The Mathematics of Arbitrage

304 14 The FTAP for Unbounded Stochastic Processes

‖Q ̃j−Q ̃j+1‖<εj,j=0,...,k− 1.

In addition we assume thatQ ̃jandQ ̃j− 1 agree “before (Tj)−and after

Tj”; this means thatQ ̃jandQ ̃j− 1 coincide on theσ-algebraF(Tj)−and that

the Radon-Nikod ́ym derivative d
Q ̃j
dQ ̃j− 1 isFTj-measurable.
Now consider the stopping timeTk+1:denoteontheset{Tk+1<∞}by
Fωthe jump measure of the jumpSTak+1−Sa(Tk+1)−conditional onF(Tk+1)−.
By (14.3) this (Ω,F(Tk+1)−,P)-measurable family of probability measures on

Rd satisfies the assumptions of Lemma 14.3.5 and we therefore may find
anF(Tk+1)−-measurable family of probability measuresGω,a.s.definedon
{Tk+1<∞}, such that

(i) Fω∼Gωand‖Fω−Gω‖<εk
(ii)EGω[‖y‖Rd]<∞and bary(Gω)=EGω[y]=0.

Letting, similarly as in the proof of Proposition 14.4.4 above,

Y(ω, x)=

dFω

dGω

(x)

be aF(Tk+1)−⊗B(Rd)-measurable version of the Radon-Nikod ́ym derivatives
dFω
dGωand defining

dQ ̃k+1

dQ ̃k

(ω)= (^1) {Tk+1<∞}Y(ω,∆STk+1(ω)) + (^1) {Tk+1=∞}
we obtain a measureQ ̃k+1∼P,sothat‖Q ̃k+1−Q ̃k‖<εk,Q ̃k+1|F(Tk+1)−=
Q ̃k|F(T
k+1)−and
dQ ̃k+1
dQ ̃k beingFTk+1-measurable. For eachM∈R+
(^1) [[Tk+1]]∩{Tk+1<∞andEGω[‖y‖]≤M}·S= (^1) [[Tk+1]]∩{Tk+1<∞andEGω[‖y‖]≤M}·Sa
is a martingale underQ ̃k+1and therefore
S(k+1):= (^1) [[Tk+1]]∩{Tk+1<∞}·S
is a sigma-martingale underQ ̃k+1.
LettingQ= limk→∞Q ̃k, each of the semi-martingalesS(k)= (^1) [[Tk]]·Sis
aQ-sigma-martingale. It follows that
Sa=

∑∞

k=1

S(k)

is aQ-sigma-martingale and therefore

S=Sa+Si

is aQ-sigma-martingale too.