14.4 The GeneralRd-valued Case 301Next observe thatQ 2 |FT−=Q 1 |FT−: indeed, we have to show thatQ 1
andQ 2 coincide on the sets of the formA∩{T>t},whereA∈Ft,asthese
sets generateFT−.NotingthatZtisequalto1on{T>t}, this becomes
obvious.
Finally we show thatSis a sigma-martingale underQ 2. First note that
Mremains a local martingale underQ 2 asMis continuous at timeT, i.e.,
MT−=MT,andQ 1 andQ 2 coincide onFT−.
As regards the remaining partXˇ+Bof the semi-martingaleSwe have by
(b) above that, fordA-almost each (ω, t),EGω,t[‖y‖Rd]<∞andEGω,t[y]=
−b(ω, t). Thisdoes not necessarilyimply thatXˇ+Bis already a martingale
(or a local martingale) underQ 2 as a glance at Example 14.2.2 reveals. We
may only conclude thatXˇ+Bis aQ 2 -sigma-martingale, as we presently
shall see.
Define
φt(ω)=(EGω,t[‖y‖Rd])−^1 ∧ 1 ,
which is a predictabledA-almost surely strictly positive process. The process
φ·(Xˇ+B) is a process ofQ 2 -integrable variation as
EQ 2[
var‖.‖Rd(φ·(Xˇ+B))]
=EQ 2 [φ·(‖y‖Rd∗Gω,t·A+‖bω,t‖Rd·A)]
≤ 2 EQ 2 [A∞]=2EQ 1 [A∞]≤ 2 ,where the last equality follows from the fact thatQ 1 andQ 2 coincide onFT−
and that,Abeing predictable,A∞isFT−-measurable.
Henceφ·(Xˇ+B) is a process of integrable variation whose compensator is
constant and thereforeφ·(Xˇ+B)isaQ 2 -martingale of integrable variation,
whence in particular aQ 2 -martingale. ThereforeXˇ+Bas well asSareQ 2 -
sigma-martingales.
Summing up: We have proved Proposition 14.4.4 under the additional
hypothesis thatSremains constant after the first timeTwhenSjumps by
at least 1 with respect to‖.‖Rd.
Step 2:Now we drop this assumption and assume w.l.g. thatS 0 =0.Let
T 0 =0,T 1 =Tand define inductively the stopping times
Tk=inf{t>Tk− 1 |‖∆St‖Rd≥ 1 },k=2, 3 ,...so that (Tk)∞k=1increases to infinity. Let
S(k)= (^1) ]]Tk− 1 ,Tk]]·S, k=1, 2 ,....
Note thatS(1) satisfies the assumptions of the first part of the proof,
where we have shown that there is a measureQ 2 ∼P, satisfying (i), (ii), (iii)
above forT=T 1. Now repeat the above argument to choose inductively, for
k=2, 3 ,...,measuresQk+1∼Psuch that
(i) ‖Qk+1−Qk‖< 2 kε− 1 ∧inf
{
2 −kQk[A]
P[A]∣
∣
∣A∈F,P[A]≥^2 −k