].Bydef-
inition ofCand by the property ofψwe have for allnthatE[(∆C)^2 τ∧Tn]=0.