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 cadl
ag, 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.