The Mathematics of Arbitrage

14.4 The GeneralRd-valued Case 297

B=b·A

ν(ω, dt, dx)=Fω,t(dx)dAt(ω)

whereb=(bi)di=1is a predictable process andFω,t(dx) a transition kernel from
(Ω×R+,P)into(Rd,B(Rd)), i.e., aP-measurable map (ω, t)→Fω,t(dx)from
Ω×R+into the non-negative Borel measures onRd. Since the processesX
andXˇare quasi left continuous, the processesAandBcan be chosen to be
continuous, but this is not really needed.
The processesνandμare such that for each non-negativeP⊗B(Rd)-
measurable functiongwe have that
∫

Ω×R+×Rd

g(ω, t, y)μ(ω, dt, dy)P(dω)=

∫

Ω×R+×Rd

g(ω, t, y)ν(ω, dt, dy)P(dω).

To stay in line with the notation used in [JS 87],Hω,t∗Fω,t,whereHis
apredictableRd-valued process andFis the kernel described above, denotes
the predictableR-valued processEFω,t[(Hω,t,.)] =

∫

Rd(Hω,t,y)Fω,t[dy].
We may assume thatAis constant afterT,Q 1 -integrable and its integral
is bounded by one, i.e.,

EQ 1 [A∞]=dA(Ω×R+)≤ 1 ,

wheredAdenotes the measure onPdefined bydA(]]T 1 ,T 2 ]] ) =EQ 1 [AT 2 −AT 1 ],
for stopping timesT 1 ≤T 2.
We now shall find aP-measurable map (ω, t)→Gω,tsuch that fordA-
almost each (ω, t),

(a)Fω,t(dx)∼Gω,t(dx),Fω,t(Rd)=Gω,t(Rd)and‖Fω,t−Gω,t‖<ε 2 ,
(b)EGω,t[‖y‖Rd]<∞andEGω,t[y]=−b(ω, t).

This is a task of the type of “martingale problem” or rather “semi-
martingale problem” as dealt with, e.g., in [JS 87, Definition III.2.4].
We apply Lemma 14.3.5 and the remark following it: as measure space
(E,E,ω)wetake(Ω×R+,P,dA) and we shall consider the map

η=(ω, t)→F ̃ω,t:=Fω,tδb(ω,t)

whereδb(ω,t)denotes the Dirac measure atb(ω, t)∈Rd,denotes convolution

and thereforeF ̃ω,tis the measureFω,tonRdtranslated by the vectorb(ω, t).

We claim that the family (F ̃ω,t)(ω,t)∈Ω×R+ satisfies the assumptions of
Lemma 14.3.5 above. Indeed, letHω,tbe anyP-measurable function such

thatdA-almost surelyHω,t∈Adm(F ̃ω,t)=Adm(Fω,t).
By multiplyingHwith a predictable strictly positive processv,wemay
eventually assume that‖Hω,t‖Rd ≤1andthat,atleastdAa.e., also the
predictable process‖〈Hω,t,.〉−‖L∞(Fω,t)is bounded by 1. That the latter
process is predictable follows from the discussion preceding the Crucial Lemma
and essentially follows from the measurable selection theorem.