14.4 The GeneralRd-valued Case 303Denote byDthe predictable setD=⋃
k≥ 1[[Tk]]⊆Ω×R+and splitSintoS=Sa+Si,where
Sa= (^1) D·S and Si= (^1) (Ω×R+)\D·S
where the letters “a” and “i” refer to “accessible” and “inaccessible”.Saand
Siare well-defined semi-martingales and in view of the above constructionSi
is quasi-left-continuous.
Denote byCaandCithe cones inL∞(Ω,F,P) associated by (14.1) toSa
andSi, and observe thatCaandCiare subsets ofC(obtained by considering
only integrands supported byDor (Ω×R+)\Drespectively) hence
EQ 1 [f]≤ 0 , forf∈Caand forf∈Ci.
HenceSisatisfies the assumptions of Proposition 14.4.4 with respect to
the probability measureQ 1 and we therefore may find a probability measure,
now denoted byQ̂,Q̂∼P,whichturnsSiinto a sigma-martingale and such
that, for each predictable stopping timeT,wehave
dQ̂|FT
dQ 1 |FT
=
dQ̂|FT−
dQ 1 |FT−. (14.2)
By assumption we have, for eachk=1, 2 ,..., and for each admissible
integrandHsupported by [[Tk]] , t h a t
EQ 1 [(H·S)∞]=EQ 1[
HTk(STk−S(Tk)−)]
≤ 0.
Noting that the inequality remains true if we replaceHbyH (^1) A, for any
F(Tk)−-measurable set A, and using (14.2) we obtain
EQ̂[(H·S)∞]=EQ̂
[
HTk(STk−S(Tk)−)]
≤ 0 (14.3)
for each admissible integrand supported by [[Tk]].
We now shall proceed inductively onk: suppose we have chosen, fork≥0,
probability measuresQ ̃ 0 =Q̂,Q ̃ 1 ,...,Q ̃ksuch that
EQ ̃k[
STj|F(Tj)−]
=S(Tj)−,j=1,...,kand such that, for
εj=ε
2 j+1∧inf{
2 −jQ ̃j[A]
P[A]∣
∣
∣
∣
∣
A∈F,P[A]≥ 2 −j}
,
we have